Spectral Shaping for Neural PDE Surrogates

Daniel E. Worrall, Miles Cranmer, J. Nathan Kutz, Peter Battaglia

Keywords: PDEs, Fluid Mechanics, Dynamical systems, Autoregressive models, Spectral methods

TL;DR: Smoothing nonlinearities improves power spectral density and stability of neural PDE surrogates

Abstract: Neural surrogates for PDE solvers suffer from an inability to model the spectrum of solutions adequately, especially in the medium to high frequency bands. This impacts not only correct spectral shapes, but also stability and long-term rollout accuracy. We identify three convergent factors that exacerbate this phenomenon, namely: distribution shift over unrolls, spectral bias of the MSE loss, and spurious high frequency noise, or _spectral junk_, introduced by the use of pointwise nonlinearities. We find that _spectral shaping_, filtering the spectrum of activations after every layer of pointwise nonlinearities, is enough to reduce spectral junk and improve long-term rollout accuracy. We show spectral shaping not only fixes the learned spectrum (down to machine precision in some cases), but also leads to very stable neural surrogates. We validate these findings on a suite of challenging fluid dynamics problems in the field of neural PDE surrogacy, promoting a clear need for more careful attention to surrogate architecture design and adding a new and simple trick to the practitioner toolbox.

Primary Area: learning on time series and dynamical systems

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Submission Number: 3280