Missing Physics Discovery through Fully Differentiable Finite Element-Based Machine Learning

TL;DR

This paper introduces FEML, a differentiable machine learning framework that learns missing physics in PDE models, enabling zero-shot generalization across different systems and discretizations.

Contribution

FEML combines PDE modeling with ML to learn operators representing missing physics, supporting zero-shot generalization and complex geometries.

Findings

Recovered nonlinear stress-strain laws from synthetic data.

Applied learned models to new scenarios without retraining.

Identified temperature-dependent conductivity in heat flow.

Abstract

Modelling complex physical systems through partial differential equations (PDEs) is central to many disciplines in science and engineering. Yet in most real applications, unknown or incomplete relationships such as constitutive or thermal laws, limits the description of the physics of interest. Existing surrogate modelling approaches aim to address this gap by learning the PDE solution directly from data (sometimes adding known physical constraints). However, these approaches are tailored to specific system configurations (e.g., geometries, boundary conditions, or discretisations) and do not directly learn the missing physics, but only the PDE solution. We introduce FEML, an end-to-end differentiable framework that combines PDE modelling of the system (known physics) with ML modelling of the operator representing the missing physics. By embedding a PDE solver into training, FEML can learn such operators from the PDE solution, even when operator outputs cannot be directly measured (e.g., stresses for learning constitutive models). FEML dissociates configuration-dependent PDE modelling from a configuration-agnostic operator shared across systems with the same hidden physics, enabling zero-shot generalisation of complex physical systems and supporting downstream study by domain specialists. Our framework uses structure-preserving operator networks (SPONs) to model the operator, preserving key continuous properties at the discrete level, learning over complex geometries and meshes, and generalising across different discretisations (mesh resolutions and/or FE discretisations). We showcase FEML and its versatility by recovering nonlinear stress-strain laws from synthetic laboratory tests, applying the learned model to a new mechanical scenario without retraining in a neat zero-shot setting, and identifying temperature-dependent conductivity in transient heat flow.