Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains

TL;DR

This work addresses the limitation of FFT-based neural operators, which require regular grids, by introducing Direct Spectral Evaluation (DSE) to compute truncated spectral transforms directly on arbitrary point clouds. DSE builds transform matrices from point locations and a small set of modes, enabling forward and backward spectral operations with complexity $O(mN)$ and enabling integration with Fourier-based neural operators on general domains, including spheres via spherical harmonics. The authors provide a PyTorch implementation and demonstrate, across multiple operators (FNO, UFNO, FFNO, SFNO) and six PDE benchmarks, that DSE yields substantial training-time speedups while maintaining or improving accuracy relative to interpolation-based and geometry-based baselines. The results show particularly strong benefits for irregular point distributions and spherical domains, broadening the applicability of spectral neural operators to real-world sensor layouts and complex geometries. Overall, DSE offers a simple, efficient, and general approach to extend spectral neural operators beyond regular grids, with significant practical impact for PDE surrogates on arbitrary domains.

Abstract

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.

Figures (12)

• Figure 1: Distributions discussed in this paper. The vanilla FNO is restricted to the regular grid, but may be applied to a general lattice distribution, random distribution, or structured point cloud via the approach outlined in Section 2.
• Figure 2: Illustration of the truncated Fourier transform on a point cloud. We construct a matrix using the values of a set of Fourier basis functions sampled at the positions of the sample points. The transformation between spatial and spectral domains is directly evaluated via a matrix-vector product, where the vector contains the training data. The subscript $\mathbf{V_j}$ refers to the $j^{th}$ row of the matrix from Equation (\ref{['eq:flattened structured matrix']}).
• Figure 3: Ablation study to compare the FNO training time using the FFT and the direct spectral evaluations (DSE) for the 1D Burgers' equation on equispaced data.
• Figure 4: The 8-point fast Fourier transform signal flow graph. $x$ and $y$ represent the signal in the physical and Fourier domain, respectively. Dashed lines represent a multiplication by -1, red elements denote a multiplication by that factor, and converging arrows represent a sum.
• Figure 5: Point distributions used in the Burgers' equation experiments. Data is selected from the uniform distribution to construct the contracting-expanding distribution and random distribution. The space between points in the contracting-expanding distribution grows from a point in both directions according to a geometric distribution.