Learning functional components of PDEs from data using neural networks

Authors

Torkel E. Loman, Yurij Salmaniw, Antonio Leon Villares, Jose A. Carrillo, and Ruth E. Baker

Journal

(Submitted)

Abstract

Partial differential equations often contain unknown functions that are difficult or impossible to measure directly, hampering our ability to derive predictions from the model. Workflows for recovering scalar PDE parameters from data are well studied: here we show how similar workflows can be used to recover functions from data. Specifically, we embed neural networks into the PDE and show how, as they are trained on data, they can approximate unknown functions with arbitrary accuracy.

Using nonlocal aggregation-diffusion equations as a case study, we recover interaction kernels and external potentials from steady state data. Specifically, we investigate how a wide range of factors, such as the number of available solutions, their properties, sampling density, and measurement noise, affect our ability to successfully recover functions.

Our approach is advantageous because it can utilise standard parameter-fitting workflows, and in that the trained PDE can be treated as a normal PDE for purposes such as generating system predictions.

Overview

This paper is about the recovery of input parameters, given access to solution profile(s). Rather than learning scalar parameters, we recover entire functional components of a PDE directly from data. That is a much harder problem, because in many applications the unknown part of the model is not just a number but something like an interaction kernel or an external potential. The main idea here is to represent those unknown functions by a simple neural network and train them using the same general fitting framework one would normally use for parameter estimation.

The case study in the paper comes from nonlocal aggregation-diffusion equations, where the goal is to recover the kernel or potential from steady-state observations. A big part of the analysis is understanding when that actually works well: for example, how much the answer depends on the number and quality of the available steady states, how densely they are sampled, and how sensitive the procedure is to noise.

Keywords

machine learning; partial differential equations; parameter identification; neural networks; inverse problems; nonlocal aggregation-diffusion

MSC 2020

68T07; 35Q84; 35Q92