Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the solution of a PDE. The last decade has seen the emergence of deep learning as a strong contender in providing efficient representations of complex functions. In the current work, we present an approach for combining deep neural networks with spectral methods to solve PDEs. In particular, we use a deep learning technique known as the Deep Operator Network (DeepONet), to identify candidate functions on which to expand the solution of PDEs. We have devised an approach which uses the candidate functions provided by the DeepONet as a starting point to construct a set of functions which have the following properties: i) they constitute a basis, 2) they are orthonormal, and 3) they are hierarchical i.e., akin to Fourier series or orthogonal polynomials. We have exploited the favorable properties of our custom-made basis functions to both study their approximation capability and use them to expand the solution of linear and nonlinear time-dependent PDEs.
翻译:光谱方法是科学计算用于解决部分差异方程式(PDEs)的武库的重要组成部分。然而,其适用性和有效性关键取决于对用于扩大PDE解决方案的基础功能的选择。过去十年中,作为提供复杂功能有效表达的强大竞争者的深刻学习出现了。在目前的工作中,我们提出了一个将深神经网络与光谱方法相结合以解决PDEs的方法。特别是,我们使用了一种叫做深操作器网络(DeepONet)的深学习技术,以确定扩大PDEs解决方案的候选功能。我们设计了一种方法,将DeepONet提供的候选功能作为建立一系列功能的起点,这些功能具有以下属性:(i) 它们构成基础,2 它们具有正态,3) 它们具有等级,即近似Fourier系列或矩形多元体。我们利用了我们定制基础功能的有利特性来研究其近似能力,并利用它们扩大线性和非线性PDE的解决方案。