WEBVTT 00:00:00.000 --> 00:00:02.460 align:middle line:90% [SQUEAKING] 00:00:02.460 --> 00:00:04.428 align:middle line:90% [RUSTLING] 00:00:04.428 --> 00:00:05.412 align:middle line:90% [CLICKING] 00:00:05.412 --> 00:00:10.225 align:middle line:90% 00:00:10.225 --> 00:00:12.850 align:middle line:84% FRANK SCHAFER: Yeah, so, yeah, I plan to split it in two parts, 00:00:12.850 --> 00:00:14.940 align:middle line:84% so the first part is about ordinary differential 00:00:14.940 --> 00:00:19.560 align:middle line:84% equations, so just a quick recap what an ordinary differential 00:00:19.560 --> 00:00:21.630 align:middle line:84% equation actually is, and then we'll 00:00:21.630 --> 00:00:24.340 align:middle line:84% go into this kind of sensitivity analysis, 00:00:24.340 --> 00:00:26.310 align:middle line:84% so how to compute derivatives of solutions 00:00:26.310 --> 00:00:28.020 align:middle line:84% of ordinary differential equations. 00:00:28.020 --> 00:00:31.410 align:middle line:84% And specifically, we will pick out one very important 00:00:31.410 --> 00:00:32.655 align:middle line:90% of these methods. 00:00:32.655 --> 00:00:35.248 align:middle line:90% 00:00:35.248 --> 00:00:37.040 align:middle line:84% So, yeah, an ordinary differential equation 00:00:37.040 --> 00:00:39.207 align:middle line:84% is basically described by this kind of initial value 00:00:39.207 --> 00:00:43.340 align:middle line:84% problem, where we have the state at a given initial time 00:00:43.340 --> 00:00:47.900 align:middle line:84% t0 is equal to u0, and then the dynamics of this state 00:00:47.900 --> 00:00:50.820 align:middle line:84% is described by this kind of time evolution. 00:00:50.820 --> 00:00:54.530 align:middle line:84% So we have that the derivative of u is-- 00:00:54.530 --> 00:00:57.650 align:middle line:84% the time derivative of u is equal to this function f 00:00:57.650 --> 00:01:00.710 align:middle line:84% that can depend on time, on the state itself, 00:01:00.710 --> 00:01:03.510 align:middle line:90% and on some parameters p. 00:01:03.510 --> 00:01:06.180 align:middle line:84% And the first natural question to ask 00:01:06.180 --> 00:01:10.980 align:middle line:84% is, basically, how do we solve such an ODE? 00:01:10.980 --> 00:01:15.630 align:middle line:84% And let's say the probably simplest ODE I could imagine 00:01:15.630 --> 00:01:19.170 align:middle line:84% basically is one of a free-falling particle 00:01:19.170 --> 00:01:22.245 align:middle line:84% or a ball, which is described by-- 00:01:22.245 --> 00:01:25.260 align:middle line:84% well, probably, you have all seen this-- 00:01:25.260 --> 00:01:28.350 align:middle line:84% by Newton's law, or something like the acceleration 00:01:28.350 --> 00:01:32.230 align:middle line:84% is equal to minus the gravitational constant. 00:01:32.230 --> 00:01:35.580 align:middle line:84% And for these kind of easy ordinary differential 00:01:35.580 --> 00:01:39.120 align:middle line:84% equations, right, we can solve that analytically, 00:01:39.120 --> 00:01:42.690 align:middle line:84% so we simply integrate that equation twice, right? 00:01:42.690 --> 00:01:48.330 align:middle line:84% So we have that z prime is equal to some 00:01:48.330 --> 00:01:54.435 align:middle line:90% offset minus g t minus t0. 00:01:54.435 --> 00:01:56.310 align:middle line:84% And then we could integrate it a second time, 00:01:56.310 --> 00:01:58.920 align:middle line:84% and we'd get, like, the full analytical solution 00:01:58.920 --> 00:02:03.505 align:middle line:84% of the height, in that case, with respect to time. 00:02:03.505 --> 00:02:06.250 align:middle line:90% 00:02:06.250 --> 00:02:07.870 align:middle line:84% Here, on this slide, I did already 00:02:07.870 --> 00:02:10.389 align:middle line:84% something that's very meaningful to solve 00:02:10.389 --> 00:02:14.500 align:middle line:84% ODEs numerically, and that's, like, we want to bring down 00:02:14.500 --> 00:02:17.320 align:middle line:84% these kind of ODEs, which are sometimes described 00:02:17.320 --> 00:02:20.050 align:middle line:84% with second-order derivatives, to just 00:02:20.050 --> 00:02:21.310 align:middle line:90% first-order derivatives. 00:02:21.310 --> 00:02:23.350 align:middle line:84% And the basic trick is to introduce 00:02:23.350 --> 00:02:26.410 align:middle line:84% this additional variable v, right, so the velocity 00:02:26.410 --> 00:02:28.840 align:middle line:84% in that case, which can be written as-- 00:02:28.840 --> 00:02:39.370 align:middle line:84% right, so v of t is equal to dz of t dt. 00:02:39.370 --> 00:02:43.090 align:middle line:84% OK, so for that kind of ODE, we know the analytical solution, 00:02:43.090 --> 00:02:44.950 align:middle line:84% and it's basically easy to solve. 00:02:44.950 --> 00:02:48.250 align:middle line:84% Now, coming back to the more general case, where 00:02:48.250 --> 00:02:54.217 align:middle line:84% we have some nice function f, but solving 00:02:54.217 --> 00:02:56.050 align:middle line:84% for the analytical solution is tough, right, 00:02:56.050 --> 00:02:57.970 align:middle line:84% because integrating is already tough 00:02:57.970 --> 00:03:00.130 align:middle line:84% and solving these kind of initial value problems 00:03:00.130 --> 00:03:02.890 align:middle line:84% is even tougher, we can fall back 00:03:02.890 --> 00:03:06.470 align:middle line:84% to some numerical routines, basically, to solve it. 00:03:06.470 --> 00:03:08.890 align:middle line:84% And the easiest numerical routine 00:03:08.890 --> 00:03:11.740 align:middle line:84% is called Euler solver, where we-- 00:03:11.740 --> 00:03:13.570 align:middle line:84% and probably all of us could have come up 00:03:13.570 --> 00:03:17.930 align:middle line:84% with that by ourselves-- so we simply discretize time. 00:03:17.930 --> 00:03:21.940 align:middle line:84% So we introduce this tn, discrete time steps 00:03:21.940 --> 00:03:25.000 align:middle line:84% of our solution, and then if we plug that 00:03:25.000 --> 00:03:27.370 align:middle line:84% into the definition of the derivative, we get something 00:03:27.370 --> 00:03:29.330 align:middle line:84% like these finite difference equations. 00:03:29.330 --> 00:03:34.090 align:middle line:84% So, right, if we see here, the du dt would be u n plus 1 00:03:34.090 --> 00:03:36.730 align:middle line:84% at the-- so that's basically the state at the time tn 00:03:36.730 --> 00:03:41.050 align:middle line:84% plus 1 minus the state at the time tn divided by delta t, 00:03:41.050 --> 00:03:44.030 align:middle line:84% and if we bring the delta t and the state to the other side, 00:03:44.030 --> 00:03:47.385 align:middle line:84% we get this explicit Euler method. 00:03:47.385 --> 00:03:48.760 align:middle line:84% STEVEN JOHNSON: And last week, we 00:03:48.760 --> 00:03:50.860 align:middle line:84% talked about finite difference approximations, 00:03:50.860 --> 00:03:53.530 align:middle line:84% so this is exactly taking that derivative 00:03:53.530 --> 00:03:55.510 align:middle line:84% and making a forward difference approximation. 00:03:55.510 --> 00:03:56.343 align:middle line:90% FRANK SCHAFER: Yeah. 00:03:56.343 --> 00:03:57.410 align:middle line:90% Yeah, exactly. 00:03:57.410 --> 00:04:00.100 align:middle line:84% And you can have more sophisticated methods, 00:04:00.100 --> 00:04:03.220 align:middle line:84% of course, to solve your ODEs, but for the rest of today, 00:04:03.220 --> 00:04:05.470 align:middle line:84% it's basically enough to just think of, 00:04:05.470 --> 00:04:07.870 align:middle line:84% if we don't know the analytical solution, 00:04:07.870 --> 00:04:09.460 align:middle line:84% we can use some numerical routine 00:04:09.460 --> 00:04:12.250 align:middle line:84% to solve for the state as a function of time. 00:04:12.250 --> 00:04:15.040 align:middle line:84% And, yeah, actually, I wanted to implement that very quickly 00:04:15.040 --> 00:04:20.620 align:middle line:84% with you, like, for the case of the free-falling ball, 00:04:20.620 --> 00:04:21.985 align:middle line:90% so for that kind of ODE. 00:04:21.985 --> 00:04:27.090 align:middle line:90% 00:04:27.090 --> 00:04:32.240 align:middle line:84% So let's define some initial height, some kind 00:04:32.240 --> 00:04:40.380 align:middle line:84% of initial velocity, the gravitational constant, 00:04:40.380 --> 00:04:45.450 align:middle line:84% and, well, maybe let's say that's basically 00:04:45.450 --> 00:04:49.930 align:middle line:90% our parameter, g. 00:04:49.930 --> 00:04:53.800 align:middle line:84% We can say our initial state, right, is then 00:04:53.800 --> 00:04:57.310 align:middle line:84% this vector where we stack together 00:04:57.310 --> 00:04:59.650 align:middle line:84% initial height and initial velocity. 00:04:59.650 --> 00:05:03.330 align:middle line:90% 00:05:03.330 --> 00:05:05.940 align:middle line:84% And we say maybe, in our task, we 00:05:05.940 --> 00:05:10.320 align:middle line:84% want to integrate that equation from some initial time 00:05:10.320 --> 00:05:16.640 align:middle line:84% to some final time, and we have to provide this kind of time 00:05:16.640 --> 00:05:18.560 align:middle line:90% discretization. 00:05:18.560 --> 00:05:22.220 align:middle line:84% Let's choose some small time discretization. 00:05:22.220 --> 00:05:24.950 align:middle line:90% 00:05:24.950 --> 00:05:26.870 align:middle line:84% Everybody's happy with that so far? 00:05:26.870 --> 00:05:30.210 align:middle line:90% 00:05:30.210 --> 00:05:35.460 align:middle line:84% So then we want to define our ODE function, which is also 00:05:35.460 --> 00:05:40.320 align:middle line:84% called the vector field, where we just insert the definitions 00:05:40.320 --> 00:05:42.090 align:middle line:90% I had on this slide, basically. 00:05:42.090 --> 00:05:48.080 align:middle line:84% So we have that the derivative of the height with respect 00:05:48.080 --> 00:05:51.080 align:middle line:84% to time is the velocity, right, which is here, in our case, now 00:05:51.080 --> 00:05:52.970 align:middle line:90% the second component. 00:05:52.970 --> 00:05:55.640 align:middle line:84% And we have that the derivative of the time 00:05:55.640 --> 00:05:58.040 align:middle line:84% is equal to-- the derivative of the velocity 00:05:58.040 --> 00:06:01.100 align:middle line:84% with respect to time is equal to minus g, which 00:06:01.100 --> 00:06:03.320 align:middle line:84% is on the right-hand side of the board, 00:06:03.320 --> 00:06:07.642 align:middle line:84% and then we return, basically, again, the vector of these two. 00:06:07.642 --> 00:06:10.100 align:middle line:84% STEVEN JOHNSON: We're going to get g from the parameters, p 00:06:10.100 --> 00:06:10.600 align:middle line:90% [INAUDIBLE]. 00:06:10.600 --> 00:06:12.540 align:middle line:90% FRANK SCHAFER: Oh, yeah. 00:06:12.540 --> 00:06:13.560 align:middle line:90% Let's say it like that. 00:06:13.560 --> 00:06:14.060 align:middle line:90% Thanks. 00:06:14.060 --> 00:06:16.410 align:middle line:90% 00:06:16.410 --> 00:06:22.320 align:middle line:84% And so now we want to implement our Euler 00:06:22.320 --> 00:06:25.110 align:middle line:90% solver, which should take-- 00:06:25.110 --> 00:06:28.200 align:middle line:84% well, the kind of function that we want to solve, 00:06:28.200 --> 00:06:32.543 align:middle line:84% we want it to take the initial condition. 00:06:32.543 --> 00:06:33.960 align:middle line:84% Let's write it a bit more general. 00:06:33.960 --> 00:06:36.560 align:middle line:90% 00:06:36.560 --> 00:06:40.860 align:middle line:84% The time span of the simulation-- 00:06:40.860 --> 00:06:42.350 align:middle line:90% STEVEN JOHNSON: tspan. 00:06:42.350 --> 00:06:46.670 align:middle line:84% FRANK SCHAFER: tspan, yeah, and the parameters, 00:06:46.670 --> 00:06:47.940 align:middle line:90% and maybe even dt. 00:06:47.940 --> 00:06:52.510 align:middle line:90% 00:06:52.510 --> 00:06:57.120 align:middle line:84% And let's say our time at some point will be-- 00:06:57.120 --> 00:07:00.130 align:middle line:84% so we make a copy so we don't overwrite into the array. 00:07:00.130 --> 00:07:03.915 align:middle line:84% So that this is preserved, we make a time grid. 00:07:03.915 --> 00:07:07.968 align:middle line:90% 00:07:07.968 --> 00:07:10.260 align:middle line:84% This is actually big enough on the screen for everyone? 