From 76fb353fa99c310f715db7ed8bc5c95e74af6a10 Mon Sep 17 00:00:00 2001 From: Ben Wells-Day Date: Mon, 17 Mar 2025 10:15:16 +0000 Subject: [PATCH] add markdown version of example --- universal-differential-equations/example.md | 351 ++++++++++++++++++ .../example_media/figure_0.png | Bin 0 -> 35640 bytes .../example_media/figure_1.png | Bin 0 -> 93060 bytes .../example_media/figure_2.png | Bin 0 -> 87821 bytes .../example_media/figure_3.png | Bin 0 -> 54971 bytes 5 files changed, 351 insertions(+) create mode 100644 universal-differential-equations/example.md create mode 100644 universal-differential-equations/example_media/figure_0.png create mode 100644 universal-differential-equations/example_media/figure_1.png create mode 100644 universal-differential-equations/example_media/figure_2.png create mode 100644 universal-differential-equations/example_media/figure_3.png diff --git a/universal-differential-equations/example.md b/universal-differential-equations/example.md new file mode 100644 index 0000000..ea656b8 --- /dev/null +++ b/universal-differential-equations/example.md @@ -0,0 +1,351 @@ + +# Discovering Differential Equations with Neural ODEs + +This example demonstrates how to use neural ODEs to discover underlying differential equations following the Universal Differential Equation \[1\] method. + + +The method is used for time\-series data originating from ordinary differential equations (ODEs), in particular when not all of the terms of the ODE are known. For example, let $\mathbf{X}(t)\in {\mathbb{R}}^n$ be a time\-series for $t\in [t_0 ,t_1 ]$ and suppose that we know $\mathbf{X}(t)$ satisfies an ODE of the form: + + $$ \frac{\mathrm{d}\mathbf{X}}{\mathrm{d}t}(t)=\mathbf{f}(t,\mathbf{X}(t)),~~t\in [t_0 ,t_1 ],~~\mathbf{f}:[t_0 ,t_1 ]\times {\mathbb{R}}^n \to {\mathbb{R}}^n ,~~(1) $$ + +with initial condition $\mathbf{X}(t_0 )={\mathbf{X}}_0 .$ + + +In the case that $\mathbf{f}$ is only partially known, for example $\mathbf{f}(t,\mathbf{X})=\mathbf{g}(t,\mathbf{X})+\mathbf{h}(t,\mathbf{X})$ where $\mathbf{g}:[t_0 ,t_1 ]\times {\mathbb{R}}^n \to {\mathbb{R}}^n$ is a known function, and $\mathbf{h}:[t_0 ,t_1 ]\times {\mathbb{R}}^n \to {\mathbb{R}}^n$ is unknown, the method proposes to model $\mathbf{h}$ by a universal approximator such as a neural network ${\mathbf{h}}_{\theta } :[t_0 ,t_1 ]\times {\mathbb{R}}^n \to {\mathbb{R}}^n$ with learnable parameters $\theta$. + + +When $\mathbf{h}$ is replaced by a neural network ${\mathbf{h}}_{\theta }$, equation (1) becomes a neural ODE which can be trained via gradient descent. + + +Once ${\mathbf{h}}_{\theta }$ has been trained, symbolic regression methods can be used to find a functional representation of ${\mathbf{h}}_{\theta }$, such as the SINDy method \[2\]. + +# Training data + +This example uses the Lotka\-Volterra equations for demonstration purposes, + + $$ \mathbf{X}(t)=(x(t),y(t)), $$ + + $$ \frac{\mathrm{d}x}{\mathrm{d}t}=x-\alpha xy, $$ + + $$ \frac{\mathrm{d}y}{\mathrm{d}t}=-y+\beta xy, $$ + +where $\alpha ,\beta$ are constants. For the purpose of the example suppose the terms $\alpha xy,\beta xy$ are unknown, and the linear terms are known. In the above notation that is $\mathbf{g}(t,(x,y))=(x,-y)$ are the known