用状态空间模型提升神经算子求解偏微分方程的效率与精度
Latent Mamba Operator for Partial Differential Equations
- 在隐空间中结合状态空间模型与核积分算子，提升表达能力
- 在多种网格和点云数据上实现32.3%的解算器性能提升
- 适合需要高效高精度求解复杂物理系统的研究者
神经算子作为求解偏微分方程（PDE）的强有力数据驱动框架，相比传统数值方法显著加速。然而现有神经算子在高维空间中存在可扩展性差、计算成本高、难以捕捉连续长程依赖等问题。为此，本文提出隐空间状态空间模型算子（LaMO），将状态空间模型（SSMs）在隐空间中的高效性与神经算子的核积分形式表达能力相结合，并建立了SSM与神经算子核积分之间的理论联系。在规则网格、结构化网格及点云上的多样化PDE基准测试中，包括固体力学与流体物理数据集，LaMO持续达到当前最优性能，解算器近似误差相较基线平均降低32.3%，验证了其对复杂PDE解建模的有效性。
原文摘要 · Abstract (English)
Neural operators have emerged as powerful data-driven frameworks for solving Partial Differential Equations (PDEs), offering significant speedups over numerical methods. However, existing neural operators struggle with scalability in high-dimensional spaces, incur high computational costs, and face challenges in capturing continuous and long-range dependencies in PDE dynamics. To address these limitations, we introduce the Latent Mamba Operator (LaMO), which integrates the efficiency of state-space models (SSMs) in latent space with the expressive power of kernel integral formulations in neural operators. We also establish a theoretical connection between state-space models (SSMs) and the kernel integral of neural operators. Extensive experiments across diverse PDE benchmarks on regular grids, structured meshes, and point clouds covering solid and fluid physics datasets, LaMOs achieve consistent state-of-the-art (SOTA) performance, with a 32.3% improvement over existing baselines in solution operator approximation, highlighting its efficacy in modeling complex PDE solutions.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。