00:07:10.260 --> 00:07:10.760 align:middle line:90% Oh, good. 00:07:10.760 --> 00:07:30.690 align:middle line:90% 00:07:30.690 --> 00:07:32.955 align:middle line:84% Right, we have to be a bit careful with the endpoint 00:07:32.955 --> 00:07:34.830 align:middle line:84% because we don't want it to step one too far. 00:07:34.830 --> 00:07:40.060 align:middle line:90% 00:07:40.060 --> 00:07:44.980 align:middle line:84% And then the update rule is basically we take u at a given 00:07:44.980 --> 00:07:49.100 align:middle line:90% time, and that's-- whoops. 00:07:49.100 --> 00:07:53.932 align:middle line:90% 00:07:53.932 --> 00:07:55.890 align:middle line:84% STEVEN JOHNSON: And you don't need the deepcopy 00:07:55.890 --> 00:07:59.275 align:middle line:84% since you're not overwriting u in place anyway. 00:07:59.275 --> 00:08:00.400 align:middle line:90% FRANK SCHAFER: That's true. 00:08:00.400 --> 00:08:04.895 align:middle line:84% I think I will need it later for one modification, so-- 00:08:04.895 --> 00:08:07.817 align:middle line:84% STEVEN JOHNSON: But you don't need any copy. 00:08:07.817 --> 00:08:09.150 align:middle line:90% FRANK SCHAFER: That's also true. 00:08:09.150 --> 00:08:09.650 align:middle line:90% OK. 00:08:09.650 --> 00:08:12.840 align:middle line:90% 00:08:12.840 --> 00:08:15.210 align:middle line:84% And let's say I'm not interested in the full solution 00:08:15.210 --> 00:08:17.880 align:middle line:84% as a function of time, but just in the endpoint as-- 00:08:17.880 --> 00:08:20.525 align:middle line:90% 00:08:20.525 --> 00:08:22.025 align:middle line:84% OK, let's see if that works already. 00:08:22.025 --> 00:08:36.289 align:middle line:90% 00:08:36.289 --> 00:08:39.380 align:middle line:90% All right, that's good. 00:08:39.380 --> 00:08:41.870 align:middle line:90% u should be at ti, for sure. 00:08:41.870 --> 00:08:49.890 align:middle line:90% 00:08:49.890 --> 00:08:53.980 align:middle line:84% Yeah, we can check if that's the correct solution by plugging it 00:08:53.980 --> 00:08:55.485 align:middle line:90% into our analytical equation. 00:08:55.485 --> 00:08:58.450 align:middle line:90% 00:08:58.450 --> 00:08:59.935 align:middle line:90% Let me just pull the form. 00:08:59.935 --> 00:09:08.470 align:middle line:90% 00:09:08.470 --> 00:09:11.080 align:middle line:84% Right, so we have now written a numerical routine that 00:09:11.080 --> 00:09:13.540 align:middle line:84% spits out the state at a given time t, 00:09:13.540 --> 00:09:15.820 align:middle line:84% and I think that looks actually good. 00:09:15.820 --> 00:09:23.560 align:middle line:84% So we have t equal to 0, so our z should be z0 minus g-- 00:09:23.560 --> 00:09:27.420 align:middle line:90% 00:09:27.420 --> 00:09:38.935 align:middle line:84% minus v0 times 1 plus v0 times 1 minus 1/2 times g. 00:09:38.935 --> 00:09:42.538 align:middle line:90% 00:09:42.538 --> 00:09:44.080 align:middle line:84% And, yeah, that looks basically good. 00:09:44.080 --> 00:09:46.247 align:middle line:84% That's the discretization error of the Euler method, 00:09:46.247 --> 00:09:48.675 align:middle line:84% and we could check the same for the second component. 00:09:48.675 --> 00:09:50.800 align:middle line:84% STEVEN JOHNSON: Yeah, so the correct answer is 0.1, 00:09:50.800 --> 00:09:52.495 align:middle line:84% and then it's-- it's, like, 0.105. 00:09:52.495 --> 00:09:53.620 align:middle line:90% FRANK SCHAFER: Yeah, it's-- 00:09:53.620 --> 00:09:56.650 align:middle line:84% yeah, exactly, it's slightly off because we have not the most 00:09:56.650 --> 00:09:59.320 align:middle line:90% sophisticated numerical solver. 00:09:59.320 --> 00:10:00.850 align:middle line:90% OK, cool. 00:10:00.850 --> 00:10:03.130 align:middle line:84% So that's how we solve ODEs numerically 00:10:03.130 --> 00:10:05.830 align:middle line:84% if we don't have an analytical solution, 00:10:05.830 --> 00:10:09.460 align:middle line:84% and, yeah, now we get into the kind of sensitivity analysis 00:10:09.460 --> 00:10:12.740 align:middle line:84% if we have a numerical routine available. 00:10:12.740 --> 00:10:16.480 align:middle line:84% So there are two different ways, fundamentally different ways, 00:10:16.480 --> 00:10:18.830 align:middle line:90% to do sensitivity analysis. 00:10:18.830 --> 00:10:21.340 align:middle line:84% One is called discrete sensitivity analysis, 00:10:21.340 --> 00:10:24.310 align:middle line:84% and one is called continuous sensitivity analysis. 00:10:24.310 --> 00:10:27.490 align:middle line:84% And the first one is basically you 00:10:27.490 --> 00:10:29.770 align:middle line:84% imagine that you wrote that solver that we just 00:10:29.770 --> 00:10:31.570 align:middle line:84% wrote in some language that's compatible 00:10:31.570 --> 00:10:33.140 align:middle line:90% with automatic differentiation. 00:10:33.140 --> 00:10:36.070 align:middle line:84% And so we first discretize, basically is the task, 00:10:36.070 --> 00:10:37.850 align:middle line:90% and then we differentiate it. 00:10:37.850 --> 00:10:40.690 align:middle line:84% And the second one is basically we 00:10:40.690 --> 00:10:43.640 align:middle line:84% write a custom rule for the sensitivity analysis, 00:10:43.640 --> 00:10:45.920 align:middle line:84% and so we first approximate, basically, 00:10:45.920 --> 00:10:50.780 align:middle line:84% the gradient that we want to compute and then differentiate 00:10:50.780 --> 00:10:51.380 align:middle line:90% it-- 00:10:51.380 --> 00:10:54.430 align:middle line:90% and then discretize it. 00:10:54.430 --> 00:10:59.230 align:middle line:84% And for each of these methods, these different paradigms, 00:10:59.230 --> 00:11:01.583 align:middle line:84% basically, in either first discretizing and then 00:11:01.583 --> 00:11:03.250 align:middle line:84% differentiating or first differentiating 00:11:03.250 --> 00:11:06.280 align:middle line:84% and then discretizing, there are two different modes 00:11:06.280 --> 00:11:09.190 align:middle line:84% that you also probably have covered in the lecture already. 00:11:09.190 --> 00:11:12.820 align:middle line:84% You can either derive a forward, or also called tangent, mode 00:11:12.820 --> 00:11:14.530 align:middle line:90% or a reverse, adjoint, mode. 00:11:14.530 --> 00:11:17.890 align:middle line:84% And depending on if you have many input parameters 00:11:17.890 --> 00:11:21.070 align:middle line:84% and a small number of outputs, one of these modes 00:11:21.070 --> 00:11:23.200 align:middle line:84% is basically preferred to each other. 00:11:23.200 --> 00:11:26.740 align:middle line:84% So in the setting where you have many parameters with respect 00:11:26.740 --> 00:11:29.350 align:middle line:84% to which you want to compute a gradient or derivative, 00:11:29.350 --> 00:11:32.230 align:middle line:84% you usually want to have some adjoint mode, 00:11:32.230 --> 00:11:35.560 align:middle line:84% and if you have high memory demands, 00:11:35.560 --> 00:11:38.570 align:middle line:84% you typically want to have these continuous sensitivity methods. 00:11:38.570 --> 00:11:41.620 align:middle line:84% So it's a very relevant case to consider 00:11:41.620 --> 00:11:44.800 align:middle line:84% these continuous adjoints, basically, and that's 00:11:44.800 --> 00:11:48.790 align:middle line:90% what we will do today. 00:11:48.790 --> 00:11:50.580 align:middle line:84% So what are sensitivities good for? 00:11:50.580 --> 00:11:53.310 align:middle line:84% Just as a recap, so sensitivity analysis 00:11:53.310 --> 00:11:57.270 align:middle line:84% basically just, well, describes what it says. 00:11:57.270 --> 00:12:00.000 align:middle line:84% So we might be interested in just describing 00:12:00.000 --> 00:12:03.780 align:middle line:84% how maybe the height of our ball would change if we slightly 00:12:03.780 --> 00:12:06.360 align:middle line:84% perturb the parameter of the differential equation, 00:12:06.360 --> 00:12:08.610 align:middle line:84% but we can also use the gradient of these ODEs 00:12:08.610 --> 00:12:11.880 align:middle line:84% to do more fancy stuff, as Steven mentioned, 00:12:11.880 --> 00:12:15.390 align:middle line:84% like parameter estimation in, what, chemical reaction 00:12:15.390 --> 00:12:19.740 align:middle line:84% networks or so, or control for whatever physical system 00:12:19.740 --> 00:12:22.740 align:middle line:84% or chemical system you have at hand. 00:12:22.740 --> 00:12:25.520 align:middle line:84% So once again, so for the free-falling particle, 00:12:25.520 --> 00:12:28.550 align:middle line:84% it might be the task that we have some terminal cost 00:12:28.550 --> 00:12:31.760 align:middle line:84% or so, that could be this G of u at the final time, 00:12:31.760 --> 00:12:34.052 align:middle line:84% and we are interested in maybe saying 00:12:34.052 --> 00:12:35.510 align:middle line:84% how the height changes with respect 00:12:35.510 --> 00:12:37.050 align:middle line:90% to the gravitational constant. 00:12:37.050 --> 00:12:40.430 align:middle line:84% And so the claim is that this is always easy 00:12:40.430 --> 00:12:43.370 align:middle line:84% if we can solve the ODE analytically, right, 00:12:43.370 --> 00:12:47.840 align:middle line:84% so if we look at the solution there of the ODE, 00:12:47.840 --> 00:12:51.020 align:middle line:84% can someone say what's the sensitivity that I wrote down, 00:12:51.020 --> 00:12:55.990 align:middle line:84% like, the sensitivity of the final height with respect to g? 00:12:55.990 --> 00:12:59.330 align:middle line:90% 00:12:59.330 --> 00:13:00.065 align:middle line:90% Yup? 00:13:00.065 --> 00:13:03.040 align:middle line:84% AUDIENCE: Negative 1/2 t minus [INAUDIBLE].. 00:13:03.040 --> 00:13:04.870 align:middle line:84% FRANK SCHAFER: Yeah, exactly, so and now we 00:13:04.870 --> 00:13:06.578 align:middle line:84% will basically cover how we do that if we 00:13:06.578 --> 00:13:09.970 align:middle line:84% don't have the analytic solution available. 00:13:09.970 --> 00:13:13.037 align:middle line:84% For this, we will have to do some derivation. 00:13:13.037 --> 00:13:13.537 align:middle line:90% Uh-huh. 00:13:13.537 --> 00:13:14.037 align:middle line:90% Yup. 00:13:14.037 --> 00:13:18.270 align:middle line:90% 00:13:18.270 --> 00:13:19.710 align:middle line:90% Let me move them. 00:13:19.710 --> 00:13:24.960 align:middle line:90% 00:13:24.960 --> 00:13:30.680 align:middle line:84% So we'll do continuous adjoins today. 00:13:30.680 --> 00:13:40.330 align:middle line:90% 00:13:40.330 --> 00:13:43.060 align:middle line:84% So let's first write down one of these cost functions 00:13:43.060 --> 00:13:45.760 align:middle line:84% which I just had in a bit more general way. 00:13:45.760 --> 00:13:48.750 align:middle line:84% So we have some cost that depends 00:13:48.750 --> 00:13:51.750 align:middle line:84% on the state as a function of time and p, 00:13:51.750 --> 00:13:57.130 align:middle line:84% where the state itself, of course, 00:13:57.130 --> 00:14:00.090 align:middle line:90% also depends on the parameters. 00:14:00.090 --> 00:14:05.205 align:middle line:84% And we'll say that this function can be written as the-- 00:14:05.205 --> 00:14:15.394 align:middle line:90% 00:14:15.394 --> 00:14:18.780 align:middle line:90% all right, so we say we can-- 00:14:18.780 --> 00:14:20.640 align:middle line:84% we don't have, maybe, just the final value, 00:14:20.640 --> 00:14:22.380 align:middle line:84% but we are interested in general in 00:14:22.380 --> 00:14:27.252 align:middle line:84% some functional on the solution of the differential equation. 00:14:27.252 --> 00:14:29.460 align:middle line:84% AUDIENCE: And you can think of capital G and little g 00:14:29.460 --> 00:14:31.000 align:middle line:90% as just being scalar functions? 00:14:31.000 --> 00:14:31.792 align:middle line:90% FRANK SCHAFER: Yes. 00:14:31.792 --> 00:14:31.950 align:middle line:90% AUDIENCE: OK. 00:14:31.950 --> 00:14:32.640 align:middle line:90% FRANK SCHAFER: Yup. 00:14:32.640 --> 00:14:33.140 align:middle line:90% Yup. 00:14:33.140 --> 00:14:35.570 align:middle line:90% 00:14:35.570 --> 00:14:38.930 align:middle line:84% And now comes basically the first nice trick 00:14:38.930 --> 00:14:41.990 align:middle line:84% in this kind of derivation, and that's related 00:14:41.990 --> 00:14:43.558 align:middle line:90% to Lagrange multipliers. 