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

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.

References in corpus (11)

• Physics-informed neural networks for solving Reynolds-averaged Navier$\unicode{x2013}$Stokes equations
• A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications
• A physics-informed variational DeepONet for predicting the crack path in brittle materials
• A Physics Informed Neural Network for Time-Dependent Nonlinear and Higher Order Partial Differential Equations
• Stress field prediction in fiber-reinforced composite materials using a deep learning approach
• Hybrid FEM-NN models: Combining artificial neural networks with the finite element method
• Theory and implementation of inelastic Constitutive Artificial Neural Networks
• Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics
• Simulating progressive intramural damage leading to aortic dissection using an operator-regression neural network
• RBF-MGN:Solving spatiotemporal PDEs with Physics-informed Graph Neural Network
• A Mathematical Guide to Operator Learning

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