# Identifying an unknown P.D.E. from solution data

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?
• You can look at model recovery literature. Giang Tran from UWaterloo works in that field, I have seen some of her talks exactly on this topic. Here is her webpage: uwaterloo.ca/scholar/g6tran/publications Commented Aug 15, 2021 at 1:52
• – Paul
Commented Aug 15, 2021 at 2:58
• Thanks for the comments. I realized my question was a bit vague, so I added a bit more detail that may help find something a bit closer. Commented Aug 15, 2021 at 3:48
• advances.sciencemag.org/content/3/4/e1602614 Might be useful
– NNN
Commented Aug 15, 2021 at 3:56
• I would try to extract a few spatial and temporal derivatives from the solution - $p_x$, $p_{xx}$, $p_{t}$, $p_{tt}$ etc. and use statistical methods (multidimensional regression) to look for relations between those. Commented Aug 16, 2021 at 3:29