论文标题
用于解决无界域问题的频谱适应物理知识的神经网络
Spectrally Adapted Physics-Informed Neural Networks for Solving Unbounded Domain Problems
论文作者
论文摘要
求解在众多物理应用中，求解在无限域中至少定义的一个变量的分析性棘手的偏微分方程（PDE）。准确地求解无界域PDE需要有效的数值方法，该方法可以在至少几个数量级上解析PDE对未绑定变量的依赖性。我们通过结合两类数值方法来解决此类问题的解决方案：（i）自适应光谱方法和（ii）物理信息信息的神经网络（PINNS）。我们开发的数值方法利用了物理知识的神经网络轻松实施高阶数值方案的能力，以在任何时空的任何时候有效地求解PDE并推断数值解决方案。然后，我们展示如何将用于光谱方法的自适应技术集成到基于PINN的PDE求解器中，以获得无界域问题的数值解决方案，这些解决方案无法有效地被标准的PINN有效地近似。通过许多示例，我们在求解PDE和估计无界域中嘈杂的观测值的模型参数方面证明了所提出的频谱适用于PINN的优势。
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.
