[论文] Learnable Splines for Unstructured Neural Operators

论文概要

研究领域: ML
作者: 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
发布时间: 2026-10-19
arXiv: 2610.22446

中文摘要

神经算子以远低于数值求解器的成本逼近偏微分方程的解算子，但 FNO 等高效网格架构假设均匀矩形网格，而工程仿真使用大规模、不规则且局部加密的网格。L-SplineFNO 把 FNO 主干与稀疏、可学习的张量积 B 样条耦合，在非结构网格与潜在笛卡尔网格之间传递；局部归一化 scatter 避免了依赖几何的解析样条变换求解，同时按构造保持紧支撑、单位剖分与样条光滑性。以全局设计与操作参数为条件的神经算子学习几何到场解的映射。六个三维工程基准上：预测精度强、内存与算力显著更低，单张 A100 训练扩展至 2660 万节点、推理 4600 万节点；支持零样本超分辨与高效的闭式空间导数。独特之处在于解析样条解码器经由伯恩斯坦凸包与泰勒模型，给出连续空间与不确定参数下的严格预测界，并在 2920 万节点的 MAST-U 托卡马克模型上予以演示。

原文摘要

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.

自动采集于 2026-10-19

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