Transform Once: Efficient Operator Learning in Frequency DomainDownload PDF

Published: 31 Oct 2022, Last Modified: 12 Mar 2024NeurIPS 2022 AcceptReaders: Everyone
Keywords: convolutions, long range dependencies, neural operators, high-resolution, frequency, transform, differential equation, dynamics, turbulence, fluid flows, PDE
Abstract: Spectral analysis provides one of the most effective paradigms for information-preserving dimensionality reduction, as simple descriptions of naturally occurring signals are often obtained via few terms of periodic basis functions. In this work, we study deep neural networks designed to harness the structure in frequency domain for efficient learning of long-range correlations in space or time: frequency-domain models (FDMs). Existing FDMs are based on complex-valued transforms i.e. Fourier Transforms (FT), and layers that perform computation on the spectrum and input data separately. This design introduces considerable computational overhead: for each layer, a forward and inverse FT. Instead, this work introduces a blueprint for frequency domain learning through a single transform: transform once (T1). To enable efficient, direct learning in the frequency domain we derive a variance preserving weight initialization scheme and investigate methods for frequency selection in reduced-order FDMs. Our results noticeably streamline the design process of FDMs, pruning redundant transforms, and leading to speedups of 3x to 10x that increase with data resolution and model size. We perform extensive experiments on learning the solution operator of spatio-temporal dynamics, including incompressible Navier-Stokes, turbulent flows around airfoils and high-resolution video of smoke. T1 models improve on the test performance of FDMs while requiring significantly less computation (5 hours instead of 32 for our large-scale experiment), with over 20% reduction in predictive error across tasks.
TL;DR: We present Transform Once (T1), a deep frequency-domain architecture for efficient learning of long-range correlations in space or time that is 3x to 20x faster than Fourier Neural Operators and improves on predictive accuracy.
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