00:14:43.558 --> 00:14:45.808 align:middle line:84% STEVEN JOHNSON: Do you want to write down the ODE that 00:14:45.808 --> 00:14:47.600 align:middle line:90% at least satisfies? 00:14:47.600 --> 00:14:51.210 align:middle line:84% FRANK SCHAFER: Yeah, probably a good idea. 00:14:51.210 --> 00:14:53.360 align:middle line:90% All right, such that-- 00:14:53.360 --> 00:15:01.430 align:middle line:90% 00:15:01.430 --> 00:15:02.420 align:middle line:90% Oh, I see. 00:15:02.420 --> 00:15:21.272 align:middle line:90% 00:15:21.272 --> 00:15:22.980 align:middle line:84% OK, so the first trick that we want to do 00:15:22.980 --> 00:15:27.150 align:middle line:84% is basically to add a 0 to this kind of function, so-- 00:15:27.150 --> 00:15:28.650 align:middle line:84% STEVEN JOHNSON: Can we pause to make 00:15:28.650 --> 00:15:30.480 align:middle line:84% sure we're understanding what you're doing? 00:15:30.480 --> 00:15:34.710 align:middle line:84% So you have some ODE, you solve for u of t, 00:15:34.710 --> 00:15:38.880 align:middle line:84% and then you're going to compute a number at the end, capital G, 00:15:38.880 --> 00:15:42.330 align:middle line:84% by keeping the ODE solution and integrating some function 00:15:42.330 --> 00:15:45.300 align:middle line:84% little g into that solution, right? 00:15:45.300 --> 00:15:48.180 align:middle line:84% And the ODE depends on some parameters, 00:15:48.180 --> 00:15:50.700 align:middle line:84% and you want to find the derivative 00:15:50.700 --> 00:15:53.550 align:middle line:84% of this final output, capital G, with respect 00:15:53.550 --> 00:15:56.110 align:middle line:90% to all those parameters of p. 00:15:56.110 --> 00:15:59.234 align:middle line:84% So little p might be a vector, it 00:15:59.234 --> 00:16:06.530 align:middle line:90% might be 10,000 reaction rates. 00:16:06.530 --> 00:16:11.710 align:middle line:84% And capital G might be the integral 00:16:11.710 --> 00:16:15.580 align:middle line:84% of the squared difference between the ODE solution 00:16:15.580 --> 00:16:18.863 align:middle line:84% and some experimental data on the chemical reaction, 00:16:18.863 --> 00:16:20.776 align:middle line:84% you know, the concentration, [INAUDIBLE].. 00:16:20.776 --> 00:16:24.192 align:middle line:90% 00:16:24.192 --> 00:16:25.400 align:middle line:90% FRANK SCHAFER: Yeah, exactly. 00:16:25.400 --> 00:16:27.110 align:middle line:84% Does everybody see why that's a 0? 00:16:27.110 --> 00:16:31.370 align:middle line:90% 00:16:31.370 --> 00:16:32.595 align:middle line:90% Right, so why-- 00:16:32.595 --> 00:16:34.970 align:middle line:84% STEVEN JOHNSON: Yeah, so, sorry, I was talking about that 00:16:34.970 --> 00:16:41.740 align:middle line:84% and then maybe you need to say what the 0 is 00:16:41.740 --> 00:16:45.070 align:middle line:84% FRANK SCHAFER: Right, so because, by definition, 00:16:45.070 --> 00:16:49.660 align:middle line:84% basically, our state fulfills this ordinary differential 00:16:49.660 --> 00:16:54.010 align:middle line:84% equation, we can introduce this kind of Lagrange multiplier, 00:16:54.010 --> 00:16:55.600 align:middle line:90% and that term is 0. 00:16:55.600 --> 00:16:59.830 align:middle line:84% So if we add that term to this kind of cost function, 00:16:59.830 --> 00:17:05.440 align:middle line:84% we basically do nothing, but we have this free Lagrange 00:17:05.440 --> 00:17:08.030 align:middle line:84% multiplier that we can choose later on. 00:17:08.030 --> 00:17:12.130 align:middle line:84% So that will be kind of the trick that we will play. 00:17:12.130 --> 00:17:15.280 align:middle line:84% And now, as Steven already said, so we're 00:17:15.280 --> 00:17:21.369 align:middle line:84% interested in computing dG over dp, 00:17:21.369 --> 00:17:24.849 align:middle line:84% right, so that's the kind of sensitivity gradient 00:17:24.849 --> 00:17:27.550 align:middle line:84% information that we want to have. 00:17:27.550 --> 00:17:30.400 align:middle line:90% 00:17:30.400 --> 00:17:33.670 align:middle line:84% There will be a long line, so what we basically do 00:17:33.670 --> 00:17:37.825 align:middle line:84% is just doing all the chain rule on that term and these terms. 00:17:37.825 --> 00:17:51.702 align:middle line:90% 00:17:51.702 --> 00:17:53.660 align:middle line:84% STEVEN JOHNSON: And now think of p as a scalar, 00:17:53.660 --> 00:17:56.180 align:middle line:84% but in general, it could be a vector or something. 00:17:56.180 --> 00:17:59.309 align:middle line:84% And if it's the gradient, of dG dp. 00:17:59.309 --> 00:18:07.850 align:middle line:90% 00:18:07.850 --> 00:18:10.630 align:middle line:84% FRANK SCHAFER: Yeah, it's a gradient transpose, yeah. 00:18:10.630 --> 00:18:13.376 align:middle line:84% STEVEN JOHNSON: So what we were calling G prime in the book. 00:18:13.376 --> 00:18:27.650 align:middle line:90% 00:18:27.650 --> 00:18:29.850 align:middle line:84% FRANK SCHAFER: Yeah, let me quickly write it down 00:18:29.850 --> 00:18:33.360 align:middle line:84% because it's easy to do mistakes on that part. 00:18:33.360 --> 00:18:37.260 align:middle line:90% 00:18:37.260 --> 00:18:40.134 align:middle line:84% Oh, minus sign from the minus we had before. 00:18:40.134 --> 00:18:57.745 align:middle line:90% 00:18:57.745 --> 00:19:00.120 align:middle line:84% STEVEN JOHNSON: Maybe you could write it a little bigger? 00:19:00.120 --> 00:19:01.260 align:middle line:90% FRANK SCHAFER: Oh. 00:19:01.260 --> 00:19:01.760 align:middle line:90% OK. 00:19:01.760 --> 00:19:23.830 align:middle line:90% 00:19:23.830 --> 00:19:26.890 align:middle line:84% OK, I'll leave out the utp's to save some space, 00:19:26.890 --> 00:19:29.928 align:middle line:84% but every u here depends on t and p. 00:19:29.928 --> 00:19:42.040 align:middle line:90% 00:19:42.040 --> 00:19:45.280 align:middle line:84% I guess that's maybe hard to read at the end, 00:19:45.280 --> 00:19:49.278 align:middle line:84% but there's a dt only missing here. 00:19:49.278 --> 00:19:52.223 align:middle line:84% Is that readable for you on that side? 00:19:52.223 --> 00:19:53.140 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:19:53.140 --> 00:19:54.057 align:middle line:90% FRANK SCHAFER: I know. 00:19:54.057 --> 00:19:56.050 align:middle line:84% I will repeat that term in a second, 00:19:56.050 --> 00:19:58.540 align:middle line:84% so you will see again what's that term. 00:19:58.540 --> 00:20:04.060 align:middle line:90% 00:20:04.060 --> 00:20:08.500 align:middle line:84% Right, so let's call that basically, 00:20:08.500 --> 00:20:12.880 align:middle line:84% like, the sensitivity, call that, and yet, let's 00:20:12.880 --> 00:20:14.270 align:middle line:90% simplify a bit the notation. 00:20:14.270 --> 00:20:15.460 align:middle line:90% So that's my small g. 00:20:15.460 --> 00:20:20.080 align:middle line:84% I'm sorry, my small g's and small f's look fairly similar. 00:20:20.080 --> 00:20:24.490 align:middle line:84% That's small gu, so I'll always denote 00:20:24.490 --> 00:20:29.878 align:middle line:84% by subscript with respect to what we differentiate. 00:20:29.878 --> 00:20:32.170 align:middle line:84% STEVEN JOHNSON: This is like in the homework, you know, 00:20:32.170 --> 00:20:36.220 align:middle line:84% has a few problems where partial derivatives are subscripts. 00:20:36.220 --> 00:20:42.340 align:middle line:84% FRANK SCHAFER: Small gp, and in this term, what we can do here 00:20:42.340 --> 00:20:44.920 align:middle line:84% is basically swap the derivative with respect 00:20:44.920 --> 00:20:47.300 align:middle line:90% to p and with respect to time. 00:20:47.300 --> 00:20:54.080 align:middle line:84% So we get here a prime, right, so because when we swap it, 00:20:54.080 --> 00:20:57.280 align:middle line:84% we have du dp, which is s, and then we 00:20:57.280 --> 00:21:00.040 align:middle line:90% have the time derivative. 00:21:00.040 --> 00:21:13.100 align:middle line:84% And we have here some fu times s, and the last term is f-- 00:21:13.100 --> 00:21:14.500 align:middle line:90% that last term is fp. 00:21:14.500 --> 00:21:19.720 align:middle line:90% 00:21:19.720 --> 00:21:22.230 align:middle line:84% OK, let me go to the center because I think it's probably 00:21:22.230 --> 00:21:23.445 align:middle line:90% better readable from there. 00:21:23.445 --> 00:21:25.320 align:middle line:84% STEVEN JOHNSON: Can you scroll the board down 00:21:25.320 --> 00:21:25.752 align:middle line:90% so we can see more? 00:21:25.752 --> 00:21:26.752 align:middle line:90% FRANK SCHAFER: Oh, Yeah. 00:21:26.752 --> 00:21:28.063 align:middle line:90% STEVEN JOHNSON: All right. 00:21:28.063 --> 00:21:28.980 align:middle line:90% FRANK SCHAFER: Oh, OK. 00:21:28.980 --> 00:21:32.385 align:middle line:90% 00:21:32.385 --> 00:21:33.600 align:middle line:90% STEVEN JOHNSON: Thank you. 00:21:33.600 --> 00:21:34.600 align:middle line:90% FRANK SCHAFER: Oh, yeah. 00:21:34.600 --> 00:21:35.990 align:middle line:90% OK. 00:21:35.990 --> 00:21:38.780 align:middle line:84% So, yeah, I wrote this second term, which was probably 00:21:38.780 --> 00:21:40.220 align:middle line:90% hard to read, a bit bigger. 00:21:40.220 --> 00:22:03.160 align:middle line:90% 00:22:03.160 --> 00:22:06.858 align:middle line:84% So that's the second term on the right-hand side and-- 00:22:06.858 --> 00:22:08.650 align:middle line:84% STEVEN JOHNSON: Could I maybe say something 00:22:08.650 --> 00:22:09.860 align:middle line:90% to try and clarify? 00:22:09.860 --> 00:22:10.930 align:middle line:90% FRANK SCHAFER: Yup. 00:22:10.930 --> 00:22:16.440 align:middle line:84% STEVEN JOHNSON: So, right, so you 00:22:16.440 --> 00:22:19.650 align:middle line:84% have this function you're trying to differentiate, 00:22:19.650 --> 00:22:23.430 align:middle line:84% and if you just differentiate this p directly, 00:22:23.430 --> 00:22:26.550 align:middle line:84% you get this term, which is easy. 00:22:26.550 --> 00:22:29.010 align:middle line:84% Right, if the function depends directly on the parameters, 00:22:29.010 --> 00:22:30.480 align:middle line:90% you can take that derivative. 00:22:30.480 --> 00:22:32.430 align:middle line:90% Often, that derivative is 0. 00:22:32.430 --> 00:22:34.260 align:middle line:84% This term, which is easy, this is 00:22:34.260 --> 00:22:37.480 align:middle line:84% your cost function, how does it depend on the solution. 00:22:37.480 --> 00:22:39.503 align:middle line:90% This term is hard. 00:22:39.503 --> 00:22:41.920 align:middle line:84% How do you get the derivative of the solution with respect 00:22:41.920 --> 00:22:43.180 align:middle line:90% to the parameters? 00:22:43.180 --> 00:22:46.055 align:middle line:84% So the goal and adding 0 here is going 00:22:46.055 --> 00:22:48.430 align:middle line:84% to be-- is going to give us this extra degree of freedom, 00:22:48.430 --> 00:22:53.470 align:middle line:84% lambda, that will-- if he's clever in how he chooses this, 00:22:53.470 --> 00:22:55.540 align:middle line:84% there'll be a trick that allows him 00:22:55.540 --> 00:22:59.230 align:middle line:84% to cancel this term this term that's hard 00:22:59.230 --> 00:23:02.110 align:middle line:84% and get a differential equation for lambda. 00:23:02.110 --> 00:23:07.030 align:middle line:84% So that's-- it's often good to sort of show-- 00:23:07.030 --> 00:23:08.410 align:middle line:84% there's kind of a magic step here 00:23:08.410 --> 00:23:10.330 align:middle line:84% where you added this lambda term, 00:23:10.330 --> 00:23:13.030 align:middle line:84% and you know, sometimes it's good to not 00:23:13.030 --> 00:23:15.580 align:middle line:84% keep people in suspense as to why 00:23:15.580 --> 00:23:17.140 align:middle line:90% the hell did you add this term. 00:23:17.140 --> 00:23:19.010 align:middle line:90% FRANK SCHAFER: That's true. 00:23:19.010 --> 00:23:19.790 align:middle line:90% Yeah, exactly. 