Learnable Splines for Unstructured Neural Operators: L-SplineFNO (arXiv 2610.22446)

Forum topic · 小凯 · 2026-10-19

Summary

L-SplineFNO is a memory-efficient neural operator that couples a Fourier Neural Operator (FNO) backbone with a sparse, learnable tensor-product B-spline transfer between unstructured engineering meshes and a latent Cartesian grid. A local normalised scatter step avoids geometry-dependent solves of analytic spline transformations while preserving compact support, partition of unity, and spline smoothness by construction. Conditioned on global design and operating parameters, the model learns geometry-to-field mappings. Across six 3D engineering benchmarks, it achieves strong predictive accuracy with substantially lower memory and compute than baselines, scaling to 26.6 million nodes in training and 46 million in inference on a single A100 GPU. It supports zero-shot super-resolution and closed-form spatial derivatives. Uniquely, its 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-million-node MAST-U tokamak model.

Paper Overview

Research Area: Machine Learning

Authors: Vignesh Gopakumar, Zhuoyuan Wang, Ander Gray, Valentin Duruisseaux, Josh Williams, William Michael Edward Ellis, Nuno Nobre, Matthew Field, James Panayis, Alexander I. Blair, Andrew Davis, Stanislas Pamela, Robert Akers, Marc Peter Deisenroth, Anima Anandkumar

Published: 2026-10-19

arXiv: 2610.22446

Summary

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.

Key Highlights

Couples an FNO backbone with sparse, learnable tensor-product B-splines to bridge unstructured meshes and a latent Cartesian grid
Local normalised scatter preserves compact support, partition of unity, and spline smoothness by construction
Scales to 26.6M nodes in training and 46M in inference on a single A100 GPU
Supports zero-shot super-resolution and closed-form spatial derivatives
Provides rigorous prediction bounds via Bernstein convex hulls and Taylor models
Demonstrated on a 29.2M-node MAST-U tokamak model

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Learnable Splines for Unstructured Neural Operators: L-SplineFNO (arXiv 2610.22446)

Summary

Paper Overview

Summary

Key Highlights

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