Conditioning Spatial Representations in Fourier Neural Operators

Geometric and operating parameters can change both the magnitude and spatial organization of a physical field. For Fourier neural operators (FNOs), this raises a design question: when should parameters participate in constructing spatial features, in addition to entering the input or modulating Fourier layers? We study this question with a parameter-conditioned sinusoidal coordinate network that supplies features to Fourier processing. Controlled heat-transport problems with translating and expanding material rings distinguish the benefit of learning spatial features from the additional benefit of conditioning them. A shared spatial representation is effective in the expanding-ring setting; conditioning provides a larger gain in the translating-ring setting, improving whole-field accuracy, localized thermal structure, and responses to geometry changes. In radio-frequency hollow cathode discharge, an industrial plasma problem in semiconductor manufacturing, the model predicts four coupled fields across operating conditions in a fixed cathode geometry and improves on the evaluated baselines. These results highlight spatial feature construction as a consequential choice in conditioning physical-field predictors.