I have a black-box simulation that produces the time evolution of a probability density function p(x, t) in 1 dimension from arbitrary initial conditions p(x, 0). The underlying simulation occurs on a discrete lattice of sites x=0, 1, 2, ... but when p(x, t) is a sufficiently smooth function of x, we expect that the behavior is approximately captured by an unknown P.D.E. (Additionally, even with non-smooth initial conditions solutions become smoother and smoother over time.)
The system is local, translationally-invariant and time-independent. However, the P.D.E. is almost certainly non-linear. What is the best strategy to identify the unknown P.D.E.? The simulation can be run on many different initial conditions at low cost.
A few sub-questions:
- is there a way to determine even just the order of a PDE (i.e. the maximum number of derivatives needed to describe the solutions, in space and in time) from solution data?
- are there ways to produce useful initial conditions to run through the simulation?