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 a P.D.E. or possibly a stochastic P.D.E.  (Additionally, non-smooth initial conditions become smoother and smoother over time.)

What is the best strategy to identify the unknown P.D.E.? The simulation can be run on many different initial conditions at relatively low cost.

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?