Data Generation-Based Operator Learning for Solving Partial Differential Equations on Unbounded Domains (Communications in Computational Physics, 2025)
Wave propagation problems are typically formulated as partial differentialequations (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 typicallyrequire significant effort, and the presence of nonlinearity in the equation makes suchdesigns even more challenging. Emerging deep learning-based methods for solvingPDEs, with the physics-informed neural networks (PINNs) method as a representative, still face significant challenges when directly used to solve PDEs on unboundeddomains. Calculations performed in a bounded domain of interest without imposingboundary constraints can lead to a lack of unique solutions thus causing the failureof PINNs. In light of this, this paper proposes a novel and effective data generation-based operator learning method for solving PDEs on unbounded domains. The keyidea 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 initialcondition and source term. Then, using these constructed data comprising exact solutions, initial conditions, and source terms, we train an operator learning model calledMIONet, which is capable of handling multiple inputs, to learn the mapping from theinitial 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 solutionof the target PDE. The effectiveness of this method is exemplified by solving the waveequation and the Schrödinger equation defined on unbounded domains. More importantly, the proposed method can deal with nonlinear problems, which has beendemonstrated by solving Burgers’ equation and Korteweg-de Vries (KdV) equation.The code is available at https://github.com/ZJLAB-AMMI/DGOL.
Keywords: Scientific machine learning; operator learning; unbounded domain; nonlinear PDEs.