LOCO: Abstracting Spectral Numerical Integrators for PDEs into Neural Operators

Lenz Pracher, Takashi Matsubara

Keywords: Fourier Neural Operator, Neural Operator, spectral convolution, burgers equation, KdV equation, Navier Stokes, Scientific Machine Learning, Neural Operators for PDEs, Scientific Foundation Models

TL;DR: We show that encoding spectral numerical integration structure into fourier neural operators yields LOCO, a slightly modified Fourier Neural Operator with strong improved single-step PDE accuracy.

Abstract: Neural operators promise fast surrogates for PDE dynamics, but their architectures rarely reflect how numerical integrators compute. We argue that importing structure from spectral time stepping into the Fourier Neural Operator (FNO) yields a beneficial inductive bias for operator learning. As a case study in this abstraction, we introduce LOCO (Local Convolutional Operator), a slightly modified FNO that (i) applies nonlinear transforms in Fourier Space to emulate pseudo-differential operators and (ii) uses convolutions in Fourier Space to approximate local field couplings. A lightweight Hybrid (FNO + LOCO) variant combines both architectures. On the 1D Burgers and KdV equation, LOCO improves single-step errors over a Fourier Neural Operator (FNO) baseline, and the Hybrid further reduces single step error on the incompressible 2D Navier-Stokes in direct comparison with FNO. However, we observe rollout instabilities in the incompressible 2D Navier-Stokes case; we do not claim LOCO is a SOTA (State Of The Art) architecture. Rather, our results support the broader position that principled abstractions of established numerical integrators provide a productive design space for neural operators.

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Submission Number: 3