Deep neural operators are recognized as an effective tool for learning solution operators of complex partial differential equations (PDEs). As compared to laborious analytical and computational tools, a single neural operator can predict solutions of PDEs for varying initial or boundary conditions and different inputs. A recently proposed Wavelet Neural Operator (WNO) is one such operator that harnesses the advantage of time-frequency localization of wavelets to capture the manifolds in the spatial domain effectively. While WNO has proven to be a promising method for operator learning, the data-hungry nature of the framework is a major shortcoming. In this work, we propose a physics-informed WNO for learning the solution operators of families of parametric PDEs without labeled training data. The efficacy of the framework is validated and illustrated with four nonlinear spatiotemporal systems relevant to various fields of engineering and science.

翻译：深度神经算子被公认为学习复杂偏微分方程解算子的有效工具。与繁琐的分析和计算工具相比，单个神经算子能够预测不同初始条件、边界条件及输入参数下的偏微分方程解。近期提出的小波神经算子利用小波的时频局部化优势，有效捕捉空间域中的流形结构。尽管小波神经算子在算子学习方面展现出潜力，但其数据密集型的框架特性仍是主要缺陷。本研究提出一种物理信息小波神经算子，可在无需标注训练数据的情况下学习参数化偏微分方程族的解算子。通过四个与工程和科学领域相关的非线性时空系统，验证并阐释了该框架的有效性。