00:23:19.790 --> 00:23:20.330 align:middle line:90% Yes. 00:23:20.330 --> 00:23:22.640 align:middle line:84% One step before we can actually do that trick 00:23:22.640 --> 00:23:26.720 align:middle line:84% to eliminate the s, and that's basically this 00:23:26.720 --> 00:23:32.030 align:middle line:84% s prime that we got, which we first have to get rid of. 00:23:32.030 --> 00:23:34.640 align:middle line:84% Anybody has an idea, maybe, how to do that, 00:23:34.640 --> 00:23:38.720 align:middle line:84% like, our plans to turn s prime into an s? 00:23:38.720 --> 00:23:45.750 align:middle line:90% 00:23:45.750 --> 00:23:48.200 align:middle line:84% STEVEN JOHNSON: 18.01 question, you 00:23:48.200 --> 00:23:50.700 align:middle line:84% have a derivative in an integral you don't like, how can you 00:23:50.700 --> 00:23:51.300 align:middle line:90% get rid of it? 00:23:51.300 --> 00:23:57.320 align:middle line:90% 00:23:57.320 --> 00:23:59.510 align:middle line:90% AUDIENCE: Integrate by parts. 00:23:59.510 --> 00:24:04.550 align:middle line:84% FRANK SCHAFER: Yeah, so we integrate that term by parts, 00:24:04.550 --> 00:24:07.980 align:middle line:90% and that gives us-- 00:24:07.980 --> 00:24:29.220 align:middle line:90% 00:24:29.220 --> 00:24:31.380 align:middle line:84% all right, so that's from the integration by parts, 00:24:31.380 --> 00:24:34.290 align:middle line:84% now the prime on the lambda, which 00:24:34.290 --> 00:24:39.870 align:middle line:84% will be exactly going towards the ODE that Steven mentioned. 00:24:39.870 --> 00:24:52.870 align:middle line:90% 00:24:52.870 --> 00:24:55.120 align:middle line:84% STEVEN JOHNSON: Time is always a time derivative? 00:24:55.120 --> 00:24:55.930 align:middle line:90% FRANK SCHAFER: Yes. 00:24:55.930 --> 00:24:57.138 align:middle line:90% Yes, I should have said that. 00:24:57.138 --> 00:25:06.273 align:middle line:90% 00:25:06.273 --> 00:25:08.940 align:middle line:84% STEVEN JOHNSON: OK, can you also comment on the shape of lambda? 00:25:08.940 --> 00:25:11.933 align:middle line:84% Like, why is lambda-- is lambda a matrix, a vector, a scalar, 00:25:11.933 --> 00:25:13.350 align:middle line:84% what is the-- why is it transpose? 00:25:13.350 --> 00:25:18.440 align:middle line:84% Like, what's the size of this lambda? 00:25:18.440 --> 00:25:22.610 align:middle line:84% FRANK SCHAFER: The size of this lambda, so it's such that-- 00:25:22.610 --> 00:25:28.140 align:middle line:84% right, this output should be a scalar, right, 00:25:28.140 --> 00:25:31.060 align:middle line:84% because we optimize the scalar cost function, 00:25:31.060 --> 00:25:34.550 align:middle line:84% so the size of lambda is basically in the-- 00:25:34.550 --> 00:25:42.980 align:middle line:84% well, in the adjoint space of f, right-- or of u prime. 00:25:42.980 --> 00:25:45.380 align:middle line:84% STEVEN JOHNSON: Yeah, so the solution here is not 00:25:45.380 --> 00:25:46.730 align:middle line:90% a scalar. u is not a scalar. 00:25:46.730 --> 00:25:49.160 align:middle line:90% It's a column vector, right? 00:25:49.160 --> 00:25:51.480 align:middle line:84% You have n equations and n variables. 00:25:51.480 --> 00:25:53.980 align:middle line:84% Lambda is a column vector with the same number of components 00:25:53.980 --> 00:25:55.270 align:middle line:90% and has the same-- 00:25:55.270 --> 00:25:56.790 align:middle line:90% it has the same shape as u. 00:25:56.790 --> 00:26:03.930 align:middle line:90% 00:26:03.930 --> 00:26:05.940 align:middle line:84% FRANK SCHAFER: Yeah, so what we want to do next 00:26:05.940 --> 00:26:08.160 align:middle line:84% is basically insert back this term 00:26:08.160 --> 00:26:12.240 align:middle line:90% into that equation, so we-- 00:26:12.240 --> 00:26:13.630 align:middle line:90% that was our term here. 00:26:13.630 --> 00:26:18.690 align:middle line:84% And so we take now this part, insert it back, 00:26:18.690 --> 00:26:21.420 align:middle line:84% and we're already going to what's our goal, 00:26:21.420 --> 00:26:26.700 align:middle line:84% and reorder all terms such that we have this s separated, 00:26:26.700 --> 00:26:27.330 align:middle line:90% basically. 00:26:27.330 --> 00:26:30.220 align:middle line:90% 00:26:30.220 --> 00:26:37.080 align:middle line:84% So what we end up with is basically this dG dp. 00:26:37.080 --> 00:26:53.570 align:middle line:90% 00:26:53.570 --> 00:26:55.820 align:middle line:84% STEVEN JOHNSON: It's a little funny because in Greek-- 00:26:55.820 --> 00:26:58.910 align:middle line:84% in linear algebra, we're used to Greek letters being scalars, 00:26:58.910 --> 00:27:02.463 align:middle line:84% but for adjoint methods, the adjoint vector 00:27:02.463 --> 00:27:04.130 align:middle line:84% is always called lambda because it comes 00:27:04.130 --> 00:27:05.600 align:middle line:90% from Lagrange multipliers. 00:27:05.600 --> 00:27:18.654 align:middle line:90% 00:27:18.654 --> 00:27:20.487 align:middle line:84% FRANK SCHAFER: Oh, I'm missing a prime here. 00:27:20.487 --> 00:27:25.160 align:middle line:90% 00:27:25.160 --> 00:27:27.230 align:middle line:84% OK, let's write that directly into one. 00:27:27.230 --> 00:27:43.410 align:middle line:90% 00:27:43.410 --> 00:27:45.810 align:middle line:84% OK, so not much happened here, basically. 00:27:45.810 --> 00:27:48.310 align:middle line:84% It's just looking at all the terms. 00:27:48.310 --> 00:27:51.420 align:middle line:90% So we leave that term here. 00:27:51.420 --> 00:27:53.790 align:middle line:90% This is that term. 00:27:53.790 --> 00:27:55.230 align:middle line:90% That is that term. 00:27:55.230 --> 00:27:59.670 align:middle line:84% We moved this term which doesn't depend on f to the front, 00:27:59.670 --> 00:28:04.570 align:middle line:84% and we had another term that was depending on S here, 00:28:04.570 --> 00:28:07.950 align:middle line:84% which we moved to the back, basically, so these two 00:28:07.950 --> 00:28:11.720 align:middle line:90% terms were, like, swapped. 00:28:11.720 --> 00:28:14.640 align:middle line:84% AUDIENCE: So again the goal here is-- 00:28:14.640 --> 00:28:16.922 align:middle line:84% the S, which is d u-- the derivative of the solution's 00:28:16.922 --> 00:28:19.380 align:middle line:84% expected parameter, that's the annoying term you don't know 00:28:19.380 --> 00:28:21.588 align:middle line:84% how to deal with, so you're going to try and group it 00:28:21.588 --> 00:28:25.620 align:middle line:84% all together and then make that term 0 so that you 00:28:25.620 --> 00:28:28.380 align:middle line:90% don't have to deal with it. 00:28:28.380 --> 00:28:37.690 align:middle line:84% FRANK SCHAFER: Yeah, so we are free to choose lambda, 00:28:37.690 --> 00:28:40.120 align:middle line:84% so we can choose lambda to be whatever we want. 00:28:40.120 --> 00:28:47.130 align:middle line:84% So why don't we take basically the choice lambda prime of T 00:28:47.130 --> 00:28:55.860 align:middle line:84% is equal to minus lambda T fu minus gu? 00:28:55.860 --> 00:28:58.920 align:middle line:84% Right, if we take that choice, we 00:28:58.920 --> 00:29:01.990 align:middle line:84% will have that this term vanishes, 00:29:01.990 --> 00:29:05.190 align:middle line:84% and we can also, since this is an ODE, 00:29:05.190 --> 00:29:09.310 align:middle line:84% specify, well, basically the initial condition. 00:29:09.310 --> 00:29:17.400 align:middle line:84% So we can specify that lambda of t should be equal to 0. 00:29:17.400 --> 00:29:21.180 align:middle line:84% If we do that, we have that this term cancels, 00:29:21.180 --> 00:29:26.580 align:middle line:84% and one of the terms from this integration by parts 00:29:26.580 --> 00:29:27.195 align:middle line:90% also cancels. 00:29:27.195 --> 00:29:29.770 align:middle line:90% 00:29:29.770 --> 00:29:35.947 align:middle line:90% Then it's the question of what-- 00:29:35.947 --> 00:29:38.400 align:middle line:84% STEVEN JOHNSON: And why did you choose capital T to be 00:29:38.400 --> 00:29:40.393 align:middle line:90% that, lambda capital T to be 0? 00:29:40.393 --> 00:29:42.310 align:middle line:84% FRANK SCHAFER: Yeah, that's the next question, 00:29:42.310 --> 00:29:43.143 align:middle line:90% and that's because-- 00:29:43.143 --> 00:29:45.870 align:middle line:90% 00:29:45.870 --> 00:29:51.760 align:middle line:84% basically, what is s of t0, which comes into that term, 00:29:51.760 --> 00:29:57.060 align:middle line:84% so we know that's s of t0, right, by definition of s, 00:29:57.060 --> 00:30:01.830 align:middle line:84% it's the sensitivity of the initial state 00:30:01.830 --> 00:30:04.083 align:middle line:90% with respect to the parameter. 00:30:04.083 --> 00:30:06.000 align:middle line:84% And, right, and we know that the initial state 00:30:06.000 --> 00:30:08.515 align:middle line:84% doesn't depend on the parameter, so that term-- 00:30:08.515 --> 00:30:09.015 align:middle line:90% 00:30:09.015 --> 00:30:09.610 align:middle line:84% STEVEN JOHNSON: Or it could, right? 00:30:09.610 --> 00:30:10.530 align:middle line:90% FRANK SCHAFER: Oh, it could. 00:30:10.530 --> 00:30:11.580 align:middle line:84% STEVEN JOHNSON: It could, but whatever it is, 00:30:11.580 --> 00:30:12.480 align:middle line:90% you know what it is. 00:30:12.480 --> 00:30:15.090 align:middle line:84% Like, you don't-- you have the initial condition in closed 00:30:15.090 --> 00:30:20.070 align:middle line:84% form, right, so even if it depends on some parameters, 00:30:20.070 --> 00:30:20.950 align:middle line:90% you know what it is. 00:30:20.950 --> 00:30:23.970 align:middle line:84% But here, here only the-- you're simplifying it 00:30:23.970 --> 00:30:27.540 align:middle line:84% to-- only the system of equations 00:30:27.540 --> 00:30:28.978 align:middle line:90% depends on the parameters. 00:30:28.978 --> 00:30:29.770 align:middle line:90% FRANK SCHAFER: Yes. 00:30:29.770 --> 00:30:31.720 align:middle line:90% Yup. 00:30:31.720 --> 00:30:35.620 align:middle line:84% Exactly, so either it's easy, or it's, well, 0, basically. 00:30:35.620 --> 00:30:40.130 align:middle line:90% 00:30:40.130 --> 00:30:44.480 align:middle line:84% Yes, nice, so we have in the end that just this term, basically, 00:30:44.480 --> 00:30:48.422 align:middle line:84% survives, so we have that dg, the derivative of g with 00:30:48.422 --> 00:30:50.880 align:middle line:84% respect to the parameters-- or the gradient, in this case-- 00:30:50.880 --> 00:30:54.180 align:middle line:84% is equal to that derivative, which is easy to compute. 00:30:54.180 --> 00:30:57.000 align:middle line:84% It depends on this lambda, and it 00:30:57.000 --> 00:31:01.530 align:middle line:84% depends on the derivative of f with respect to p. 00:31:01.530 --> 00:31:03.960 align:middle line:90% That's a Jacobian, basically. 00:31:03.960 --> 00:31:08.820 align:middle line:90% So let me let this here. 00:31:08.820 --> 00:31:11.970 align:middle line:84% Right here, so what's kind of the procedure 00:31:11.970 --> 00:31:14.160 align:middle line:84% that we need to do to solve it, right? 00:31:14.160 --> 00:31:20.190 align:middle line:84% So when we solve that integral, we 00:31:20.190 --> 00:31:22.800 align:middle line:84% realize that we first have to solve 00:31:22.800 --> 00:31:25.120 align:middle line:84% this ordinary differential equation, which 00:31:25.120 --> 00:31:26.370 align:middle line:90% gets a special name, actually. 00:31:26.370 --> 00:31:27.990 align:middle line:90% So this is the adjoint ODE. 00:31:27.990 --> 00:31:36.720 align:middle line:90% 00:31:36.720 --> 00:31:40.500 align:middle line:84% But we also see that the forward solution 00:31:40.500 --> 00:31:43.500 align:middle line:90% is embedded in that ODE, right? 00:31:43.500 --> 00:31:47.950 align:middle line:84% So to solve for the quantity which we're interested in, 00:31:47.950 --> 00:31:52.390 align:middle line:84% we basically have a three-step procedure. 00:31:52.390 --> 00:32:02.538 align:middle line:90% So we first solve for u of t. 00:32:02.538 --> 00:32:05.350 align:middle line:84% Well, that's at least the easiest way to think about it. 00:32:05.350 --> 00:32:09.270 align:middle line:84% We first would solve for the ODE for u of t, 00:32:09.270 --> 00:32:11.040 align:middle line:90% then we have that information. 00:32:11.040 --> 00:32:17.805 align:middle line:84% In the next step, we solve for lambda of T. I can also-- 00:32:17.805 --> 00:32:19.680 align:middle line:84% STEVEN JOHNSON: Can you write down ODEs again 00:32:19.680 --> 00:32:21.090 align:middle line:90% and just these steps? 