Learnable Splines for Unstructured Neural Operators

Abstract

Neural operators approximate solution operators of partial differential equations at a fraction of the cost of numerical solvers. Efficient grid-based architectures such as the Fourier neural operator (FNO) assume uniform rectangular grids, whereas engineering simulations use large, irregular, and locally refined meshes. We introduce L-SplineFNO, a memory-efficient neural operator that couples an FNO backbone with a sparse, learnable tensor-product B-spline transfer between unstructured meshes and a latent Cartesian grid. A local normalised scatter avoids the geometry-dependent solve of analytic spline transformation while preserving compact support, partition of unity, and spline smoothness by construction. The neural operator, conditioned on global design and operating parameters, learns the mapping of geometry to the field solution. Across six three-dimensional engineering benchmarks, L-SplineFNO achieves strong predictive accuracy while requiring substantially less memory and compute than baseline operators, scaling to 26.6 million nodes in training and 46 million in inference on a single A100 GPU. It also supports zero-shot super-resolution and efficient closed-form spatial derivatives. Uniquely, the analytic spline decoder yields rigorous prediction bounds over continuous space and uncertain parameters, via Bernstein convex hulls and Taylor models, demonstrated on a 29.2-million-node MAST-U tokamak model.