Matlab code of Neural delay differential equation NDDE

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Muhammad 2024-1-2

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https://ww2.mathworks.cn/matlabcentral/answers/2065831-matlab-code-of-neural-delay-differential-equation-ndde

评论： Muhammad 2024-1-5

I have written a code of NDDE, but it is not correct and I am not able to simulate this code, because I am getting errors from it

my code

%% Dynamical System Modeling Using Neural ODE

% Parameters

par = [4; 2; 9.65];

tau = 1;

tau_max = 1.5;

num_trajectories = 5;

T = 30;

% Generating 100 trajectories

colors = jet(num_trajectories);

figure; % Create a single figure

for i = 0:(num_trajectories - 1)

c = 0.5 + i/99;

phi = @(t) c * ones(size(t));

g = @(t, y, Z) par(1) * Z / (1 + Z^par(3)) - par(2) * y;

% Solve the DDE using dde23 for baseline data

sol = dde23(@(t, y, Z) g(t, y, Z), tau, phi, [0 T]);

% Plot the current trajectory on the same figure

plot(sol.x, sol.y, 'Color', colors(i+1, :));

hold on;

end

title('Mackey-Glass Equation Trajectories');

xlabel('Time');

ylabel('x(t)');

hold off

numTimeSteps = 1000;

t = linspace(0, T, numTimeSteps);

X=deval(sol,t);

X_delay=interp1(t,X,t-tau);

xTrain = X(1,:)'; % Transpose to match your shape

% Plot x vs. delayed term

figure;

plot(X(1, :), X_delay);

xlabel('x(t)');

ylabel('x(t-\tau)');

title('x vs. delayed term');

grid on;

neuralOdeTimesteps = 10;

dt = t(2);

timesteps = (0:neuralOdeTimesteps)*dt;

nd=1;

tau = dlarray(tau_max*rand(1,nd));

nx = 1;

hiddenSize = 5;

% output_size = 1;

NDDE = struct;

NDDE.fc1 = struct;

sz = [hiddenSize nx*(1+nd)];

NDDE.fc1.Weights = initializeGlorot(sz);

NDDE.fc1.Bias = initializeZeros([sz(1) 1]);

NDDE.fc2 = struct;

sz = [hiddenSize hiddenSize];

NDDE.fc2.Weights = initializeGlorot(sz);

NDDE.fc2.Bias = initializeZeros([sz(1) 1]);

NDDE.fc3 = struct;

sz = [nx hiddenSize];

NDDE.fc3.Weights = initializeGlorot(sz);

tau = min(max(1e-5,tau),tau_max-1e-5);

gradDecay = 0.9;

sqGradDecay = 0.999;

learnRate = 0.002;

numIter = 1200;

miniBatchSize = 200;

plotFrequency = 10;

averageGrad_p = [];

averageSqGrad_p = [];

monitor = trainingProgressMonitor(Metrics="Loss",Info=["Iteration","LearnRate"],XLabel="Iteration");

Error using matlab.internal.lang.capability.Capability.require
This functionality is not available on remote platforms.

Error in matlab.ui.internal.uifigureImpl (line 32)
Capability.require(Capability.WebWindow);

Error in uifigure (line 34)
window = matlab.ui.internal.uifigureImpl(varargin{:});

Error in deepmonitor.internal.DLTMonitorView/createGUIComponents (line 159)
this.Figure = uifigure("Tag", "DEEPMONITOR_UIFIGURE");

Error in deepmonitor.internal.DLTMonitorView (line 116)
this.createGUIComponents();

Error in deepmonitor.internal.DLTMonitorFactory/createStandaloneView (line 8)
view = deepmonitor.internal.DLTMonitorView(model, this);

Error in deep.TrainingProgressMonitor/set.Visible (line 175)
this.View = this.Factory.createStandaloneView(this.Model);

Error in trainingProgressMonitor (line 140)
monitor.Visible = args.Visible;

numTrainingTimesteps = numTimeSteps;

trainingTimesteps = 1:numTrainingTimesteps;

plottingTimesteps = 2:numTimeSteps;

iteration = 0;

while iteration < numIter && ~monitor.Stop

iteration = iteration + 1;

% Create batch

[X, targets] = createMiniBatch(numTrainingTimesteps, neuralOdeTimesteps, miniBatchSize, xTrain);

% Evaluate network and compute loss and gradients

[loss,grad_par,grad_tau] = dlfeval(@modelLoss,timesteps,X,par,targets);

