Data Generation-based Operator Learning for Solving Partial Differential Equations on Unbounded Domains

TL;DR

This work tackles the challenge of solving PDEs on unbounded domains by introducing a data-generation-based operator learning framework. It leverages MIONet to learn the map from initial data $\varphi$ and source term $f$ to the solution $u$ on a bounded, tractable domain, using a generated family of approximate analytical solutions to create training data with low cost. The approach demonstrates accurate results for linear and nonlinear PDEs, including 1D/2D wave equations, Burgers' equation, KdV, and Schrödinger equations, and supports solving multiple related PDEs with a single trained model. While effective, the method relies on suitably constructed analytical data and may require physics-informed training to ensure solution reliability, highlighting both practical benefits and avenues for future refinement in unbounded-domain PDE computation.

Abstract

Wave propagation problems are typically formulated as partial differential equations (PDEs) on unbounded domains to be solved. The classical approach to solving such problems involves truncating them to problems on bounded domains by designing the artificial boundary conditions or perfectly matched layers, which typically require significant effort, and the presence of nonlinearity in the equation makes such designs even more challenging. Emerging deep learning-based methods for solving PDEs, with the physics-informed neural networks (PINNs) method as a representative, still face significant challenges when directly used to solve PDEs on unbounded domains. Calculations performed in a bounded domain of interest without imposing boundary constraints can lead to a lack of unique solutions thus causing the failure of PINNs. In light of this, this paper proposes a novel and effective operator learning-based method for solving PDEs on unbounded domains. The key idea behind this method is to generate high-quality training data. Specifically, we construct a family of approximate analytical solutions to the target PDE based on its initial condition and source term. Then, using these constructed data comprising exact solutions, initial conditions, and source terms, we train an operator learning model called MIONet, which is capable of handling multiple inputs, to learn the mapping from the initial condition and source term to the PDE solution on a bounded domain of interest. Finally, we utilize the generalization ability of this model to predict the solution of the target PDE. The effectiveness of this method is exemplified by solving the wave equation and the Schrodinger equation defined on unbounded domains. More importantly, the proposed method can deal with nonlinear problems, which has been demonstrated by solving Burger's equation and Korteweg-de Vries (KdV) equation.

Figures (29)

• Figure 1.1: Solving a 1D Burgers' equation using PINNs with and without boundary conditions: Left: Training loss. Right: Exact solution and predicted solutions at time $t=1$. It is evident that in the absence of boundary conditions, the PINN solution exhibits a substantial error even after achieving training stability. (The experimental details can be found in the Appendix D.)
• Figure 2.1: Architectures of vanilla DeepONet and MIONet.
• Figure 3.1: Flowchart of the proposed method.
• Figure 4.1: Solving a 1D wave equation (case 1). Left: Training loss and test errors for 80,000 iterations. Right: Test errors on the target PDE w/ and w/o noise.
• Figure 4.2: Solving a 1D wave equation (case 1): Exact solution versus the predictions of the trained MIONet.

Theorems & Definitions (2)

• Remark 3.1
• Remark 3.2