AI 中文总结
研究如何准确求解任意几何形状和各种网格上的偏微分方程，提出自适应曼巴神经算子，通过构建竹内 - 马尔姆奎斯特系统集成再生核，在多个领域的基准问题上，其相对\(L^2\)误差优于现有求解器，为神经算子框架设计提供新范例。
AI 中文摘要
准确求解任意几何形状和各种网格上的偏微分方程（PDEs）是科学与工程应用中的重要任务。本文提出自适应曼巴神经算子（AMO），它集成了状态空间模型（SSMs）的再生核而非SSMs的核积分公式，通过为PDEs构建竹内 - 马尔姆奎斯特系统实现。AMO提供了与自适应傅里叶分解（AFD）理论契合的新表示，能在多种几何形状和网格上近似PDEs的解流形。在流体物理、固体物理和金融领域几个具有挑战性的基准PDE问题上，AMO在相对\(L^2\)误差方面始终优于现有求解器。这项工作为设计可解释的神经算子框架提供了新范例。
英文摘要
Accurately solving partial differential equations (PDEs) on arbitrary geometries and a variety of meshes is an important task in science and engineering applications. In this paper, we propose Adaptive Mamba Neural Operators (AMO), which integrates reproducing kernels for state-space models (SSMs) rather than the kernel integral formulation of SSMs. This is achieved by constructing Takenaka-Malmquist systems for the PDEs. AMO offers new representations that align well with the adaptive Fourier decomposition (AFD) theory and can approximate the solution manifold of PDEs on a wide range of geometries and meshes. In several challenging benchmark PDE problems in the fields of fluid physics, solid physics, and finance on point clouds, structured meshes, regular grids, and irregular domains, AMO consistently outperforms state-of-the-art solvers in terms of relative $L^2$ error. Overall, this work presents a new paradigm for designing explainable neural operator frameworks.
Comments22 pages, accepted by ICLR 2026 conference (https://openreview.net/forum?id=OenyzvFZPs)
