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

Modelling complex physical systems through partial differential equations (PDEs) is central to many disciplines in science and engineering. However, in most real applications, missing physics within the PDE model, expressed as 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 knowledge gap by learning the PDE solution directly from data, and in some cases, by also adding known physical constraints. However, these approaches are tailored and tied 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 for learning missing physics that combines the PDE modelling of the system of interest (known physics) with ML modelling of the operator representing the missing physics. By embedding a PDE solver into training, our approach allows one to train such operators directly from the PDE solution, which unlocks the learning of unknown relationships when the operator output cannot be directly measured (e.g., stress signals for learning constitutive models). FEML dissociates the PDE modelling, which is tied to the system configuration considered, from the operator representing the missing physics, which is agnostic to system configurations and common across all physical systems that share the same physical properties (i.e., the hidden physics). Consequently, our framework naturally allows for zero-shot generalisation of complex physical systems that share the same hidden physics. It also enables downstream study of the learned model by domain specialists. Our framework uses structure-preserving operator networks (SPONs) to model the missing physics operator, which allows one to preserve key continuous properties at the discrete level, to learn over complex geometries and meshes, and to achieve zero-shot generalisation across different discretisations (i.e., different mesh resolutions and/or FE discretisations). We showcase our framework and its versatility by recovering nonlinear stress–strain laws from synthetic laboratory tests, applying the learned model to a new mechanical scenario without retraining, and identifying temperature-dependent conductivity in transient heat flow.