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

arXiv: https://arxiv.org/abs/2507.15787 · HTML: https://mlsanity.com/paper/2507.15787/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jul 2025.

## Built on by (most cited first, 1 of 1)

- [Integrating Mechanistic and Data-Driven Models for Neurological Disorders through Differentiable Programming](https://mlsanity.com/paper/2606.06094/) (2026)

## Abstract (arXiv)

Modelling physical systems with partial differential equations (PDEs) is central to science and engineering, yet in most real applications the PDE model is incomplete: relationships such as constitutive or thermal laws are unknown. Existing surrogate approaches close this gap by learning the PDE solution from data, but remain tied to a specific configuration (geometry, boundary conditions, discretisation) and recover the solution rather than the missing physics itself. We introduce FEML, an end-to-end differentiable framework that couples the known PDE with a machine-learned operator for the missing physics. Embedding the PDE solver into training lets this operator be learned directly from the PDE solution, even when its own output cannot be measured - for example, stress in constitutive laws. Because the operator is independent of the system configuration, a law learned in one setting transfers zero-shot to new geometries, boundary conditions, and discretisations, and can be inspected by domain specialists. FEML represents the operator with structure-preserving operator networks (SPONs), which retain key continuous properties at the discrete level. We demonstrate FEML across solid mechanics and thermal transport. From synthetic data we progressively discover an elastoplastic law - the nonlinear elastic response, then the plastic hardening law - and compose them into a foundation constitutive model that transfers zero-shot to a 3D torsion problem. Moving to real data, we learn coupled plastic-hardening and ductile-damage laws from a benchmark shear-coupon test, reproducing the measured response, including post-peak softening, to within the experimental scatter. Finally, we recover a temperature-dependent conductivity from transient heat-flow data and apply symbolic regression to the learned operator to extract a closed-form law matching the ground truth.