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Machine Learning: Deep Neural Architectures and Foundation ModelsReasoning under Uncertainty: Relational Probabilistic ModelsComputer Vision: Diffusion Models for Vision
Missing Physics Discovery through Fully Differentiable Finite Element-Based Machine Learning
🤖 AI Summary
Many partial differential equation (PDE) models in science and engineering suffer from inaccurate or non-generalizable predictions due to unknown constitutive relations—e.g., nonlinear stress–strain laws or temperature-dependent thermal conductivity. To address this, we propose a fully differentiable finite element machine learning framework that embeds trainable physical operators into a general-purpose finite element method (FEM) solver, enabling end-to-end gradient propagation while preserving variational consistency. Our approach employs an encode-process-decode neural network architecture that directly models the unknown physical mapping on FEM degrees of freedom and jointly optimizes the neural operator with the FEM solver. Experiments demonstrate high-fidelity inversion of nonlinear constitutive laws from sparse experimental data and successful transfer to unseen geometries and boundary conditions in novel mechanical and thermal scenarios. The framework significantly enhances PDE models’ physical interpretability, predictive accuracy, and cross-scenario generalization capability.
Technology Category
Application Category
📝 Abstract
Although many problems in science and engineering are modelled by well-established PDEs, they often involve unknown or incomplete relationships, such as material constitutive laws or thermal response, that limit accuracy and generality. Existing surrogate-modelling approaches directly approximate PDE solutions but remain tied to a specific geometry, boundary conditions, and set of physical constraints. To address these limitations, we introduce a fully differentiable finite element-based machine learning (FEBML) framework that embeds trainable operators for unknown physics within a state-of-the-art, general FEM solver, enabling true end-to-end differentiation. At its core, FEBML represents each unknown operator as an encode-process-decode pipeline over finite-element degrees of freedom: field values are projected to nodal coefficients, transformed by a neural network, and then lifted back to a continuous FE function, ensuring the learned physics respects the variational structure. We demonstrate its versatility by recovering nonlinear stress-strain laws from laboratory tests, applying the learned model to a new mechanical scenario without retraining, and identifying temperature-dependent conductivity in transient heat flow.
Problem
Research questions and friction points this paper is trying to address.
Identify unknown material constitutive laws in PDE models
Overcome geometry-specific limitations in surrogate modeling
Discover missing physics through differentiable FEM-ML integration
Innovation
Methods, ideas, or system contributions that make the work stand out.
Fully differentiable FEM-based machine learning framework
Trainable operators embedded in general FEM solver
Encode-process-decode pipeline for unknown physics
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