Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains
This work addresses a bottleneck for researchers and practitioners in scientific computing who need efficient neural operators for PDEs on non-uniform grids, representing an incremental improvement over existing methods.
The paper tackled the limitation of neural operators relying on fast Fourier transforms for equispaced grids by proposing a method to extend them to arbitrary domains using direct spectral evaluations, achieving significant gains in training speed while maintaining or improving accuracy compared to Fourier neural operators.
The computational efficiency of many neural operators, widely used for learning solutions of PDEs, relies on the fast Fourier transform (FFT) for performing spectral computations. As the FFT is limited to equispaced (rectangular) grids, this limits the efficiency of such neural operators when applied to problems where the input and output functions need to be processed on general non-equispaced point distributions. Leveraging the observation that a limited set of Fourier (Spectral) modes suffice to provide the required expressivity of a neural operator, we propose a simple method, based on the efficient direct evaluation of the underlying spectral transformation, to extend neural operators to arbitrary domains. An efficient implementation* of such direct spectral evaluations is coupled with existing neural operator models to allow the processing of data on arbitrary non-equispaced distributions of points. With extensive empirical evaluation, we demonstrate that the proposed method allows us to extend neural operators to arbitrary point distributions with significant gains in training speed over baselines while retaining or improving the accuracy of Fourier neural operators (FNOs) and related neural operators.