By learning the map between function spaces using carefully designed deep neural networks, the operator learning become a focused field in recent several years, and have shown considerable efficiency over traditional numerical methods on solving complicated problems such as differential equations, but the method is still disturbed with the concern of its accuracy and reliability. In this paper, combined with the structures and technologies of a popular numerical method, i.e. the spectral method, a general learning-based architecture named Spectral Operator Learning is introduced. One of its variants, Orthogonal Polynomials Neural Operator designed for partial differential equations with Dirichlet, Neumann and Robin boundary conditions is proposed later, of which the effectiveness, efficacy and accuracy of boundary conditions are illustrated by numerical experiments. The code will be available at https://github.com/liu-ziyuan-math/spectral_operator_learning after all the numerical results are summarised.

翻译：通过使用精心设计的深神经网络在功能空间之间学习地图,操作员学习成为近年来一个重点领域,在解决不同方程式等复杂问题的传统数字方法上显示出相当的效率,但这种方法仍然受到其准确性和可靠性的担忧。在本文中,结合流行数字方法的结构和技术,即光谱法,引入了一个名为光谱操作员学习的一般基于学习的结构。它的一个变体,即为Drichlet、Neumann和Robin边界条件部分差异方程式设计的Orthogonal多线性神经操作员,后来提出用数字实验来说明边界条件的有效性、效力和准确性。在对数字结果进行总结后,该代码将在https://githu.com/liu-ziyuan-math/光谱_operator_学习网站查阅。