Enhancing Solutions for Complex PDEs: Introducing Translational Equivariant Attention in Fourier Neural Operators
Keywords: Attentive Equivariant Convolution, Fourier Neural Operator
TL;DR: A PDE-informed neural network algorithm for resolving complex problems of physical dynamics (AI for physics)
Abstract: Neural operators extend conventional neural networks by expanding their functional mapping capabilities across various function spaces, thereby promoting the solving of partial differential equations (PDEs). A particularly notable method within this framework is the Fourier Neural Operator (FNO), which draws inspiration from Green's function method to directly approximate operator kernels in the frequency domain. However, after empirical observation and theoretical validation, we demonstrate that the FNO predominantly approximates operator kernels within the low-frequency domain. This limitation results in a restricted capability to solve complex PDEs, particularly those characterized by rapidly changing coefficients and highly oscillatory solution spaces. To address this challenge, inspired by the attentive equivariant convolution, we propose a novel \textbf{T}ranslational \textbf{E}quivariant \textbf{F}ourier \textbf{N}eural \textbf{O}perator (\textbf{TE-FNO}) which utilizes equivariant attention to enhance the ability of FNO to capture high-frequency features. We perform experiments on forward and reverse problems of multiscale elliptic equations, Navier-Stokes equations, and other physical scenarios. The results demonstrate that the proposed approach achieves superior performance across these benchmarks, particularly for equations characterized by rapid coefficient variations.
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Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)
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Submission Number: 4220
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