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.

翻译：通过偏微分方程建模复杂物理系统是科学与工程众多学科的核心。然而在大多数实际应用中，未知或不完整的关系（如本构关系或热力学定律）限制了对目标物理现象的描述。现有代理建模方法旨在通过直接从数据中学习偏微分方程解（有时加入已知物理约束）来弥补这一缺陷。但这些方法仅针对特定系统配置（如几何形状、边界条件或离散化方案）设计，并未直接学习缺失的物理规律，而仅学习偏微分方程解。本文提出FEML——一个端到端可微框架，将系统的偏微分方程建模（已知物理）与表征缺失物理的算子机器学习建模相结合。通过将偏微分方程求解器嵌入训练过程，FEML能够从偏微分方程解中学习此类算子，即使在算子输出无法直接测量的情况下（如学习本构模型时的应力）。FEML将配置依赖的偏微分方程建模与跨具有相同隐藏物理规律系统共享的配置无关算子解耦，实现了复杂物理系统的零样本泛化，并支持领域专家进行下游研究。本框架采用结构保持算子网络建模算子，在离散层面保持关键连续性质，支持复杂几何与网格的学习，并能泛化至不同离散化方案（网格分辨率及/或有限元离散方案）。我们通过三个案例展示FEML的通用性：从合成实验室测试中恢复非线性应力-应变定律，在零样本设置中将习得模型应用于全新力学场景而无需重新训练，以及识别瞬态热流中温度相关的导热系数。