Can anyone point me to some good examples (code, papers, tutorials) of a dynamical physical system that are modelled with UDEs, i.e., ODEs where parts are known and other parts are unknown and modelled by a neural network, i.e. on the form:
\frac{dx}{dt} = f_{known}(t,x(t), u(t),p) + NN(t,x(t),u(t),p),
where x(t) is the state(s) of the system and u(t) is the time-varying exogenous inputs.
The UDEs could, e.g., describe mass or energy balances.
I have looked alot into these two examples:
- Lotka-Volterra UDE (no exogenous inputs, simulated data):
Automatically Discover Missing Physics by Embedding Machine Learning into Differential Equations · Overview of Julia's SciML - Hammerstein system w. exogenous inputs (rhs is pure NN):
Handling Exogenous Input Signals · SciMLSensitivity.jl
But other than that, it is difficult to find good examples to get inspiration from. Maybe I am just looking the wrong places.
Thx! 