EXTENDING FOURIER NEURAL OPERATORS FOR PARAMETERIZED AND COUPLED PDES

Scientific and engineering simulations often require solving parameterized and coupled partial differential equations (PDEs) repeatedly. While Fourier Neural Operators (FNOs) are efficient surrogates, they are not natively optimized for systems where dynamics depend on physical parameters or involve multiple interdependent variables. This paper addresses these limitations by proposing minimal, principled architectural extensions to FNOs that maintain their computational efficiency.

The authors introduce two primary modifications to the standard FNO architecture:

1. Parameterized Dynamics: They implement a hypernetwork-based modulation. Instead of simply concatenating parameters to the input, a lightweight hypernetwork generates parameter-dependent shifts that act as biases within each Fourier layer. This allows the model to adapt its internal dynamics to specific physical parameters without requiring a massive increase in model size.

2. Coupled Systems: The authors conduct a systematic exploration of how to enable cross-variable interactions. They propose a spectral-domain coupling mechanism where variables are transformed into the Fourier domain, mixed via a shallow encoder-decoder network, and then mapped back to the data space. This approach avoids the overhead of complex spatial-domain exchanges while capturing long-range correlations between variables.

The proposed methods, collectively referred to as FNOx, were evaluated on benchmark problems including the one-dimensional capacitively coupled plasma (CCP) equations and the Gray-Scott system. The results demonstrate that the hypernetwork modulation and spectral coupling significantly outperform existing baselines, including standard FNOs and specialized multi-physics models, by reducing normalized root mean square error (nRMSE) by 55% to 72%. The authors conclude that these modular, lightweight modifications provide a robust framework for complex PDE surrogate modeling.