terms and $\mathbf{h}(t,(x,y)=(-\alpha xy,\beta xy)$ is unknown. + +```matlab +function dXdt = lotkaVolterraODE(X,nvp) +arguments + X + nvp.Alpha (1,1) double = 0.6 + nvp.Beta (1,1) double = 0.4 +end +x = X(1); +y = X(2); +dxdt = x*(1 - nvp.Alpha*y); +dydt = y*(-1 + nvp.Beta *x); +dXdt = [dxdt; dydt]; +end +``` + +Set $\alpha =0.6$, $\beta =0.4$, $t_0 =0$ and $\mathbf{X}(t_0 )={\mathbf{X}}_0 =(1,1)$. + +```matlab +alpha = 0.6; +beta = 0.4; +X0 = [1;1]; +``` + +Solve the ODE using `ode`. For the purpose of the example, add random noise to the solution, as realistic is often noisy. + +```matlab +function X = noisySolve(F,ts,nvp) +arguments + F (1,1) ode + ts (1,:) double + nvp.NoiseMagnitude (1,1) double = 2e-2 +end +S = solve(F,ts); +X = S.Solution; +% Add noise to the solution, but not the initial condition. +noise = nvp.NoiseMagnitude * randn([size(X,1),size(X,2)-1]); +X(:,2:end) = X(:,2:end) + noise; +end + +F = ode(... + ODEFcn = @(t,X) lotkaVolterraODE(X, Alpha=alpha, Beta=beta),... + InitialValue = X0); + +ts = linspace(0,15,250); +X = noisySolve(F,ts); +scatter(ts,X(1,:),"."); +hold on +scatter(ts,X(2,:),"."); +title("Time-series data $\mathbf{X}(t) = (x(t),y(t))$", Interpreter="latex") +xlabel("$t$",Interpreter="latex") +legend(["$x(t)$","$y(t)$"], Interpreter="latex") +hold off +``` + +
figure_0.png
+ + +Take only the data $\mathbf{X}(t),t\in [0,3]$ as training data. + +```matlab +tsTrain = ts(ts<=3); +XTrain = X(:,ts<=3); +``` +# Design Universal ODE + +Recall $\mathbf{f}(t,\mathbf{X})=\mathbf{g}(t,\mathbf{X})+\mathbf{h}(t,\mathbf{X})$ where $\mathbf{g}(t,\mathbf{X})=\mathbf{g}(t,(x,y))=(x,-y)$ is known, and $\mathbf{h}$ is to be approximated by a neural network ${\mathbf{h}}_{\theta }$. Let ${\mathbf{f}}_{\theta } (t,\mathbf{X})=\mathbf{g}(t,\mathbf{X})+{\mathbf{h}}_{\theta } (t,\mathbf{X})$. By representing $\mathbf{g}$ as a neural network layer it is possible to represent ${\mathbf{f}}_{\theta }$ as a neural network. + +```matlab +gFcn = @(X) [X(1,:); -X(2,:)]; +gLayer = functionLayer(gFcn, Acceleratable=true, Name="g"); +``` + +Define a neural network for ${\mathbf{h}}_{\theta }$. + +```matlab +activationLayer = functionLayer(@softplusActivation,Acceleratable=true); +depth = 3; +hiddenSize = 5; +stateSize = size(X,1); +hLayers = [ + featureInputLayer(stateSize,Name="X") + repmat([fullyConnectedLayer(hiddenSize); activationLayer],[depth-1,1]) + fullyConnectedLayer(stateSize,Name="h")]; +hNet = dlnetwork(hLayers); +``` + +Add the layer representing $\mathbf{g}$, connect the $\mathbf{X}$ input to $\mathbf{g}(\mathbf{X})$. Also add an `additionLayer` to perform the addition in ${\mathbf{f}}_{\theta } =\mathbf{g}+{\mathbf{h}}_{\theta }$ and connect ${\mathbf{h}}_{\theta }$ to this. + +```matlab +fNet = addLayers(hNet, [gLayer; additionLayer(2,Name="add")]); +fNet = connectLayers(fNet,"X","g"); +fNet = connectLayers(fNet,"h","add/in2"); +``` + +Analyse the network. + +```matlab +analyzeNetwork(fNet) +``` + +The `dlnetwork` specified by `fNet` represents the function ${\mathbf{f}}_{\theta } (t,\mathbf{X})$ in the neural ODE + + $$ \frac{\mathrm{d}\mathbf{X}}{\mathrm{d}t}(t)={\mathbf{f}}_{\theta } (t,\mathbf{X}(t)). $$ + +To solve the neural ODE, place `fNet` inside a `neuralODELayer`, and