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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?
beables
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  • 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: https://uwaterloo.ca/scholar/g6tran/publications – Abdullah Ali Sivas Aug 15 '21 at 01:52
  • Related: https://scicomp.stackexchange.com/questions/28302/inverse-problem-in-linear-ode – Paul Aug 15 '21 at 02:58
  • 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. – beables Aug 15 '21 at 03:48
  • https://advances.sciencemag.org/content/3/4/e1602614 Might be useful – NNN Aug 15 '21 at 03:56
  • Thanks Nachiket, that does look very useful. They have code at https://github.com/snagcliffs/PDE-FIND/ with examples that look like they do exactly this. – beables Aug 15 '21 at 04:14
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    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. – Maxim Umansky Aug 16 '21 at 03:29
  • Karniadakis at Brown University has been publishing a ton in this area in the past 3-4 years. Look up "physics-informed neural networks". – coolguy1000000 Aug 16 '21 at 12:49

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