用自适应框架提升复杂几何下偏微分方程求解精度
Adaptive Mamba Neural Operators
- 基于重构核构建状态空间模型，适配多种网格与几何
- 在流体、固体力学及金融领域的多类问题上误差显著更低
- 兼具可解释性，适合需要高精度与理论支撑的工程场景
在任意几何和多种网格上精确求解偏微分方程（PDEs）是科学与工程中的关键任务。本文提出自适应马尔萨模型神经算子（AMO），通过引入再生核而非传统核积分形式构建状态空间模型，并基于PDE构造Takenaka-Malmquist系统。该方法与自适应傅里叶分解（AFD）理论高度契合，能有效逼近各类几何与网格下的PDE解流形。在点云、结构化网格、规则格点及不规则域上的流体物理、固体力学和金融领域多个挑战性基准问题中，AMO在相对$L^2$误差上持续优于现有最先进求解器。本工作提出了可解释神经算子设计的新范式。
原文摘要 · Abstract (English)
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.
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