Missing physics discovery through fully differentiable finite element-based machine learning

Abstract 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, sometimes with added physical constraints, but they 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 (the system’s known physics) 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 when learning constitutive laws. Because the operator, unlike the PDE model, 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 and enable learning over complex geometries and meshes. We demonstrate FEML across solid mechanics and thermal transport. From synthetic experiments we progressively discover an elastoplastic law-the nonlinear elastic response, then the plastic hardening law-and compose the two operators into a foundation constitutive model that transfers zero-shot to a three-dimensional torsion problem. Moving to real data, we learn coupled plastic-hardening and ductile-damage laws directly 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.