Extending Fourier Neural Operators for Modeling Parameterized and Coupled PDEs

2026-07-26

AI Summary

Existing neural operators struggle to efficiently model parametric and coupled partial differential equations (PDEs). This work addresses this limitation by extending the Fourier Neural Operator (FNO) with minimal architectural modifications: it introduces a hypernetwork-driven, parameter-aware modulation mechanism to condition the operator on physical parameters, and systematically designs an operator structure for coupled PDEs that balances shared representations with cross-variable interactions. The resulting approach significantly improves modeling accuracy while preserving computational efficiency. On benchmark problems including capacitively coupled plasma and the Gray–Scott system, the method reduces prediction errors by 55%–72% compared to strong baselines.

Abstract

Parameterized and coupled partial differential equations (PDEs) are central to modeling phenomena in science and engineering, yet neural operator methods that address both aspects remain limited. We extend Fourier neural operators (FNOs) with minimal architectural modifications along two directions. For parameterized dynamics, we propose a hypernetwork-based modulation that conditions the operator on physical parameters. For coupled systems, we conduct a systematic exploration of architectural choices, examining how operator components can be adapted to balance shared structure with cross-variable interactions while retaining the efficiency of standard FNOs. Evaluations on benchmark PDEs, including the one-dimensional capacitively coupled plasma equations and the Gray-Scott system, show that our methods achieve up to 55-72% lower errors than strong baselines, demonstrating the effectiveness of principled modulation and systematic design exploration.

Problem

• parameterized PDEs
• coupled PDEs
• neural operators
• Fourier neural operators

Innovation

• Fourier Neural Operators
• parameterized PDEs
• coupled PDEs
• hypernetwork modulation
• neural operator architecture