arXiv:2602.11630cs.AI2026-02

提出可发现PDE通解的神经符号框架,兼具高精度与可解释性。

Neuro-Symbolic Multitasking: A Unified Framework for Discovering Generalizable Solutions to PDE Families

  • 多任务联合优化,同时寻找PDE族的解析解。
  • 引入仿射迁移机制,计算效率提升35.7%以上。
  • 适合需要科学洞察力的物理建模与工程仿真场景。

求解偏微分方程(PDE)是众多科学与工程领域的重要基础。面对具有相同数学结构但参数不同的PDE族,传统数值方法(如有限元法)需逐个求解,计算成本高昂。尽管机器学习类PDE求解器具备高速与高精度优势,但其“黑箱”特性导致缺乏解析表达式,难以提供深层科学洞见。为此,本文提出神经辅助多任务符号求解框架NMIPS,通过多因子优化并行发现PDE族的解析解。为提升效率,设计仿射转移方法,在同族PDE间迁移已学数学结构,避免从零求解。在多个案例上的实验表明,该方法相较现有基线实现最高约35.7%的精度提升,同时输出可解释的解析解。

原文摘要 · Abstract (English)

Solving Partial Differential Equations (PDEs) is fundamental to numerous scientific and engineering disciplines. A common challenge arises from solving the PDE families, which are characterized by sharing an identical mathematical structure but varying in specific parameters. Traditional numerical methods, such as the finite element method, need to independently solve each instance within a PDE family, which incurs massive computational cost. On the other hand, while recent advancements in machine learning PDE solvers offer impressive computational speed and accuracy, their inherent ``black-box" nature presents a considerable limitation. These methods primarily yield numerical approximations, thereby lacking the crucial interpretability provided by analytical expressions, which are essential for deeper scientific insight. To address these limitations, we propose a neuro-assisted multitasking symbolic PDE solver framework for PDE family solving, dubbed NMIPS. In particular, we employ multifactorial optimization to simultaneously discover the analytical solutions of PDEs. To enhance computational efficiency, we devise an affine transfer method by transferring learned mathematical structures among PDEs in a family, avoiding solving each PDE from scratch. Experimental results across multiple cases demonstrate promising improvements over existing baselines, achieving up to a $\sim$35.7% increase in accuracy while providing interpretable analytical solutions.

PDE求解神经符号可解释性多任务学习

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