# Variational system identification of the partial differential equations governing pattern-forming physics: Inference under varying fidelity and noise

## TL;DR

This paper introduces a variational system identification method for PDEs that models pattern formation physics, accounting for data noise and fidelity, to determine the most accurate mathematical description of complex biological and material systems.

## Contribution

It advances PDE system identification by integrating variational frameworks and statistical tests, improving robustness under data noise and varying fidelity conditions.

## Key findings

- Effective identification of PDEs from noisy data
- Robustness to data fidelity variations demonstrated
- Improved model selection accuracy over previous methods

## Abstract

We present a contribution to the field of system identification of partial differential equations (PDEs), with emphasis on discerning between competing mathematical models of pattern-forming physics. The motivation comes from developmental biology, where pattern formation is central to the development of any multicellular organism, and from materials physics, where phase transitions similarly lead to microstructure. In both these fields there is a collection of nonlinear, parabolic PDEs that, over suitable parameter intervals and regimes of physics, can resolve the patterns or microstructures with comparable fidelity. This observation frames the question of which PDE best describes the data at hand. This question is particularly compelling because identification of the closest representation to the true PDE, while constrained by the functional spaces considered relative to the data at hand, immediately delivers insights to the physics underlying the systems. While building on recent work that uses stepwise regression, we present advances that leverage the variational framework and statistical tests. We also address the influences of variable fidelity and noise in the data.