solve for the times `tsTrain`. + +```matlab +neuralODE = [ + featureInputLayer(stateSize,Name="X0") + neuralODELayer(fNet, tsTrain, GradientMode="adjoint")]; +neuralODE = dlnetwork(neuralODE); +``` +# Train the neural ODE + +Set the network to `double` precision using `dlupdate`. + +```matlab +neuralODE = dlupdate(@double, neuralODE); +``` + +Specify `trainingOptions` for ADAM and train. For a small neural ODE, often training on the CPU is faster than the GPU, as there is not sufficient parallelism in the neural ODE to make up for the overhead of sending data to the GPU. + +```matlab +opts = trainingOptions("adam",... + Plots="training-progress",... + MaxEpochs=600,... + ExecutionEnvironment="cpu",... + InputDataFormats="CB",... + TargetDataFormats="CTB"); +neuralODE = trainnet(XTrain(:,1),XTrain(:,2:end),neuralODE,"l2loss",opts); +``` + +```matlabTextOutput + Iteration Epoch TimeElapsed LearnRate TrainingLoss + _________ _____ ___________ _________ ____________ + 1 1 00:00:00 0.001 2.2743 + 50 50 00:00:03 0.001 1.3138 + 100 100 00:00:05 0.001 0.97882 + 150 150 00:00:07 0.001 0.73854 + 200 200 00:00:09 0.001 0.56192 + 250 250 00:00:12 0.001 0.43058 + 300 300 00:00:14 0.001 0.33261 + 350 350 00:00:16 0.001 0.2598 + 400 400 00:00:18 0.001 0.20618 + 450 450 00:00:21 0.001 0.16721 + 500 500 00:00:23 0.001 0.13936 + 550 550 00:00:25 0.001 0.1198 + 600 600 00:00:27 0.001 0.10621 +Training stopped: Max epochs completed +``` + +
figure_1.png
+ + +Next train with L\-BFGS to optimize the training loss further. + +```matlab +opts = trainingOptions("lbfgs",... + MaxIterations = 400,... + Plots="training-progress",... + ExecutionEnvironment="cpu",... + GradientTolerance=1e-8,... + StepTolerance=1e-8,... + InputDataFormats="CB",... + TargetDataFormats="CTB"); +neuralODE = trainnet(XTrain(:,1),XTrain(:,2:end),neuralODE,"l2loss",opts); +``` + +```matlabTextOutput + Iteration TimeElapsed TrainingLoss GradientNorm StepNorm + _________ ___________ ____________ ____________ ________ + 1 00:00:00 0.10012 1.5837 0.072727 + 50 00:00:07 0.001433 0.024051 0.10165 + 100 00:00:14 0.00083738 0.015113 0.071645 + 150 00:00:23 0.00062907 0.013306 0.037026 + 200 00:00:34 0.00058742 0.00016328 0.00036442 + 250 00:01:12 0.00058725 0.000197 6.464e-05 +Training stopped: Stopped manually +``` + +
figure_2.png
+ +# Identify equations for the universal approximator + +Extract ${\mathbf{h}}_{\theta }$ from `neuralODE`. + +```matlab +fNetTrained = neuralODE.Layers(2).Network; +hNetTrained = removeLayers(fNetTrained,["g","add"]); +lrn = hNetTrained.Learnables; +lrn = dlupdate(@dlarray, lrn); +hNetTrained = initialize(hNetTrained); +hNetTrained.Learnables = lrn; +``` + +The SINDy method \[2\] takes a library of basis functions ${\mathbf{e}}_i (t,\mathbf{X}):[t_0 ,t_1 ]\times {\mathbb{R}}^n \to {\mathbb{R}}^n$ for $i=1,2,\ldots,N$ and sample points $(\tau_j ,{\mathbf{X}}_j )\in [t_0 ,t_1 ]\times {\mathbb{R}}^n$. The goal is to identify which terms ${\mathbf{e}}_i$ represent $h_{\theta }$. + + +Let $\mathbf{E}(t,\mathbf{X})=({\mathbf{e}}_1 (t,\mathbf{X}),\ldots,{\mathbf{e}}_N (t,\mathbf{X}))$ denote the matrix formed by concatenating all of the evaluations of the basis functions ${\mathbf{e}}_i$. Identifying terms ${\mathbf{e}}_i$ that make up ${\mathbf{h}}_{\theta }$ can be written as the linear problem ${\mathbf{h}}_{\theta } (t,\mathbf{X})=\mathbf{W}\mathbf{E}(t,\mathbf{X})$ for a matrix $\mathbf{W}$. + + +The SINDy method proposes to use a sparse regression method to solve for $\mathbf{W}$. The sequentially thresholded least squares method \[2\] iteratively solves for $\mathbf{W}$ in the above problem, and zeros out the terms in $\mathbf{W}$ with absolute value below a specified threshold. + + +Use the training data `XTrain` as the sample points ${\mathbf{X}}_j$ and evaluate ${\mathbf{h}}_{\theta } ({\mathbf{X}}_j )$. + +```matlab +Xextra = interp1(tsTrain,XTrain.', linspace(tsTrain(1), tsTrain(2), 100)).'; +XSample = [XTrain,Xextra]; +hEval = predict(hNetTrained,XSample,InputDataFormats="CB"); +``` + +Denote $\mathbf{X}=(x,y)$ and specify the basis functions ${\mathbf{e}}_1 (x,y)=1,{\mathbf{e}}_2 (x,y)=x^2 ,{\mathbf{e}}_3 (x,y)=xy,{\mathbf{e}}_4 (x,y)=y^2$. Note that the linear functions $(x,y)\to x$ and $(x,y)\to y$ are not included as the linear terms in $\mathbf{f}$ are already known. + +```matlab +e1 = @(X) ones(1,size(X,2)); +e2 = @(X) X(1,:).^2; +e3 = @(X) X(1,:).*X(2,:); +e4 = @(X) X(2,:).^2; +E = @(X) [e1(X); e2(X); e3(X); e4(X)]; +``` + +Evaluate the basis functions at the sample points. + +```matlab +EEval = E(XSample); +``` + +Sequentially solve ${\mathbf{h}}_{\theta } =\mathbf{W}\mathbf{E}$ for 10 iterations, and set terms with absolute value less that $0.05$ to $0$. + +```matlab +iters = 10; +threshold = 0.1; +Ws = cell(iters,1); +W = hEval/EEval; +Ws{1} = W; +for iter = 2:iters + belowThreshold = abs(W)figure_3.png + +# Helper Functions +```matlab +function x = softplusActivation(x) +x = max(x,0) + log(1 + exp(-abs(x))); +end +``` + +References + + +\[1\] Christopher Rackauckas, Yingbo Ma, Julius Martensen, Collin Warner, Kirill Zubov, Rohit Superkar, Dominic Skinner, Ali Ramadhan, and Alan Edelman. "Universal Differential Equations for Scientific Machine Learning". Preprint, submitted January 13, 2020. [https://arxiv.org/abs/2001.04385](https://arxiv.org/abs/2001.04385) + + +\[2\] Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. "Discovering governing equations from data by sparse identification of nonlinear dynamical systems". Proceedings of the National Academy of Sciences, 113 (15) 3932\-3937, March 28, 2016. + diff --git a/universal-differential-equations/example_media/figure_0.png b/universal-differential-equations/example_media/figure_0.png new file mode 100644 index 0000000000000000000000000000000000000000..6aca8c703bdc51d2eb94218a4132471d0e1ee916 GIT binary patch literal 35640 zcmb@uc{r5q8$XPu3CX@M84OZm-}jxduN5M)Z;`b`mZmHbW8b%oHEoEbkV=NJi|ooS zOCgf&y>2}{&+~oW_jkOHgdozQWxX4X3Zb!pX&ew()Q2VIUFx*D8ibONB4@9V% zM=w7>9U`~1Ol&A@<9!<*Hr@=g-7dF1AMk_q+{MS%Za-Cs zNT|b!UrLXpYp?5eCJ*-~ z`vP*WPXx_1AES~`{`rO~Nm+RZZozm||Prnxq zeHJv-FyS=uYp^aW{q?fb+=EylG28v6i^-pcY{PDCY}GZ5znWKE6KzTPNUki)Mt%V)#CR5HkBq*lsO-c?~C;MB?uJz&T!T1Qo zC)n*KwrcA#9pYtMt#5q3t8>EWAawcJhr!i_AgXdibJXdZZTi_#f}&`9SY%|RoLism z;jfE+Z%;{Qhm24;{z*3)Xl&7johUUMZb@M$vzU;AodvP*b zu#2XCbJM{O;p&N-Ujy<`_NSHh7mbF*(0zsbUDq2O0zX|jvEhBV>X-42-sNh4jau0Uag-C z>JGl_(aPqnPpHl)tnBj5

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