00:32:21.090 --> 00:32:24.000 align:middle line:84% Just to summarize, you're solving an ODE for u of t, 00:32:24.000 --> 00:32:27.470 align:middle line:84% and write down the ODE for lambda, maybe. 00:32:27.470 --> 00:32:42.618 align:middle line:90% 00:32:42.618 --> 00:32:49.320 align:middle line:84% Maybe transpose it back, so just a lambda prime. 00:32:49.320 --> 00:32:51.150 align:middle line:84% Usually, you would transpose it, right? 00:32:51.150 --> 00:32:51.900 align:middle line:90% FRANK SCHAFER: OK. 00:32:51.900 --> 00:32:54.540 align:middle line:90% 00:32:54.540 --> 00:32:55.410 align:middle line:90% Then it's f. 00:32:55.410 --> 00:32:56.610 align:middle line:84% STEVEN JOHNSON: Just transposing both sides 00:32:56.610 --> 00:32:57.825 align:middle line:90% of that equation for lambda. 00:32:57.825 --> 00:33:06.110 align:middle line:90% 00:33:06.110 --> 00:33:07.215 align:middle line:90% FRANK SCHAFER: gu, right? 00:33:07.215 --> 00:33:08.536 align:middle line:90% STEVEN JOHNSON: gu transpose. 00:33:08.536 --> 00:33:11.718 align:middle line:90% 00:33:11.718 --> 00:33:13.760 align:middle line:84% And there's an initial condition, right, so for-- 00:33:13.760 --> 00:33:16.010 align:middle line:84% FRANK SCHAFER: Yeah, so that has an initial condition. 00:33:16.010 --> 00:33:26.318 align:middle line:90% 00:33:26.318 --> 00:33:28.360 align:middle line:84% Right, the initial condition is kind of important 00:33:28.360 --> 00:33:31.810 align:middle line:84% because we see, the first ODE, we solve forwards 00:33:31.810 --> 00:33:34.750 align:middle line:84% in time, and then the adjoint ODE, 00:33:34.750 --> 00:33:37.390 align:middle line:84% as you are now used to probably from reverse-mode AD, 00:33:37.390 --> 00:33:39.470 align:middle line:84% we have to solve backwards in time. 00:33:39.470 --> 00:33:42.460 align:middle line:84% So we have also this forward-backwards workflow, 00:33:42.460 --> 00:33:45.490 align:middle line:84% and then once we know u and lambda, 00:33:45.490 --> 00:33:59.795 align:middle line:84% we can actually solve the integration for dG dp. 00:33:59.795 --> 00:34:01.420 align:middle line:84% STEVEN JOHNSON: Can you write that out? 00:34:01.420 --> 00:34:06.370 align:middle line:84% Just because it simplifies now, so several of the terms 00:34:06.370 --> 00:34:07.910 align:middle line:90% have dropped out. 00:34:07.910 --> 00:34:09.770 align:middle line:90% FRANK SCHAFER: Oh, just g-- 00:34:09.770 --> 00:34:11.020 align:middle line:90% STEVEN JOHNSON: And write big. 00:34:11.020 --> 00:34:21.320 align:middle line:90% 00:34:21.320 --> 00:34:23.060 align:middle line:90% FRANK SCHAFER: OK, that's a-- 00:34:23.060 --> 00:34:25.429 align:middle line:84% STEVEN JOHNSON: No, no, wait, there's no-- 00:34:25.429 --> 00:34:27.050 align:middle line:84% I thought you wanted to write it in terms of lambda, right? 00:34:27.050 --> 00:34:28.050 align:middle line:90% FRANK SCHAFER: Oh, yeah. 00:34:28.050 --> 00:34:29.900 align:middle line:84% STEVEN JOHNSON: Yeah, there's no s anymore. 00:34:29.900 --> 00:34:30.733 align:middle line:90% FRANK SCHAFER: Wait. 00:34:30.733 --> 00:34:32.650 align:middle line:90% Yeah, exactly. 00:34:32.650 --> 00:34:39.915 align:middle line:84% And g-- yeah, that should be right. 00:34:39.915 --> 00:34:42.540 align:middle line:84% STEVEN JOHNSON: And then plus a boundary-- oh, no, the boundary 00:34:42.540 --> 00:34:43.800 align:middle line:90% term is 0 in your case. 00:34:43.800 --> 00:34:45.505 align:middle line:84% FRANK SCHAFER: Yeah, I set it to 0. 00:34:45.505 --> 00:34:47.130 align:middle line:84% STEVEN JOHNSON: So before you continue, 00:34:47.130 --> 00:34:48.172 align:middle line:90% let me just compare this. 00:34:48.172 --> 00:34:50.639 align:middle line:84% We talked about adjoint methods in class before, 00:34:50.639 --> 00:34:54.570 align:middle line:84% but for solving, like, linear systems or nonlinear 00:34:54.570 --> 00:34:57.690 align:middle line:84% equations where the dependence on parameter, and it 00:34:57.690 --> 00:34:59.130 align:middle line:90% was of the same flavor. 00:34:59.130 --> 00:35:01.770 align:middle line:84% So first, you solve the forward equations. 00:35:01.770 --> 00:35:04.230 align:middle line:84% You solve ax equals b for x, right, 00:35:04.230 --> 00:35:06.840 align:middle line:84% and then we solved an adjoint equation 00:35:06.840 --> 00:35:11.190 align:middle line:84% that involved a transpose or a transpose of a Jacobian, right? 00:35:11.190 --> 00:35:12.630 align:middle line:84% But whatever the adjoint equation 00:35:12.630 --> 00:35:15.415 align:middle line:90% was of the same kind of-- 00:35:15.415 --> 00:35:18.540 align:middle line:84% same flavor and same size as the original problem, but it 00:35:18.540 --> 00:35:19.647 align:middle line:90% was linear, right? 00:35:19.647 --> 00:35:21.480 align:middle line:84% And then once you have the forward solution, 00:35:21.480 --> 00:35:23.730 align:middle line:84% the adjoint solution, you just take some dot products. 00:35:23.730 --> 00:35:25.890 align:middle line:84% So here, you solve an ODE, right, 00:35:25.890 --> 00:35:28.470 align:middle line:84% and you're using whatever your favorite ODE solver is. 00:35:28.470 --> 00:35:33.090 align:middle line:84% Then you solve an adjoint ODE, which is sort of transposed, 00:35:33.090 --> 00:35:36.660 align:middle line:84% and it's linear now, even if the original one was linear, 00:35:36.660 --> 00:35:38.830 align:middle line:84% and it's backwards in time if the original one was 00:35:38.830 --> 00:35:39.730 align:middle line:90% forwards in time. 00:35:39.730 --> 00:35:42.520 align:middle line:84% And then once you have the forward solution u of t 00:35:42.520 --> 00:35:44.560 align:middle line:84% and the adjoint solution lambda of T, 00:35:44.560 --> 00:35:48.340 align:middle line:84% you take some dot products, which for functions, the analog 00:35:48.340 --> 00:35:50.230 align:middle line:90% of that is integrals, right? 00:35:50.230 --> 00:35:53.740 align:middle line:84% And so everything in that sense becomes easy now. 00:35:53.740 --> 00:35:56.800 align:middle line:84% Independent-- even if you have 10,000 parameters, 00:35:56.800 --> 00:36:01.570 align:middle line:84% you solve one forward ODE, one adjoint ODE of the same size, 00:36:01.570 --> 00:36:03.170 align:middle line:90% and then some dot products. 00:36:03.170 --> 00:36:05.560 align:middle line:84% So this is why you can compute derivatives 00:36:05.560 --> 00:36:11.290 align:middle line:84% with a huge number of parameters efficiently, 00:36:11.290 --> 00:36:14.140 align:middle line:84% and they're kind of always of the same flavor, these adjoint, 00:36:14.140 --> 00:36:16.155 align:middle line:90% reverse-mode things. 00:36:16.155 --> 00:36:18.280 align:middle line:84% FRANK SCHAFER: Yes, and now there are a few tricks, 00:36:18.280 --> 00:36:20.830 align:middle line:84% basically, to evaluate that efficiently, 00:36:20.830 --> 00:36:24.550 align:middle line:84% or different styles to evaluate that equation. 00:36:24.550 --> 00:36:27.220 align:middle line:84% The first thing to notice is basically 00:36:27.220 --> 00:36:31.090 align:middle line:84% that we don't necessarily, for this continuous adjoint method, 00:36:31.090 --> 00:36:35.830 align:middle line:84% have to store and compute the full continuous solution of u 00:36:35.830 --> 00:36:41.240 align:middle line:84% of t because an ODE is basically reversible. 00:36:41.240 --> 00:36:44.830 align:middle line:84% And so we can solve the ODE forwards, 00:36:44.830 --> 00:36:48.070 align:middle line:84% just save the final value of that ODE, 00:36:48.070 --> 00:36:52.300 align:middle line:84% and then solve it basically in lockstep backwards together 00:36:52.300 --> 00:36:54.790 align:middle line:90% with lambda to have-- 00:36:54.790 --> 00:36:58.030 align:middle line:84% well, to avoid this kind of storing the continuous solution 00:36:58.030 --> 00:37:01.450 align:middle line:90% of u of t at all times. 00:37:01.450 --> 00:37:04.990 align:middle line:84% And everybody happy with that kind of trick, right? 00:37:04.990 --> 00:37:05.695 align:middle line:90% So-- 00:37:05.695 --> 00:37:08.195 align:middle line:84% STEVEN JOHNSON: I'm not happy with that kind of trick, like, 00:37:08.195 --> 00:37:10.211 align:middle line:84% if it's chaotic or something like that. 00:37:10.211 --> 00:37:10.270 align:middle line:84% FRANK SCHAFER: But ODEs are reversible-- 00:37:10.270 --> 00:37:11.990 align:middle line:84% STEVEN JOHNSON: Solving forwards might be exponentially 00:37:11.990 --> 00:37:13.610 align:middle line:84% decaying, and solving it backwards 00:37:13.610 --> 00:37:14.902 align:middle line:90% might be exponentially growing. 00:37:14.902 --> 00:37:16.850 align:middle line:84% So that trick doesn't always work, right? 00:37:16.850 --> 00:37:18.017 align:middle line:90% FRANK SCHAFER: Yes, exactly. 00:37:18.017 --> 00:37:21.180 align:middle line:84% That trick has to be done with care, and there are some-- 00:37:21.180 --> 00:37:23.840 align:middle line:84% right, so that trick, just to visualize it again, 00:37:23.840 --> 00:37:27.350 align:middle line:84% it basically says, if we have some ODE at u of t0 00:37:27.350 --> 00:37:32.120 align:middle line:84% and we solve it to some final time, u of capital T, 00:37:32.120 --> 00:37:35.150 align:middle line:84% I was saying, you can just store this value 00:37:35.150 --> 00:37:37.640 align:middle line:90% and start solving it backwards. 00:37:37.640 --> 00:37:40.340 align:middle line:84% But depending on your physical system, 00:37:40.340 --> 00:37:42.170 align:middle line:84% well, your trajectory might not-- 00:37:42.170 --> 00:37:44.540 align:middle line:84% because your numerical solver might not be reversible, 00:37:44.540 --> 00:37:47.673 align:middle line:84% you might deviate from your true solution. 00:37:47.673 --> 00:37:49.340 align:middle line:84% And then if you solve it for long times, 00:37:49.340 --> 00:37:52.110 align:middle line:84% again, backwards, that might be numerically very bad, 00:37:52.110 --> 00:37:54.260 align:middle line:84% and so the first thing people are usually doing 00:37:54.260 --> 00:37:58.880 align:middle line:84% is introducing so-called checkpoints, which only means, 00:37:58.880 --> 00:38:02.060 align:middle line:84% once you solve it back and you reach your checkpoint, you 00:38:02.060 --> 00:38:05.930 align:middle line:84% basically reset your solution to the state which 00:38:05.930 --> 00:38:06.650 align:middle line:90% you had forwards. 00:38:06.650 --> 00:38:09.860 align:middle line:84% So that reduces a bit the memory consumption. 00:38:09.860 --> 00:38:13.280 align:middle line:84% And the second trick people are doing, if it's really unstable 00:38:13.280 --> 00:38:16.190 align:middle line:84% and your ODE solver is not working, then what you would do 00:38:16.190 --> 00:38:17.900 align:middle line:84% is maybe also have these checkpoints 00:38:17.900 --> 00:38:20.240 align:middle line:84% but always just store a continuous solution 00:38:20.240 --> 00:38:24.600 align:middle line:84% within the checkpoints that you solve forwards again in time. 00:38:24.600 --> 00:38:27.330 align:middle line:84% STEVEN JOHNSON: So just another comment on that, 00:38:27.330 --> 00:38:29.060 align:middle line:84% so one thing we have to be always 00:38:29.060 --> 00:38:30.560 align:middle line:84% careful about in presenting things 00:38:30.560 --> 00:38:33.000 align:middle line:84% is explaining the solution before we explain the problem, 00:38:33.000 --> 00:38:33.500 align:middle line:90% right? 00:38:33.500 --> 00:38:36.410 align:middle line:84% So the issue here that you're trying 00:38:36.410 --> 00:38:38.750 align:middle line:84% to solve with going backwards and reversing 00:38:38.750 --> 00:38:41.240 align:middle line:84% and so forth is-- so in principle, 00:38:41.240 --> 00:38:44.390 align:middle line:84% you solve it forward, right, to find u of t, 00:38:44.390 --> 00:38:46.850 align:middle line:84% and then you solve it backwards to find lambda T, 00:38:46.850 --> 00:38:48.770 align:middle line:84% and then you take some dot products, right? 