% Update network

[par,averageGrad_p,averageSqGrad_p] = adamupdate(par,grad_par,averageGrad_p,averageSqGrad_p,iteration, learnRate,gradDecay,sqGradDecay);

[tau,averageGrad_t,averageSqGrad_t] = adamupdate(tau,grad_tau,averageGrad_t,averageSqGrad_t,iteration, learnRate,gradDecay,sqGradDecay);

% Plot loss

recordMetrics(monitor,iteration,Loss=loss);

% Plot predicted vs. real dynamics

if mod(iteration,plotFrequency) == 0 || iteration == 1

% Use ode45 to compute the solution

y=dde23(fun,par,delay,hist,time);

plot(xTrain(1,plottingTimesteps),xTrain(2,plottingTimesteps),"r--")

hold on

plot(y(1,:),y(2,:),"b-")

hold off

xlabel("x(1)")

ylabel("x(2)")

title("Predicted vs. Real Dynamics")

legend("Training Ground Truth", "Predicted")

drawnow

end

updateInfo(monitor,Iteration=iteration,LearnRate=learnRate);

monitor.Progress = 100*iteration/numIter;

end

function X = model(fun,par,delay,hist,time)

X = dde23(fun,par,delay,hist,time);

end

function dx=fun(t,y,Z,par)

dx = par.fc3.Weights*tanh(par.fc2.Weights*tanh(par.fc1.Weights*[y;Z]+par.fc1.Bias)+par.fc2.Bias);

end

function [loss,grad_par,grad_tau] = modelLoss(tspan,X0,par,targets)

% % Compute predictions.

X = model(fun,par,delay,hist,time);

% Compute L1 loss.

loss = l1loss(X,targets);

% Compute gradients.

grad_par = dlgradient(loss,par);

grad_tau = dlgradient(loss,tau);

end

function [x0, targets] = createMiniBatch(numTimesteps,numTimesPerObs,miniBatchSize,X)

% Create batches of trajectories.

s = randperm(numTimesteps - numTimesPerObs, miniBatchSize);

x0 = dlarray(X(s,:));

targets = zeros([size(X,1) miniBatchSize numTimesPerObs]);

for i = 1:miniBatchSize

targets( 1:numTimesPerObs,i) = X( s(i) + 1:(s(i) + numTimesPerObs),:);

end

function bias = initializeZeros(sz)

bias = zeros(sz,'single');

bias = dlarray(bias);

end

function weights = initializeGlorot(sz)

Z = 2*rand(sz,'single') - 1;

bound = sqrt(6 / (sz(2)+ sz(1)));

weights = bound * Z;

weights = dlarray(weights);

end

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Answer 1

Ben 2024-1-5

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2065831-matlab-code-of-neural-delay-differential-equation-ndde#answer_1383841

I notice that the model function uses dde23. Unfortunately dde23 is not supported by dlarray and so you can't use this with automatic differentiation. The dlode45 solver is the only ODE solver we have that supports dlarray and automatic differentiation.

It may be possible to work something out with dlode45 but it could be quite complex. For example, for a delay differential equation with 1 delay

and

for

, it's possible to reduce this to an ordinary differential equation for

by writing

for

. Here's a proof of concept using

.

p = 0.1;

tau = 1;

phi = @(t) 1;

F = @(t,y,z,p) -y + p*z;

% Solve with dde23 for fixed p on [0,tau]

sol = dde23(@(t,y,z) F(t,y,z,p), tau, phi, [0,tau]);

% Construct equivalent ODE on [0,tau]

G = @(t,y,p) F(t,y,phi(t),p);

% Solve for fixed p

odeSol = ode45(@(t,y) G(t,y,p), [0,tau], phi(0));

% Compute gradients with dlode45

% Here I compute gradients of y(tau) with respect to params as a proof of

% concept.

[y,grad] = dlfeval(@solveAndGradient,G,[0,tau],dlarray(phi(0)),dlarray(p));

function [y,grad] = solveAndGradient(G,tspan,y0,params)

y = dlode45(G,tspan,y0,params,DataFormat="CB");

grad = dlgradient(y,params);

end

To extend beyond

would get a little bit complex - potentially you could interpolate the solution on

to define

for

using interp1 (since it is supported by dlarray and automatic differentiation) and use this to solve for

for

using dlode45. However I expect this could get quite difficult to implement.

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Muhammad 2024-1-5

Thank you for nice explaination and I got a point where I am doing mistake

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Matlab code of Neural delay differential equation NDDE

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