Neural operators approximate the solution operators of PDEs at a fraction of the cost of traditional numerical solvers. Efficient grid‑based designs such as the Fourier Neural Operator (FNO) assume uniform rectangular grids, while engineering simulations typically employ large, irregular, locally refined meshes. L‑SplineFNO addresses this gap by coupling an FNO backbone with a sparse, learnable tensor‑product B‑spline transfer that maps between unstructured meshes and a latent Cartesian grid. A local normalised scatter eliminates the geometry‑dependent analytic spline solve yet preserves compact support, partition of unity, and spline smoothness by construction. Conditioned on global design and operating parameters, the neural operator learns the mapping from geometry to field solution. Across six 3‑D engineering benchmarks, L‑SplineFNO attains strong predictive accuracy while drastically reducing memory and compute requirements, scaling to 26.6 M training nodes and 46 M inference nodes on a single A100 GPU. It also enables zero‑shot super‑resolution and efficient closed‑form spatial derivatives. Uniquely, the analytic spline decoder provides rigorous prediction bounds over continuous space and uncertain parameters via Bernstein convex hulls and Taylor models, demonstrated on a 29.2‑M‑node MAST‑U tokamak model.
Review