提出新框架，让神经算子在任意曲面上传播动态信号。
Geometric Laplace Neural Operator
- 用指数基函数和极点-残差分解建模非周期衰减动态
- 在任意黎曼流形上实现算子学习，无需规则网格或周期性
- 适合物理模拟、几何信号处理等需曲面建模的场景
神经算子已成为学习函数空间间映射的强大工具，可高效求解跨不同输入与域的偏微分方程。尽管取得成功，现有方法在非周期激励、瞬态响应及定义于不规则或非欧几里得几何上的信号方面仍面临挑战。为此，我们提出一种基于极点-残差分解并融合指数基函数的广义算子学习框架，可有效建模非周期与衰减动态。在此基础上，引入几何拉普拉斯神经算子（GLNO），将拉普拉斯谱表示嵌入拉普拉斯-贝尔特拉米算子的特征基中，使算子学习扩展至任意黎曼流形，无需周期性或均匀网格。我们进一步设计了网格不变网络架构（GLNONet）以实现该框架。在偏微分方程/常微分方程及真实世界数据集上的大量实验表明，本方法在性能上优于其他最先进模型。
原文摘要 · Abstract (English)
Neural operators have emerged as powerful tools for learning mappings between function spaces, enabling efficient solutions to partial differential equations across varying inputs and domains. Despite the success, existing methods often struggle with non-periodic excitations, transient responses, and signals defined on irregular or non-Euclidean geometries. To address this, we propose a generalized operator learning framework based on a pole-residue decomposition enriched with exponential basis functions, enabling expressive modeling of aperiodic and decaying dynamics. Building on this formulation, we introduce the Geometric Laplace Neural Operator (GLNO), which embeds the Laplace spectral representation into the eigen-basis of the Laplace-Beltrami operator, extending operator learning to arbitrary Riemannian manifolds without requiring periodicity or uniform grids. We further design a grid-invariant network architecture (GLNONet) that realizes GLNO in practice. Extensive experiments on PDEs/ODEs and real-world datasets demonstrate our robust performance over other state-of-the-art models.
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