00:38:48.770 --> 00:38:52.400 align:middle line:84% The problem you can run into is that then you 00:38:52.400 --> 00:38:56.090 align:middle line:84% need to store u of t not just at the endpoint 00:38:56.090 --> 00:38:59.900 align:middle line:84% but at all the intermediate times, which is fine if u of t 00:38:59.900 --> 00:39:01.730 align:middle line:84% is just a small number of parameters. 00:39:01.730 --> 00:39:04.520 align:middle line:84% And you just store it at a bunch of points, 00:39:04.520 --> 00:39:07.820 align:middle line:84% and then you basically can interpolate 00:39:07.820 --> 00:39:09.320 align:middle line:90% with polynomials in between. 00:39:09.320 --> 00:39:12.230 align:middle line:84% Right, this is how ODE solvers work. 00:39:12.230 --> 00:39:14.810 align:middle line:84% But this is a problem if you're solving 00:39:14.810 --> 00:39:18.380 align:middle line:84% a huge system of ODEs, where, say, it's really 00:39:18.380 --> 00:39:20.790 align:middle line:90% a discretized PDE, for example. 00:39:20.790 --> 00:39:24.290 align:middle line:84% So you have u is, say, the electric field at every point 00:39:24.290 --> 00:39:27.830 align:middle line:84% in space as a function of time in some big computational cell, 00:39:27.830 --> 00:39:29.390 align:middle line:90% then solving-- 00:39:29.390 --> 00:39:32.390 align:middle line:84% then usually when you solve it, you march it forward in time, 00:39:32.390 --> 00:39:34.340 align:middle line:84% and you kind of just only store the solution 00:39:34.340 --> 00:39:35.420 align:middle line:90% at the current time. 00:39:35.420 --> 00:39:39.290 align:middle line:84% But here, you need to store all the solutions 00:39:39.290 --> 00:39:42.330 align:middle line:84% at all the intermediate times all the way until the end 00:39:42.330 --> 00:39:45.360 align:middle line:84% and then go backwards, and you can run out of memory. 00:39:45.360 --> 00:39:48.620 align:middle line:84% And so this is the basic problem with adjoint methods 00:39:48.620 --> 00:39:52.430 align:middle line:84% and when you have, like, evolving in time 00:39:52.430 --> 00:39:54.140 align:middle line:84% or going through big recurrences, 00:39:54.140 --> 00:39:56.210 align:middle line:84% is you need to store all the intermediate ones. 00:39:56.210 --> 00:39:58.760 align:middle line:84% And so there's all these tricks to try 00:39:58.760 --> 00:40:00.800 align:middle line:84% and only store some of the intermediate values 00:40:00.800 --> 00:40:04.100 align:middle line:84% at some checkpoints and kind of reconstruct things on the fly 00:40:04.100 --> 00:40:09.560 align:middle line:84% as you integrate lambda T backwards so that you don't 00:40:09.560 --> 00:40:13.580 align:middle line:84% need to store everything, but that's sort of the issue 00:40:13.580 --> 00:40:15.200 align:middle line:84% that you're struggling with here. 00:40:15.200 --> 00:40:19.850 align:middle line:84% But for small ODEs, to only have a few things at any time, 00:40:19.850 --> 00:40:21.740 align:middle line:84% you just store the whole solution. 00:40:21.740 --> 00:40:22.545 align:middle line:90% It's not that hard. 00:40:22.545 --> 00:40:23.420 align:middle line:90% FRANK SCHAFER: Right. 00:40:23.420 --> 00:40:25.220 align:middle line:90% True. 00:40:25.220 --> 00:40:27.950 align:middle line:84% Yeah, well, for small ODEs that contain 00:40:27.950 --> 00:40:31.470 align:middle line:84% some parametric function, that's very expensive to evaluate, 00:40:31.470 --> 00:40:31.970 align:middle line:90% right? 00:40:31.970 --> 00:40:34.190 align:middle line:84% Like, what Chris is basically doing 00:40:34.190 --> 00:40:36.990 align:middle line:84% is putting a neural network inside the drift function, 00:40:36.990 --> 00:40:40.310 align:middle line:84% so even if the ODE itself looks very simple, 00:40:40.310 --> 00:40:42.600 align:middle line:84% if you have a neural network in each timestep, 00:40:42.600 --> 00:40:45.380 align:middle line:84% you will have to store all intermediate computations also, 00:40:45.380 --> 00:40:46.730 align:middle line:90% and outputs of that. 00:40:46.730 --> 00:40:51.560 align:middle line:84% So, yeah, that makes the continuous adjoint 00:40:51.560 --> 00:40:54.490 align:middle line:90% very meaningful. 00:40:54.490 --> 00:41:00.830 align:middle line:84% Yeah, right, so there's a third kind of trick 00:41:00.830 --> 00:41:06.822 align:middle line:84% which can be used to simplify that workflow even a bit more. 00:41:06.822 --> 00:41:09.280 align:middle line:84% Basically, that's the first trick, and as we now discussed, 00:41:09.280 --> 00:41:10.750 align:middle line:90% it's not always applicable. 00:41:10.750 --> 00:41:14.750 align:middle line:84% But if it's applicable, it can be nice and very efficient. 00:41:14.750 --> 00:41:20.050 align:middle line:84% The third trick is basically to look at this integration term 00:41:20.050 --> 00:41:28.280 align:middle line:84% and basically realize that an integration can also 00:41:28.280 --> 00:41:33.596 align:middle line:84% be phrased as a solution to an ordinary differential equation. 00:41:33.596 --> 00:41:44.770 align:middle line:84% So I'm not sure if that was covered or-- 00:41:44.770 --> 00:41:47.800 align:middle line:84% but, yeah, let's say I have a function 00:41:47.800 --> 00:41:53.510 align:middle line:84% F. I want to integrate it as [INAUDIBLE] prime to delta t. 00:41:53.510 --> 00:41:57.225 align:middle line:90% 00:41:57.225 --> 00:41:59.350 align:middle line:84% But everybody agrees, that's kind of the same form? 00:41:59.350 --> 00:42:04.780 align:middle line:84% So we have some integration, and now the claim 00:42:04.780 --> 00:42:07.780 align:middle line:84% is that we can write that as an ordinary differential 00:42:07.780 --> 00:42:13.000 align:middle line:90% equation small f of t. 00:42:13.000 --> 00:42:16.360 align:middle line:84% And, yeah, now we want to basically apply 00:42:16.360 --> 00:42:21.320 align:middle line:84% Euler's method to that, and so if we apply Euler's method-- 00:42:21.320 --> 00:42:24.640 align:middle line:84% STEVEN JOHNSON: So, wait, stop here for a second. 00:42:24.640 --> 00:42:26.660 align:middle line:84% So before you again give the solution, 00:42:26.660 --> 00:42:29.050 align:middle line:84% so you're going to write a differential equation w 00:42:29.050 --> 00:42:30.460 align:middle line:90% prime equals F of T, right? 00:42:30.460 --> 00:42:34.910 align:middle line:84% So wt is going to be a new variable. 00:42:34.910 --> 00:42:37.330 align:middle line:84% What's the initial condition of that, that you're 00:42:37.330 --> 00:42:38.205 align:middle line:90% going to want to use? 00:42:38.205 --> 00:42:40.413 align:middle line:84% Again, at the end, you're going to want to write down 00:42:40.413 --> 00:42:42.670 align:middle line:84% a differential equation whose solution gives that 00:42:42.670 --> 00:42:44.815 align:middle line:90% integral, so what-- 00:42:44.815 --> 00:42:46.690 align:middle line:84% you're going to write a differential equation 00:42:46.690 --> 00:42:49.510 align:middle line:90% dw dt equals little f of t. 00:42:49.510 --> 00:42:51.265 align:middle line:84% What initial condition should you use? 00:42:51.265 --> 00:42:54.340 align:middle line:90% 00:42:54.340 --> 00:42:56.220 align:middle line:90% Any guesses? 00:42:56.220 --> 00:42:58.150 align:middle line:90% To get-- to turn it back-- 00:42:58.150 --> 00:43:00.120 align:middle line:84% so its solution will give you the same thing 00:43:00.120 --> 00:43:03.090 align:middle line:84% as that definite integral from t0 to t. 00:43:03.090 --> 00:43:08.535 align:middle line:90% 00:43:08.535 --> 00:43:09.530 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:43:09.530 --> 00:43:12.828 align:middle line:84% STEVEN JOHNSON: 0, OK, so but at what time? 00:43:12.828 --> 00:43:13.370 align:middle line:90% AUDIENCE: t0. 00:43:13.370 --> 00:43:22.530 align:middle line:84% STEVEN JOHNSON: At t0, so w of t0 equals 0, and then what's-- 00:43:22.530 --> 00:43:24.510 align:middle line:84% capital F of T equals what, then? 00:43:24.510 --> 00:43:28.140 align:middle line:84% So now the integral you want, the first line, 00:43:28.140 --> 00:43:30.930 align:middle line:90% capital-- is that an I or an F? 00:43:30.930 --> 00:43:31.973 align:middle line:90% I can't-- 00:43:31.973 --> 00:43:33.390 align:middle line:84% FRANK SCHAFER: That's an F. Sorry. 00:43:33.390 --> 00:43:34.470 align:middle line:84% STEVEN JOHNSON: That's a capital F, OK. 00:43:34.470 --> 00:43:35.520 align:middle line:90% FRANK SCHAFER: Sorry. 00:43:35.520 --> 00:43:38.250 align:middle line:84% STEVEN JOHNSON: It has the second line going kind 00:43:38.250 --> 00:43:39.450 align:middle line:90% of-- looks a little bit-- 00:43:39.450 --> 00:43:42.120 align:middle line:84% OK, so anyway, so your capital F of T, your integral that you 00:43:42.120 --> 00:43:44.040 align:middle line:90% want is what in terms of w? 00:43:44.040 --> 00:43:45.920 align:middle line:84% So if you have-- the differential equation, 00:43:45.920 --> 00:43:48.840 align:middle line:84% its initial condition t equals 0 is 0-- 00:43:48.840 --> 00:43:49.950 align:middle line:90% t0 is 0. 00:43:49.950 --> 00:43:54.770 align:middle line:84% Its derivative is little f of t, so then the integral 00:43:54.770 --> 00:43:56.686 align:middle line:90% is what now? 00:43:56.686 --> 00:43:58.120 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:43:58.120 --> 00:44:00.820 align:middle line:84% FRANK SCHAFER: Yeah, right, so I can write that integral. 00:44:00.820 --> 00:44:02.860 align:middle line:84% Let me write one step in between, basically, 00:44:02.860 --> 00:44:04.180 align:middle line:90% and kind of slide it. 00:44:04.180 --> 00:44:06.862 align:middle line:90% 00:44:06.862 --> 00:44:09.070 align:middle line:84% STEVEN JOHNSON: You're kind of discretizing it first, 00:44:09.070 --> 00:44:10.528 align:middle line:84% so before you discretize it, I want 00:44:10.528 --> 00:44:12.987 align:middle line:84% to make sure they understand the connection between the ODE 00:44:12.987 --> 00:44:13.800 align:middle line:90% and the integral. 00:44:13.800 --> 00:44:16.440 align:middle line:84% So don't discretize it yet, all right? 00:44:16.440 --> 00:44:19.773 align:middle line:84% Just, this is an exact relationship, right? 00:44:19.773 --> 00:44:22.315 align:middle line:84% FRANK SCHAFER: Oh, so you just want to have plugged in the w, 00:44:22.315 --> 00:44:22.570 align:middle line:90% so-- 00:44:22.570 --> 00:44:24.362 align:middle line:84% STEVEN JOHNSON: Yeah, yeah, it's w of what? 00:44:24.362 --> 00:44:27.200 align:middle line:90% 00:44:27.200 --> 00:44:28.200 align:middle line:90% No, there's no integral. 00:44:28.200 --> 00:44:29.550 align:middle line:90% It's equal to w of-- 00:44:29.550 --> 00:44:34.270 align:middle line:90% 00:44:34.270 --> 00:44:36.130 align:middle line:90% it's w of capital T, right? 00:44:36.130 --> 00:44:37.030 align:middle line:90% That's-- 00:44:37.030 --> 00:44:37.600 align:middle line:90% FRANK SCHAFER: Oh, right. 00:44:37.600 --> 00:44:37.830 align:middle line:90% STEVEN JOHNSON: Right. 00:44:37.830 --> 00:44:38.450 align:middle line:84% FRANK SCHAFER: OK, now I see what 00:44:38.450 --> 00:44:39.700 align:middle line:90% you want you want me to write. 00:44:39.700 --> 00:44:41.710 align:middle line:90% STEVEN JOHNSON: Right. 00:44:41.710 --> 00:44:44.050 align:middle line:84% So the point is, if you have a differential equation 00:44:44.050 --> 00:44:47.740 align:middle line:84% w prime equals f of t, the solution is just the integral. 00:44:47.740 --> 00:44:50.600 align:middle line:84% w is just the integral of f, which is what you want, 00:44:50.600 --> 00:44:53.350 align:middle line:84% and then you put the initial condition at t0 equals 0. 00:44:53.350 --> 00:44:55.480 align:middle line:84% That's the left edge of the integral, 00:44:55.480 --> 00:44:58.360 align:middle line:84% and then the final integral you want is just w of t. 00:44:58.360 --> 00:45:01.990 align:middle line:84% So that's-- he's turning-- the integral you want is just 00:45:01.990 --> 00:45:06.210 align:middle line:84% a differential equation, is this differential equation, 00:45:06.210 --> 00:45:09.518 align:middle line:84% and now you're going to discretize it, right? 