Spectral operator learning for parametric PDEs without data reliance

TL;DR

This paper introduces SCLON, a spectral operator learning method that solves parametric PDEs accurately without relying on training data, combining spectral expansions with neural networks for efficient and boundary-condition-fulfilling solutions.

Contribution

The paper presents a novel data-free operator learning framework using spectral methods and neural networks, enabling accurate solutions of complex parametric PDEs without extensive data or paired training examples.

Findings

Achieves high accuracy with fewer grid points.

Successfully predicts solutions for complex PDEs like Navier-Stokes.

Outperforms existing scientific machine learning techniques.

Abstract

In this paper, we introduce the Spectral Coefficient Learning via Operator Network (SCLON), a novel operator learning-based approach for solving parametric partial differential equations (PDEs) without the need for data harnessing. The cornerstone of our method is the spectral methodology that employs expansions using orthogonal functions, such as Fourier series and Legendre polynomials, enabling accurate PDE solutions with fewer grid points. By merging the merits of spectral methods - encompassing high accuracy, efficiency, generalization, and the exact fulfillment of boundary conditions - with the prowess of deep neural networks, SCLON offers a transformative strategy. Our approach not only eliminates the need for paired input-output training data, which typically requires extensive numerical computations, but also effectively learns and predicts solutions of complex parametric PDEs, ranging from singularly perturbed convection-diffusion equations to the Navier-Stokes equations. The proposed framework demonstrates superior performance compared to existing scientific machine learning techniques, offering solutions for multiple instances of parametric PDEs without harnessing data. The mathematical framework is robust and reliable, with a well-developed loss function derived from the weak formulation, ensuring accurate approximation of solutions while exactly satisfying boundary conditions. The method's efficacy is further illustrated through its ability to accurately predict intricate natural behaviors like the Kolmogorov flow and boundary layers. In essence, our work pioneers a compelling avenue for parametric PDE solutions, serving as a bridge between traditional numerical methodologies and cutting-edge machine learning techniques in the realm of scientific computation.