Tackling multiphysics problems via finite element-guided physics-informed operator learning

TL;DR

This work presents a finite element-guided physics-informed operator learning framework for multiphysics problems with coupled partial differential equations (PDEs) on arbitrary domains, and investigates the potential of several neural operator backbones, including Fourier neural operators, deep operator networks, and a newly proposed implicit finite operator learning (iFOL) approach based on conditional neural fields.

Abstract

This work presents a finite element-guided physics-informed operator learning framework for multiphysics problems with coupled partial differential equations (PDEs) on arbitrary domains. Implemented with Folax, a JAX-based operator-learning platform, the proposed framework learns a mapping from the input parameter space to the solution space with a weighted residual formulation based on the finite element method, enabling discretization-independent prediction beyond the training resolution without relying on labaled simulation data. The present framework for multiphysics problems is verified on nonlinear thermo-mechanical problems. Two- and three-dimensional representative volume elements with varying heterogeneous microstructures, and a close-to-reality industrial casting example under varying boundary conditions are investigated as the example problems. We investigate the potential of several neural operator backbones, including Fourier neural operators (FNOs), deep operator networks (DeepONets), and a newly proposed implicit finite operator learning (iFOL) approach based on conditional neural fields. The results demonstrate that FNOs yield highly accurate solution operators on regular domains, where the global topology can be efficiently learned in the spectral domain, and iFOL offers efficient parametric operator learning capabilities for complex and irregular geometries. Furthermore, studies on training strategies, network decomposition, and training sample quality reveal that a monolithic training strategy using a single network is sufficient for accurate predictions, while training sample quality strongly influences performance. Overall, the present approach highlights the potential of physics-informed operator learning with a finite element-based loss as a unified and scalable approach for coupled multiphysics simulations.

Figures (25)

• Figure 1: Overall idea of the present research: addressing multiphysics problems across scales with a novel and scalable physics-informed operator learning framework, where finite element residuals are directly used for parametric training of deep learning models. The final goal would be to enable strong coupling of multiphysics problems between macro and microscale as illustrated on the bottom. As a first step, the present work demonstrates the performance of the proposed framework on typical multiphysics problems on both scales.
• Figure 1: Relative L2 error statistics over 50 samples on four different test cases on different numbers of Fourier modes in FNO.
• Figure 2: Temperature-dependent material properties utilized in this study.
• Figure 2: Relative L2 error statistics over 50 samples on four different test cases on different numbers of FNO layers.
• Figure 3: Schematic of the three operator learning backbones employed in this study: Fourier Neural Operator (FNO), Deep Operator Network (DeepONet), and implicit Finite Operator Learning (iFOL).