00:45:09.518 --> 00:45:11.560 align:middle line:84% FRANK SCHAFER: Right, now if I'm discretizing it, 00:45:11.560 --> 00:45:15.180 align:middle line:84% I will see the same, basically, as for-- 00:45:15.180 --> 00:45:16.533 align:middle line:90% yeah, for an Euler step. 00:45:16.533 --> 00:45:17.700 align:middle line:90% I should write it down, or-- 00:45:17.700 --> 00:45:19.560 align:middle line:84% STEVEN JOHNSON: Sure, if you want to discretize it. 00:45:19.560 --> 00:45:21.870 align:middle line:84% Yeah, I just wanted to-- because not everyone here 00:45:21.870 --> 00:45:24.522 align:middle line:84% has done a lot of differential equations, right, 00:45:24.522 --> 00:45:25.980 align:middle line:84% or maybe hasn't done them recently, 00:45:25.980 --> 00:45:29.730 align:middle line:84% so I want to make sure they understand, 00:45:29.730 --> 00:45:31.230 align:middle line:90% understood that trick. 00:45:31.230 --> 00:45:49.115 align:middle line:90% 00:45:49.115 --> 00:45:50.140 align:middle line:90% It's good. 00:45:50.140 --> 00:45:51.986 align:middle line:90% Take it all the way up now. 00:45:51.986 --> 00:45:52.962 align:middle line:90% FRANK SCHAFER: Oh, OK. 00:45:52.962 --> 00:45:54.426 align:middle line:90% STEVEN JOHNSON: [INAUDIBLE]. 00:45:54.426 --> 00:46:02.240 align:middle line:90% 00:46:02.240 --> 00:46:06.470 align:middle line:84% FRANK SCHAFER: Right, so I'm basically writing down 00:46:06.470 --> 00:46:09.110 align:middle line:84% what the Euler steps would be in the discretization 00:46:09.110 --> 00:46:10.130 align:middle line:90% for that integral. 00:46:10.130 --> 00:46:13.620 align:middle line:90% 00:46:13.620 --> 00:46:19.590 align:middle line:84% That's the first, and then I just keep iterating. 00:46:19.590 --> 00:46:32.010 align:middle line:90% 00:46:32.010 --> 00:46:33.545 align:middle line:90% Let's think a moment for-- 00:46:33.545 --> 00:46:46.410 align:middle line:90% 00:46:46.410 --> 00:46:50.670 align:middle line:84% So these are n Euler steps iterated on the function. 00:46:50.670 --> 00:46:54.620 align:middle line:90% 00:46:54.620 --> 00:46:57.043 align:middle line:84% There's an i missing, probably, somewhere. 00:46:57.043 --> 00:47:01.780 align:middle line:90% 00:47:01.780 --> 00:47:03.910 align:middle line:84% Right, because once I have my first step, 00:47:03.910 --> 00:47:08.200 align:middle line:84% the time changes, and then that's flagged back also 00:47:08.200 --> 00:47:09.850 align:middle line:90% into the function. 00:47:09.850 --> 00:47:12.100 align:middle line:84% STEVEN JOHNSON: OK, so this is really an approximation 00:47:12.100 --> 00:47:13.350 align:middle line:90% for the original thing, right? 00:47:13.350 --> 00:47:18.190 align:middle line:84% So you're writing "equals" here, but just to be clear, 00:47:18.190 --> 00:47:19.990 align:middle line:84% right, the original equation is-- 00:47:19.990 --> 00:47:22.200 align:middle line:84% you have an integral and the differential equation. 00:47:22.200 --> 00:47:24.378 align:middle line:84% That's an exact relationship, and now you're 00:47:24.378 --> 00:47:26.170 align:middle line:84% going to write the discrete analog of that, 00:47:26.170 --> 00:47:27.420 align:middle line:90% or the discrete approximation. 00:47:27.420 --> 00:47:30.460 align:middle line:84% So w of t plus delta t is really approximately 00:47:30.460 --> 00:47:34.710 align:middle line:84% equal to w of t plus t delta t, and this is approximately. 00:47:34.710 --> 00:47:36.940 align:middle line:84% FRANK SCHAFER: Yeah, if I do it again-- 00:47:36.940 --> 00:47:40.300 align:middle line:90% something like an Euler step. 00:47:40.300 --> 00:47:44.560 align:middle line:84% STEVEN JOHNSON: So you can approximate the derivative 00:47:44.560 --> 00:47:47.830 align:middle line:84% by a difference, and then the integral of your-- 00:47:47.830 --> 00:47:50.125 align:middle line:84% is essentially approximated by a sum. 00:47:50.125 --> 00:47:55.670 align:middle line:90% 00:47:55.670 --> 00:47:57.710 align:middle line:84% FRANK SCHAFER: Right, and so this now has, 00:47:57.710 --> 00:48:00.740 align:middle line:84% then, also, I mean, the typical form 00:48:00.740 --> 00:48:04.070 align:middle line:84% of what you would do numerically to solve an integral, right? 00:48:04.070 --> 00:48:11.490 align:middle line:84% So it's-- yeah, you have the function value at different 00:48:11.490 --> 00:48:14.700 align:middle line:84% points, and then with the discretization that you use 00:48:14.700 --> 00:48:16.382 align:middle line:90% and summed it all up. 00:48:16.382 --> 00:48:18.090 align:middle line:84% STEVEN JOHNSON: This is a very simple way 00:48:18.090 --> 00:48:21.710 align:middle line:84% to do an approximate integral numerically, just like-- 00:48:21.710 --> 00:48:25.470 align:middle line:84% Euler integration for ODEs and this approximate sum 00:48:25.470 --> 00:48:28.260 align:middle line:84% for an integral, you should never, ever do this in reality, 00:48:28.260 --> 00:48:31.350 align:middle line:84% almost, because there's exponentially more 00:48:31.350 --> 00:48:34.200 align:middle line:84% accurate ways to integrate approximate integrals. 00:48:34.200 --> 00:48:37.800 align:middle line:84% But it's a nice way to really simply think of-- 00:48:37.800 --> 00:48:40.080 align:middle line:84% if you're confused by integrals and derivatives, 00:48:40.080 --> 00:48:43.110 align:middle line:84% you can think of them as differences and sums or sums 00:48:43.110 --> 00:48:44.740 align:middle line:90% and differences, right? 00:48:44.740 --> 00:48:46.490 align:middle line:84% FRANK SCHAFER: Yeah, right, that statement 00:48:46.490 --> 00:48:48.947 align:middle line:84% is true for both, right, the Euler solver for ODEs 00:48:48.947 --> 00:48:50.780 align:middle line:84% and the kind of derivation for the integral. 00:48:50.780 --> 00:48:54.223 align:middle line:90% 00:48:54.223 --> 00:48:56.390 align:middle line:84% Yeah, that's actually the full derivation, I'm sure. 00:48:56.390 --> 00:48:58.807 align:middle line:84% STEVEN JOHNSON: Yeah, so does everyone see that basically, 00:48:58.807 --> 00:49:00.860 align:middle line:84% then, if you plug in, the initial t is 0. 00:49:00.860 --> 00:49:02.903 align:middle line:90% w of t0 is 0. 00:49:02.903 --> 00:49:05.320 align:middle line:84% Right, so and then why don't you take it one step further? 00:49:05.320 --> 00:49:10.650 align:middle line:84% So then w of t0, if w of t0 is 0, the first term is 0. 00:49:10.650 --> 00:49:14.400 align:middle line:84% The second term is approximately the integral, right, 00:49:14.400 --> 00:49:16.530 align:middle line:84% so it's just a restating the same thing, just 00:49:16.530 --> 00:49:17.650 align:middle line:90% with sums and differences. 00:49:17.650 --> 00:49:22.548 align:middle line:84% So this is-- so the solution of the ODE is really the integral. 00:49:22.548 --> 00:49:23.340 align:middle line:90% FRANK SCHAFER: Yes. 00:49:23.340 --> 00:49:27.260 align:middle line:90% 00:49:27.260 --> 00:49:29.300 align:middle line:84% Yeah, to summarize that, so we can have, 00:49:29.300 --> 00:49:33.005 align:middle line:84% then, basically all the three methods above. 00:49:33.005 --> 00:49:34.880 align:middle line:84% STEVEN JOHNSON: So can you finish and write-- 00:49:34.880 --> 00:49:35.750 align:middle line:90% so the point at the-- 00:49:35.750 --> 00:49:37.250 align:middle line:84% what you're going to do then is just 00:49:37.250 --> 00:49:41.717 align:middle line:84% merge the solution of the adjoint ODE with that integral 00:49:41.717 --> 00:49:44.300 align:middle line:84% in the third step, so instead of doing them in separate steps, 00:49:44.300 --> 00:49:47.630 align:middle line:84% so you can maybe just write down the derivative. 00:49:47.630 --> 00:49:48.618 align:middle line:90% It has a-- 00:49:48.618 --> 00:49:49.910 align:middle line:90% FRANK SCHAFER: That one, right? 00:49:49.910 --> 00:49:51.327 align:middle line:84% STEVEN JOHNSON: Yeah, so what's w? 00:49:51.327 --> 00:49:56.448 align:middle line:84% w is now going to be-- no, no, w is going to be lambda and-- 00:49:56.448 --> 00:49:57.365 align:middle line:90% OK, it doesn't matter. 00:49:57.365 --> 00:50:03.652 align:middle line:90% 00:50:03.652 --> 00:50:05.860 align:middle line:84% FRANK SCHAFER: Yeah, depending on the other side of-- 00:50:05.860 --> 00:50:07.140 align:middle line:84% STEVEN JOHNSON: You want to go in the other direction, right? 00:50:07.140 --> 00:50:08.190 align:middle line:90% Yeah. 00:50:08.190 --> 00:50:10.720 align:middle line:84% FRANK SCHAFER: Probably want to solve it like that. 00:50:10.720 --> 00:50:12.095 align:middle line:84% STEVEN JOHNSON: But the point is, 00:50:12.095 --> 00:50:14.460 align:middle line:84% you think of this as a coupled ODE, for w 00:50:14.460 --> 00:50:16.990 align:middle line:84% along with the adjoint ODE for lambda prime. 00:50:16.990 --> 00:50:19.830 align:middle line:84% So if you write down the lambda prime underneath, right-- 00:50:19.830 --> 00:50:32.610 align:middle line:90% 00:50:32.610 --> 00:50:33.360 align:middle line:90% FRANK SCHAFER: gu. 00:50:33.360 --> 00:50:43.047 align:middle line:90% 00:50:43.047 --> 00:50:45.630 align:middle line:84% STEVEN JOHNSON: So you can think of this as one system of ODEs 00:50:45.630 --> 00:50:52.050 align:middle line:84% for a vector that contains w and lambda, right, 00:50:52.050 --> 00:50:54.330 align:middle line:84% so you just do the two things together 00:50:54.330 --> 00:50:57.885 align:middle line:90% instead of the separate steps. 00:50:57.885 --> 00:51:01.680 align:middle line:84% Oh, and then I guess, yeah, you could also 00:51:01.680 --> 00:51:03.870 align:middle line:84% integrate the original equation backwards as well. 00:51:03.870 --> 00:51:07.310 align:middle line:90% 00:51:07.310 --> 00:51:09.865 align:middle line:84% All right, so now your unknowns are w, lambda, and u, so, 00:51:09.865 --> 00:51:12.365 align:middle line:84% like, one big vector of those three things stacked together. 00:51:12.365 --> 00:51:18.930 align:middle line:90% 00:51:18.930 --> 00:51:20.700 align:middle line:84% And the advantage of this is that you 00:51:20.700 --> 00:51:23.730 align:middle line:84% don't need to store the lambdas, I guess, 00:51:23.730 --> 00:51:26.070 align:middle line:84% and maybe the u's. If you were able to integrate 00:51:26.070 --> 00:51:28.770 align:middle line:84% the whole thing backwards, you don't need to store anything. 00:51:28.770 --> 00:51:32.520 align:middle line:84% You just pass this to your ODE integrator, 00:51:32.520 --> 00:51:34.980 align:middle line:84% and it only keeps track of your current step. 00:51:34.980 --> 00:51:36.840 align:middle line:84% And it just walks from beginning to end, 00:51:36.840 --> 00:51:39.540 align:middle line:84% and at the end of the day, your w 00:51:39.540 --> 00:51:40.950 align:middle line:90% is the integral that you want-- 00:51:40.950 --> 00:51:42.785 align:middle line:90% is the derivative. 00:51:42.785 --> 00:51:45.750 align:middle line:84% FRANK SCHAFER: Yeah, so that's the most memory-efficient one 00:51:45.750 --> 00:51:51.150 align:middle line:84% to basically solve for the sensitivity dG dp in the end. 00:51:51.150 --> 00:51:55.060 align:middle line:90% 00:51:55.060 --> 00:51:55.660 align:middle line:90% Right. 00:51:55.660 --> 00:51:56.327 align:middle line:90% That's all for-- 00:51:56.327 --> 00:51:58.900 align:middle line:84% STEVEN JOHNSON: So I think we should maybe 00:51:58.900 --> 00:52:00.620 align:middle line:84% take a break for the-- or in a minute, 00:52:00.620 --> 00:52:03.250 align:middle line:84% but are you basically-- is this basically it? 00:52:03.250 --> 00:52:04.900 align:middle line:84% Or are you going to show some examples? 00:52:04.900 --> 00:52:06.692 align:middle line:84% FRANK SCHAFER: I would have the integration 00:52:06.692 --> 00:52:08.810 align:middle line:84% of that numerically, but it takes another 10, 00:52:08.810 --> 00:52:10.155 align:middle line:90% 15 minutes at least. 00:52:10.155 --> 00:52:12.655 align:middle line:84% STEVEN JOHNSON: Yeah, I see you want to show an example of-- 00:52:12.655 --> 00:52:14.530 align:middle line:84% yeah, I mean, we can do that after the break, 00:52:14.530 --> 00:52:17.290 align:middle line:84% but maybe we can just take a break for a couple minutes now. 00:52:17.290 --> 00:52:19.510 align:middle line:84% But do people have questions before we-- 00:52:19.510 --> 00:52:21.220 align:middle line:90% about this step? 00:52:21.220 --> 00:52:23.372 align:middle line:84% You have forward ODE, adjoint ODE, and then 00:52:23.372 --> 00:52:25.330 align:middle line:84% a dot product, which is basically the integral. 