Hi, I am new to the SciML ecosystem, and I apologise if my question is too basic. I have a 1D differential equation system, dx/dt = f(x; p), where the parameter p is a function of the state variable. Given data for the state variable, can I use a Universal Differential Equation to estimate the function p(x) with uncertainty quantification? Could I please receive some guidance on the best approaches to do this? Thanks in advance.
Hello Jack and welcome to the forum 
You say that the parameter p is a function of the state x, p(x), this is slightly non-standard terminology that may be foreign to some, so I’m asking a few questions to clarify things.
- Would it be fair to say that you are trying to estimate f(x) and that you know some aspects of f but not others, e.g., f(x) = g(x) + p(x) where g is known and p is unknown?
- What do you know about p(x)? Do you know some functional form, but are missing the exact values of (static) parameters \theta, e.g., p(x, \theta)?, Or do you know nothing at all?
Hi @baggepinnen, thanks for your reply and apologies for any ambiguity. Here is a clearer plain text version of the problem:
We have a state function f that includes a parameter which is a function of the state variable. For instance, in an RC circuit, the resistance R depends on the temperature T. The system dynamics are described by the following differential equations:
For temperature: dT/dt = -h * (T - T0) + sin(wt) * V * R(T)
For voltage: dV/dt = -[V + sin(wt) * R(T)] / [R(T) * C]
Initial conditions: T(0)=T0, and V(0)=0
In these equations, T0, h, w, and C are known scalar constants. We have data for both state variables (temperature and voltage) collected from multiple experiments with different initial conditions. The goal is to estimate the function R(T) from this data and to determine uncertainty bounds for the estimate.