Learning functional components of PDEs from data using neural networks

TL;DR

This paper introduces a neural network-based method to recover unknown functions within PDEs from data, enabling better predictions and understanding of complex systems.

Contribution

It demonstrates how neural networks can be embedded into PDEs to accurately recover unknown functions from steady state data, extending existing parameter estimation workflows.

Findings

Neural networks can approximate unknown PDE functions with arbitrary accuracy.

The method successfully recovers interaction kernels and potentials from data.

Performance depends on data quality, sampling, and noise levels.

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.