Higher-Order Fourier Neural Operator: Explicit Mode Mixer for Nonlinear PDEs

Alex Colagrande, Paul Caillon, Eva Feillet, Alexandre Allauzen

Keywords: Fourier Neural Operator, Operator Learning, Partial Differential Equations

TL;DR: We introduce a higher-order spectral mixer that adds explicit m-linear mode interactions to FNO, boosting nonlinear PDE accuracy with a negligible additional cost.

Abstract:

Neural operators provide resolution-equivariant deep learning models for learning mappings between function spaces. Among them, the Fourier Neural Operator (FNO) is particularly effective: its spectral convolution combines a low-dimensional Fourier representation with strong empirical performance, enabling generalization across resolutions.
While this design aligns with settings where the Fourier basis diagonalizes the underlying operator, such as linear, constant-coefficient PDEs on periodic domains, in which Fourier modes evolve independently, nonlinear PDEs exhibit structured interactions between modes governed by polynomial nonlinearities. To capture this inductive bias, we introduce the **Higher-Order Spectral Convolution**, a spectral mixer that extends FNO from diagonal modulation to explicit $n$-linear mode mixing aligned with nonlinear PDE dynamics. Across benchmarks, including Burgers and Navier-Stokes equations, our method consistently improves accuracy in nonlinear regimes, achieving lower error while retaining the efficiency of FFT-based architectures.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

Submission Number: 19617