Solving analytically intractable partial differential equations (PDEs) that involve at least one variable defined on an unbounded domain arises in numerous physical applications. Accurately solving unbounded domain PDEs requires efficient numerical methods that can resolve the dependence of the PDE on the unbounded variable over at least several orders of magnitude. We propose a solution to such problems by combining two classes of numerical methods: (i) adaptive spectral methods and (ii) physics-informed neural networks (PINNs). The numerical approach that we develop takes advantage of the ability of physics-informed neural networks to easily implement high-order numerical schemes to efficiently solve PDEs and extrapolate numerical solutions at any point in space and time. We then show how recently introduced adaptive techniques for spectral methods can be integrated into PINN-based PDE solvers to obtain numerical solutions of unbounded domain problems that cannot be efficiently approximated by standard PINNs. Through a number of examples, we demonstrate the advantages of the proposed spectrally adapted PINNs in solving PDEs and estimating model parameters from noisy observations in unbounded domains.

翻译：[翻译摘要] 求解包含至少一个定义于无界域变量的解析难解偏微分方程(PDEs)是众多物理应用中的常见问题。精确求解无界域PDEs需要高效的数值方法，能够解析PDE对无界变量在至少数个数量级上的依赖性。我们提出结合两类数值方法来解决此类问题：(i)自适应谱方法与(ii)物理信息神经网络(PINNs)。所发展的数值方法利用物理信息神经网络易于实现高阶数值格式的特性，能够高效求解PDEs并外推任意时空点的数值解。随后我们展示了如何将近期提出的自适应谱方法技术集成至基于PINN的PDE求解器中，以获得标准PINN无法高效逼近的无界域问题数值解。通过多个数值算例，我们证明了所提出的谱自适应PINN在求解无界域PDEs及从含噪观测数据中估计模型参数方面的优势。