Wavelet Neural Operator for solving parametric partial differential equations in computational mechanics problems

• Published in: Computer methods in applied mechanics and engineering Vol. 404; p. 115783
• Main Authors: Tripura, Tapas, Chakraborty, Souvik
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.02.2023
• Subjects: Nonlinear mappings; Operator learning; Scientific machine learning; Wavelet; Wavelet neural operator; Wavelet; Wavelet neural operator; Scientific machine learning; Nonlinear mappings; Operator learning
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2022.115783

With massive advancements in sensor technologies and Internet-of-things (IoT), we now have access to terabytes of historical data; however, there is a lack of clarity on how to best exploit the data to predict future events. One possible alternative in this context is to utilize an operator learning algorithm that directly learns the 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 the 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. A comparative study with respect to existing operator learning frameworks is presented. Finally, the proposed approach is used to build a digital twin capable of predicting Earth’s air temperature based on available historical data. •A neural operator for solution to computational mechanics problems is proposed.•The proposed operator learns the underlying differential in wavelet space.•A multiresolution representation for the spatial domain is created using wavelets.•The parameterization in the wavelet space is performed using coevolution kernels.•A variety of time-independent and time-dependent boundary value problems are considered.