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

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 Galerkin discretized weak formulation is employed to incorporate physics into the loss function, termed finite operator learning (FOL), along with the implicit Euler time integration scheme for temporal discretization. A transient thermal conduction problem is considered to benchmark the performance, where FOL takes a temperature field at the current time step as input and predicts a temperature field at the next time step. Upon training, the network successfully predicts the temperature evolution over time for any initial temperature field at high accuracy compared to the solution by the finite element method (FEM) even with 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 large data 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. Additionally, 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 in optimizing weights and biases while accepting possible discretization errors. Finally, thanks to the interpolation power of FEM, any arbitrary geometry with heterogeneous microstructure can be handled with FOL, which is crucial to addressing various engineering application scenarios.