Neural ODE for dynamic systems with input signals

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Hi! Community!
Mathworks provided a nice example here for modeling dynamic systems through neural ODE.
Is it possible to consider input signals in training? That is, to define the differential equation to be:
where is the input signal.
However, dlode45 will not allow the ODE function to be with more than three inputs.
So is there any other possible approach to incorporate the input signal?
Thanks a lot!

Accepted Answer

Ben
Ben on 25 Jan 2022
Hi Bowei,
You should be able to create a new ODE function that has only three inputs as required. Let me show a few cases.
Case 1 -
In this case you can define . Assuming you have f as a function handle you can define g in code with:
g = @(t,x,theta) f(t,x,theta,e(t))
Then solve using g in dlode45.
Case 2 -
This is a special case of case 1:
g = @(t,x,theta) f(t,x,theta) + e(t)
Call dlode45 with g.
Case 3 - for
In this case you have an extra hyperparameter i which you just have to select a specific value for. For example let and . You could write this in code as:
e = @(t,i) cos(i*t);
f = @(t,x,A,i) A*x + e(t,i);
x0 = dlarray(randn());
tspan = [0,1];
A = dlarray(randn());
i = 3;
x = dlode45(@(t,x,A) f(t,x,A,i), tspan, x0, A, DataFormat="CB");
Note that in this case you can loop over the values you want for i.
Hope that helps,
Ben

More Answers (1)

David Willingham
David Willingham on 24 Jan 2022
Hi Bowei,
Thanks for the feedback on our neural ode example! For your request, can you elloborate on what type of signal e(t) might be and what use cases you're looking to apply neural ode's to?
David
  1 Comment
Bowei Li
Bowei Li on 25 Jan 2022
Edited: Bowei Li on 25 Jan 2022
Thanks David!
I was thinking about training neural ode for predicting states of dynamic systems given some arbitrary input signals, such as:
where is the system state vector, a function of time t; the input signal is also a function of time t.
For example, can be like or or a random process .
The example here corresponding to the case of and training a neural ode for a set of initial conditions.
Is it possible to train a neural ode for a fixed initial condition, but for a set of input signals .

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