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Fourier layers

Fourier layers are neural network layers designed to learn mappings between continuous functions, commonly used in neural operators to model physical systems governed by partial differential equations. Within a Fourier layer, input feature representations are transformed into the frequency domain via the Fast Fourier Transform, where learnable weight matrices are multiplied across a truncated set of frequency modes to perform global convolutions efficiently. The filtered representation is then converted back to physical space using the inverse Fast Fourier Transform, combined with a parallel local linear transform of the input, and evaluated with a nonlinear activation function. Because the learnable parameters operate on spectral modes rather than fixed coordinate grids, Fourier layers achieve discretization invariance, allowing models to evaluate functions and make predictions across arbitrary grid resolutions without retraining.

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