# Equivalent of Neural ODE for discrete time state space models

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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 !

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### Answers (1)

Arkadiy Turevskiy
on 31 Jan 2023

##### 3 Comments

Ben
on 2 Feb 2023

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.

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