Wavelet neural operator: a neural operator for parametric partial differential equations

TL;DR

The paper introduces Wavelet Neural Operator (WNO), a novel operator learning algorithm that combines wavelet transforms with neural operators to improve the accuracy and resolution of parametric PDE solutions, demonstrated on various complex equations and a climate prediction task.

Contribution

It presents the Wavelet Neural Operator (WNO), a new method that integrates wavelet transforms into neural operators for enhanced learning of complex parametric PDEs.

Findings

WNO achieves high spatial and frequency resolution in PDE solutions.

WNO outperforms existing operator learning frameworks in accuracy.

WNO successfully predicts Earth's air temperature using historical data.

Abstract

With massive advancements in sensor technologies and Internet-of-things, we now have access to terabytes of historical data; however, there is a lack of clarity in how to best exploit the data to predict future events. One possible alternative in this context is to utilize operator learning algorithm that directly learn nonlinear mapping between two functional spaces; this facilitates real-time prediction of naturally arising complex evolutionary dynamics. In this work, we introduce a novel operator learning algorithm referred to as the Wavelet Neural Operator (WNO) that blends integral kernel with wavelet transformation. WNO harnesses the superiority of the wavelets in time-frequency localization of the functions and enables accurate tracking of patterns in spatial domain and effective learning of the functional mappings. Since the wavelets are localized in both time/space and frequency, WNO can provide high spatial and frequency resolution. This offers learning of the finer details of the parametric dependencies in the solution for complex problems. The efficacy and robustness of the proposed WNO are illustrated on a wide array of problems involving Burger's equation, Darcy flow, Navier-Stokes equation, Allen-Cahn equation, and Wave advection equation. Comparative study with respect to existing operator learning frameworks are presented. Finally, the proposed approach is used to build a digital twin capable of predicting Earth's air temperature based on available historical data.