A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains

TL;DR

This paper introduces finite operator learning (FOL), a physics-informed, FEM-based framework to learn time-stepping operators for spatiotemporal PDEs on arbitrary domains. By embedding the discretized weak form of the transient heat equation into the loss function and using backward-Euler time integration, FOL trains unsupervised, node-specific neural networks to map the current temperature field ${T}^n$ to the next state ${T}^{n+1}$, while enforcing Dirichlet boundaries as hard constraints. The approach is validated on homogeneous and heterogeneous conductivities, including irregular geometries, and demonstrates high accuracy relative to FEM with substantial inference speedups (over 10× in some cases) after lengthy but once-off training (tens of hours on standard GPUs). The work offers a scalable surrogate for complex, parameterized PDEs, with potential extensions to additional PDEs and higher-order temporal schemes, supported by FEM-based discretization and the operator-learning paradigm. The unsupervised, FEM-grounded training and the ability to handle irregular domains and material heterogeneity are the key strengths and practical contributions for engineering applications.

Abstract

We propose a novel finite element-based physics-informed operator learning framework that allows for predicting spatiotemporal dynamics governed by partial differential equations (PDEs). The proposed framework employs a loss function inspired by the finite element method (FEM) with the implicit Euler time integration scheme. A transient thermal conduction problem is considered to benchmark the performance. The proposed operator learning framework takes a temperature field at the current time step as input and predicts a temperature field at the next time step. The Galerkin discretized weak formulation of the heat equation is employed to incorporate physics into the loss function, which is coined finite operator learning (FOL). Upon training, the networks successfully predict the temperature evolution over time for any initial temperature field at high accuracy compared to the FEM solution. The framework is also confirmed to be applicable to a heterogeneous thermal conductivity and arbitrary geometry. The advantages of FOL can be summarized as follows: First, the training is performed in an unsupervised manner, avoiding the need for a large data set prepared from costly simulations or experiments. Instead, random temperature patterns generated by the Gaussian random process and the Fourier series, combined with constant temperature fields, are used as training data to cover possible temperature cases. Second, shape functions and backward difference approximation are exploited for the domain discretization, resulting in a purely algebraic equation. This enhances training efficiency, as one avoids time-consuming automatic differentiation when optimizing weights and biases while accepting possible discretization errors. Finally, thanks to the interpolation power of FEM, any arbitrary geometry can be handled with FOL, which is crucial to addressing various engineering application scenarios.

Figures (33)

• Figure 1: Schematic of training and evaluation parts in the proposed finite element-based physics-informed operator learning framework termed finite operator learning (FOL).
• Figure 1: Summary of hyperparameters and material parameters
• Figure 2: Comparison of vanilla PINNs, physics-informed DeepONet (PI-DeepONet), and finite operator learning (FOL).
• Figure 3: Dimensions of the problem domain and the boundary conditions.
• Figure 4: Two types of thermal conductivity distributions considered in this study.