Integration of physics-informed operator learning and finite element method for parametric learning of partial differential equations

TL;DR

The paper introduces Finite Operator Learning (FOL), a physics-informed operator-learning framework that uses a discretized weak form from finite element ideas to train neural networks without automatic differentiation. Targeting parametric steady-state heat diffusion in two-phase microstructures, the method predicts nodal temperatures by leveraging separate neural networks per node and an algebraic, FE-inspired loss based on energy terms and Dirichlet constraints. Compared with data-driven and FEM baselines, FOL achieves accurate predictions for unseen morphologies and offers substantial speedups in evaluation (and potential gains for nonlinear problems) while operating in a data-free, parametric setting. This approach enables rapid, parametric solutions to PDEs in heterogeneous media and provides a foundation for extending physics-driven operator learning to broader classes of problems.

Abstract

We present a method that employs physics-informed deep learning techniques for parametrically solving partial differential equations. The focus is on the steady-state heat equations within heterogeneous solids exhibiting significant phase contrast. Similar equations manifest in diverse applications like chemical diffusion, electrostatics, and Darcy flow. The neural network aims to establish the link between the complex thermal conductivity profiles and temperature distributions, as well as heat flux components within the microstructure, under fixed boundary conditions. A distinctive aspect is our independence from classical solvers like finite element methods for data. A noteworthy contribution lies in our novel approach to defining the loss function, based on the discretized weak form of the governing equation. This not only reduces the required order of derivatives but also eliminates the need for automatic differentiation in the construction of loss terms, accepting potential numerical errors from the chosen discretization method. As a result, the loss function in this work is an algebraic equation that significantly enhances training efficiency. We benchmark our methodology against the standard finite element method, demonstrating accurate yet faster predictions using the trained neural network for temperature and flux profiles. We also show higher accuracy by using the proposed method compared to purely data-driven approaches for unforeseen scenarios.

Figures (19)

• Figure 1: Utilizing physics-informed operator learning, one can enhance the classical data-driven approach and bypass the need for supervised learning for forward problems.
• Figure 2: Difference between data and physics driven neural networks. The derivations for constructing differential equations can be approximated by different approaches.
• Figure 3: Network architecture for finite operator learning, where information about the input parameter at each grid point goes in and the unknown is evaluated utilizing a series of separate deep neural networks.
• Figure 4: The physical space and the parent space. Gaussian integration and shape functions are executed and defined specifically within the parent space.
• Figure 5: Downsampling the grid size involves using convolutional layers combined with max-pooling, resulting in a reduced space of $11 \times 11 = 121$ grid points. The training will be conducted based on an 11 by 11 grid size, resembling the concept of a latent space.