Equivalent of Neural ODE for discrete time state space models

Dear forum members,

I have read recently the interesting example on how to train neural ODE to identify dynamical system :

https://mathworks.com/help/deeplearning/ug/dynamical-system-modeling-using-neural-ode.html

This example talks about continuous time models and I was wondering if there was any equivalent tutorial related to discrete time models ?

I know it is possible to create one with the network function, but I would like to implement my own training loop.

Thanks in advance for any help you can provide !

Answer 1

[Arkadiy Turevskiy]

We added idNeuralStateSpace object that support both continuous and discrete time model. Maybe this could be useful. It was created to simplify code you have to write, so it would not allow you to write your own training loop though.

[M.]

Thank you for the answer.

It is not exactly what I am looking for but maybe you could help me to find a solution.

I have designed a discrete-time state space model using neural network implemented thanks to the network function.

It has the fom:

where f is a one-hidden-layer neural network.

For some reason, I would like to be able to evaluate f with two different sets of inputs such that:

The layer f is applied twice but with different inputs.

Is this somehow possible ?

I hope this makes sense !

Thanks again.

[Ben]

Hi M.

I'm not sure if this is possible with the shallow network functions but it can be done with the dlnetwork and custom training loops since these we allow you to write your own model function that re-uses the same network on 2 different inputs. Here's some example code with dummy data - in practice you may need to tweak the training and network hyperparameters to get optimal performance.

% share a neural net across multiple calls

% create some fake data

% predict x(t+1) = F(x(1,t),u(1,t)) + F(x(2,t),u(2,t)) for some unknown F

numSteps = 100;

t = linspace(0,2*pi,numSteps);

F = @(x,u) sqrt(x+u+1);

x = [0;1];

u = [cos(t);sin(t)];

for i = 2:numSteps

x(:,i) = F(x(1,i-1),u(1,i-1)) + F(x(2,i-1),u(2,i-1));

end

% create a network to model F

% it needs to have two inputs, for x and u.

hiddenSize = 5000;

inputSize = 1;

outputSize = 2;

layers = [

featureInputLayer(inputSize,Name="x")

concatenationLayer(1,2,Name="concat");

fullyConnectedLayer(hiddenSize)

reluLayer

fullyConnectedLayer(outputSize)];

net = dlnetwork(layers,Initialize=false);

net = addLayers(net,featureInputLayer(1,Name="u"));

net = connectLayers(net,"u","concat/in2");

net = initialize(net);

% train with custom training loop

numEpochs = 1000;

vel = [];

x = dlarray(x,"CB");

u = dlarray(u,"CB");

learnRate = 0.1;

for epoch = 1:numEpochs

[loss,gradient] = dlfeval(@modelLoss,x,u,net);

lossValue = extractdata(loss);

fprintf("Epoch: %d, Loss %.4f\n", epoch, lossValue);

[net,vel] = sgdmupdate(net,gradient,vel,learnRate);

end

function [loss,gradient] = modelLoss(x,u,net)

% predict x(:,2:end) from x(:,1:end-1) and u(:,1:end-1)

xtarget = x(:,2:end);

xpred = model(x(:,1:end-1),u(:,1:end-1),net);

loss = mse(xtarget,xpred);

gradient = dlgradient(loss,net.Learnables);

end

function xpred = model(x,u,net)

% model xpred = x(t+1) = f(x(1,t),u(1,t)) + f(x(2,t),u(2,t)) where f is a neural net.

xpred = forward(net,x(1,:),u(1,:)) + forward(net,x(2,:),u(2,:));

end

Hope that helps.

[M.]

Thank you very much.

It is very close to what I am looking for.

The last thing is that I would like is to explicitely write the network function as in the continuous time example inside odeModel:

y = tanh(theta.fc1.Weights*y + theta.fc1.Bias);
y = theta.fc2.Weights*y + theta.fc2.Bias;

I tried for example to replace your example "xpred = forward(net,x(1,:),u(1,:))" with the above line replacing theta by net.Learnables.Value but when I try this with your code I get the following error :

One input argument can have dimension labels only when the other input argument is an
unformatted scalar. Use .* for element-wise multiplication.

Any idea why this is not working ?

Thanks again.