00:52:25.330 --> 00:52:28.210 align:middle line:84% And then there's these tricks to try and basically combine them 00:52:28.210 --> 00:52:30.460 align:middle line:84% into one ODE that you solve instead 00:52:30.460 --> 00:52:33.280 align:middle line:84% of having to store u of T, store lambda of T, 00:52:33.280 --> 00:52:34.345 align:middle line:90% and then do the integral. 00:52:34.345 --> 00:52:37.552 align:middle line:90% 00:52:37.552 --> 00:52:41.448 align:middle line:84% AUDIENCE: So say you have some data point that you're 00:52:41.448 --> 00:52:47.210 align:middle line:84% comparing to something you're looking for, would you-- 00:52:47.210 --> 00:52:50.130 align:middle line:84% so if the system's not chaotic, you'd probably 00:52:50.130 --> 00:52:52.130 align:middle line:84% want to start it at the final data point 00:52:52.130 --> 00:52:56.018 align:middle line:84% and integrate backwards with the lambda equation, 00:52:56.018 --> 00:52:58.680 align:middle line:84% but if it is, then would you just 00:52:58.680 --> 00:53:01.826 align:middle line:84% want to integrate forward to get the full solution 00:53:01.826 --> 00:53:05.214 align:middle line:90% and then just solve that? 00:53:05.214 --> 00:53:07.375 align:middle line:84% STEVEN JOHNSON: Yeah, so suppose capital G-- wait, 00:53:07.375 --> 00:53:11.020 align:middle line:84% so suppose capital G is a sum and not an integral, right? 00:53:11.020 --> 00:53:13.668 align:middle line:84% So you're-- or you have a bunch of delta functions, 00:53:13.668 --> 00:53:15.460 align:middle line:84% is another way of thinking about it, right? 00:53:15.460 --> 00:53:18.850 align:middle line:84% So basically, you're comparing the solution u of T 00:53:18.850 --> 00:53:21.520 align:middle line:84% to, say, some known solution, some measured data, 00:53:21.520 --> 00:53:23.350 align:middle line:90% at only a discrete set of times. 00:53:23.350 --> 00:53:26.320 align:middle line:84% So now G is a sum, right, so one way 00:53:26.320 --> 00:53:29.380 align:middle line:84% to do it would be to compute u of T, lambda of T, 00:53:29.380 --> 00:53:32.260 align:middle line:90% and then just do the sum. 00:53:32.260 --> 00:53:36.010 align:middle line:84% But is there a way to combine that into a single ODE 00:53:36.010 --> 00:53:39.160 align:middle line:84% if now capital G is not an integral, 00:53:39.160 --> 00:53:43.480 align:middle line:84% it's a discrete summation over a set of times? 00:53:43.480 --> 00:53:45.880 align:middle line:84% FRANK SCHAFER: Yeah, basically, you can write it-- 00:53:45.880 --> 00:53:48.963 align:middle line:84% go to this kind of discrete formulation. 00:53:48.963 --> 00:53:50.380 align:middle line:84% Probably the easiest way, I think, 00:53:50.380 --> 00:53:54.190 align:middle line:84% to think about it is that these describe 00:53:54.190 --> 00:53:56.170 align:middle line:84% the kind of continuous changes that you want 00:53:56.170 --> 00:54:01.630 align:middle line:84% to have in the cost function, and so 00:54:01.630 --> 00:54:04.840 align:middle line:84% what you will have if you just have your cost 00:54:04.840 --> 00:54:08.350 align:middle line:84% function compared at discrete time points you-- 00:54:08.350 --> 00:54:10.300 align:middle line:84% right, you basically have your solution. 00:54:10.300 --> 00:54:15.950 align:middle line:84% And then at some point, you compare your cost, 00:54:15.950 --> 00:54:19.600 align:middle line:84% and so what will happen is that you just solve-- 00:54:19.600 --> 00:54:22.780 align:middle line:84% you will have a backward solution 00:54:22.780 --> 00:54:28.340 align:middle line:84% starting from here for the adjoint and for u. 00:54:28.340 --> 00:54:32.680 align:middle line:84% And then at these time points where you have your comparison, 00:54:32.680 --> 00:54:36.590 align:middle line:84% you basically have to-- you can update your state 00:54:36.590 --> 00:54:40.910 align:middle line:84% by the change that comes in just from those nodes, basically. 00:54:40.910 --> 00:54:45.510 align:middle line:84% So we'll just have, like, partial derivatives 00:54:45.510 --> 00:54:49.570 align:middle line:84% added at those time points to your adjoint state. 00:54:49.570 --> 00:54:51.558 align:middle line:84% STEVEN JOHNSON: Yeah, so, I mean, ODE solvers, 00:54:51.558 --> 00:54:53.350 align:middle line:84% you can tell them to stop at certain times, 00:54:53.350 --> 00:54:54.808 align:middle line:84% so as you go backwards in time, you 00:54:54.808 --> 00:54:56.523 align:middle line:84% can just have it stop at that time 00:54:56.523 --> 00:54:58.690 align:middle line:84% and then update something and then stop at that time 00:54:58.690 --> 00:54:59.560 align:middle line:90% and add-- 00:54:59.560 --> 00:55:01.660 align:middle line:84% at the next time and add something to 00:55:01.660 --> 00:55:03.520 align:middle line:84% accumulate your sum as you go along. 00:55:03.520 --> 00:55:06.190 align:middle line:84% But mathematically, you could basically 00:55:06.190 --> 00:55:09.260 align:middle line:84% think of it as jump conditions that occur at those times, 00:55:09.260 --> 00:55:14.530 align:middle line:84% and so there's also ways to include those in differential 00:55:14.530 --> 00:55:17.050 align:middle line:84% equation solvers, where, oh, every once in a while, 00:55:17.050 --> 00:55:19.630 align:middle line:90% these times, something jumps. 00:55:19.630 --> 00:55:21.470 align:middle line:84% There's a discontinuity in the solution, 00:55:21.470 --> 00:55:24.978 align:middle line:84% and then in between, it's not changing at all, in this case. 00:55:24.978 --> 00:55:26.808 align:middle line:90% FRANK SCHAFER: Yup. 00:55:26.808 --> 00:55:28.350 align:middle line:84% STEVEN JOHNSON: Probably, it's easier 00:55:28.350 --> 00:55:32.527 align:middle line:84% just to say let me integrate it up to this time, 00:55:32.527 --> 00:55:34.110 align:middle line:84% get that solution up to the next time, 00:55:34.110 --> 00:55:36.240 align:middle line:84% and just have a callback at those discrete times 00:55:36.240 --> 00:55:37.560 align:middle line:90% and just accumulate the sum. 00:55:37.560 --> 00:55:39.907 align:middle line:90% 00:55:39.907 --> 00:55:41.907 align:middle line:84% AUDIENCE: One more question about that, so maybe 00:55:41.907 --> 00:55:44.590 align:middle line:84% cycling back to the idea of matrix calculus, so here, 00:55:44.590 --> 00:55:46.360 align:middle line:90% everything is still-- 00:55:46.360 --> 00:55:48.850 align:middle line:84% basically, if you want to generalize it, 00:55:48.850 --> 00:55:50.710 align:middle line:84% say you have a matrix-valued ODE, 00:55:50.710 --> 00:55:55.404 align:middle line:84% then your lambdas become some matrix-valued thing. 00:55:55.404 --> 00:55:58.446 align:middle line:84% So they live in the dual state to whatever you 00:55:58.446 --> 00:56:01.390 align:middle line:84% want to take the derivative of, and as long as it sort of makes 00:56:01.390 --> 00:56:06.420 align:middle line:84% sense, the ODEs that come out at the other end, that will 00:56:06.420 --> 00:56:07.920 align:middle line:90% give you the right sensitivity? 00:56:07.920 --> 00:56:08.590 align:middle line:90% Is that-- 00:56:08.590 --> 00:56:09.465 align:middle line:90% STEVEN JOHNSON: Yeah. 00:56:09.465 --> 00:56:11.690 align:middle line:84% Yeah, so if you have a more general space-- 00:56:11.690 --> 00:56:14.710 align:middle line:84% so u, u lives in some arbitrary vector space, 00:56:14.710 --> 00:56:17.920 align:middle line:84% right, then you have to replace the lambda transpose there 00:56:17.920 --> 00:56:20.230 align:middle line:90% with an inner product, right? 00:56:20.230 --> 00:56:27.190 align:middle line:84% And then this fu transpose is some more general adjoint 00:56:27.190 --> 00:56:31.630 align:middle line:84% operator in that Hilbert space, right, so all of this 00:56:31.630 --> 00:56:32.180 align:middle line:90% still works. 00:56:32.180 --> 00:56:34.097 align:middle line:84% So you can think of this just-- it's literally 00:56:34.097 --> 00:56:35.140 align:middle line:90% the same derivation. 00:56:35.140 --> 00:56:36.730 align:middle line:84% You can think of it as just a change. 00:56:36.730 --> 00:56:39.220 align:middle line:84% It's just, lambda transpose is now 00:56:39.220 --> 00:56:42.430 align:middle line:90% notation for that inner product. 00:56:42.430 --> 00:56:43.900 align:middle line:84% It's just-- you have to understand 00:56:43.900 --> 00:56:46.030 align:middle line:84% what the notation means in a more general sense, 00:56:46.030 --> 00:56:49.547 align:middle line:84% but it's literally the same, same thing. 00:56:49.547 --> 00:56:51.380 align:middle line:84% I don't know, have you done anything where-- 00:56:51.380 --> 00:56:53.250 align:middle line:84% I guess you're doing quantum systems, 00:56:53.250 --> 00:56:56.837 align:middle line:84% so probably, your states live in some more-- 00:56:56.837 --> 00:56:58.670 align:middle line:84% FRANK SCHAFER: Density matrices, yeah, sure. 00:56:58.670 --> 00:57:00.200 align:middle line:84% STEVEN JOHNSON: --live in more general Hilbert spaces 00:57:00.200 --> 00:57:00.980 align:middle line:90% all the time. 00:57:00.980 --> 00:57:03.677 align:middle line:90% FRANK SCHAFER: Yeah. 00:57:03.677 --> 00:57:05.760 align:middle line:84% Yeah, and numerically, you can also then vectorize 00:57:05.760 --> 00:57:06.843 align:middle line:90% the density matrix, right? 00:57:06.843 --> 00:57:09.580 align:middle line:90% And they have-- yeah. 00:57:09.580 --> 00:57:11.790 align:middle line:84% Yeah, rewrite, basically, also the definitions 00:57:11.790 --> 00:57:14.220 align:middle line:84% of f to deal with vectors depending 00:57:14.220 --> 00:57:16.690 align:middle line:84% on your implementation, but, yeah, right. 00:57:16.690 --> 00:57:18.940 align:middle line:84% STEVEN JOHNSON: Yeah, so we talked about that as well. 00:57:18.940 --> 00:57:21.510 align:middle line:84% So one way of dealing with matrices and other things 00:57:21.510 --> 00:57:23.700 align:middle line:84% is just to turn them into column vectors, right, 00:57:23.700 --> 00:57:28.200 align:middle line:84% but if it's more natural in your system space 00:57:28.200 --> 00:57:30.510 align:middle line:84% to talk about matrices, then you'd 00:57:30.510 --> 00:57:33.180 align:middle line:84% really rather deal with that Hilbert space directly. 00:57:33.180 --> 00:57:34.055 align:middle line:90% FRANK SCHAFER: Right. 00:57:34.055 --> 00:57:37.290 align:middle line:84% STEVEN JOHNSON: And if these are PDEs, so u is now not-- 00:57:37.290 --> 00:57:39.337 align:middle line:84% is like a function of space, right? 00:57:39.337 --> 00:57:40.920 align:middle line:84% Of course, on a computer, you probably 00:57:40.920 --> 00:57:43.770 align:middle line:84% discretized it, but it's nice conceptually 00:57:43.770 --> 00:57:45.540 align:middle line:84% to think about this as a function of space 00:57:45.540 --> 00:57:47.100 align:middle line:90% and this inner-- 00:57:47.100 --> 00:57:49.770 align:middle line:84% lambda is also a function of space, for example, 00:57:49.770 --> 00:57:51.510 align:middle line:84% and then this inner product is some kind 00:57:51.510 --> 00:57:53.470 align:middle line:90% of integral over a space. 00:57:53.470 --> 00:57:56.970 align:middle line:84% So it is really powerful to be able to not 00:57:56.970 --> 00:57:59.097 align:middle line:84% have to turn everything into a column vector 00:57:59.097 --> 00:58:00.555 align:middle line:84% when you want to do linear algebra. 00:58:00.555 --> 00:58:06.490 align:middle line:90% 00:58:06.490 --> 00:58:10.330 align:middle line:84% Got a five-minute break, so we'll come back at 12:10. 00:58:10.330 --> 00:58:13.460 align:middle line:84% And I could go then, or if you want to show some more 00:58:13.460 --> 00:58:15.080 align:middle line:90% examples, you can do that. 00:58:15.080 --> 00:58:16.630 align:middle line:90% But-- 00:58:16.630 --> 00:58:20.000 align:middle line:90%