arXiv 2602.079702026-02cs.CEcs.AIcs.LGcs.NAmath.NA

基于学习的Kansa配点法用于超越线性的正向和反向PDEs

Learning-guided Kansa collocation for forward and inverse PDEs beyond linearity

Department of Computer Science and Technology（计算机科学与技术系）；Department of Applied Mathematics and Theoretical Physics（应用数学与理论物理系）；University of Cambridge（剑桥大学）

Zheyuan Hu, Weitao Chen, Cengiz Öztireli, Chenliang Zhou, Fangcheng Zhong

备注：: Accepted for poster presentation at the ICLR 2026 Artificial Intelligence and Partial Differential Equations (AI&PDE) Workshop. Fangcheng Zhong and Chenliang Zhou are co-corresponding authors

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AI总结：

本文提出基于学习的Kansa配点法，用于解决超越线性的正向和反向PDEs问题，通过扩展CNF框架并应用到耦合和非线性场景，实现高效求解与自调技术。

摘要：

偏微分方程在建模物理、生物和图形现象方面非常精确。然而，数值方法却受到维度诅咒、高计算成本和领域特定离散化的限制。我们旨在探索不同PDE求解器的优缺点，并将其应用于特定的科学模拟问题，包括正向求解、反问题和方程发现。特别是，我们将最近的CNF（NeurIPS 2023）框架求解器扩展到耦合和非线性设置，并结合下游应用。结果包括所选方法的实现、自调技术、在基准问题上的评估以及神经PDE求解器和科学模拟应用的全面调查。

英文摘要：

Partial Differential Equations are precise in modelling the physical, biological and graphical phenomena. However, the numerical methods suffer from the curse of dimensionality, high computation costs and domain-specific discretization. We aim to explore pros and cons of different PDE solvers, and apply them to specific scientific simulation problems, including forwarding solution, inverse problems and equations discovery. In particular, we extend the recent CNF (NeurIPS 2023) framework solver to coupled and non-linear settings, together with down-stream applications. The outcomes include implementation of selected methods, self-tuning techniques, evaluation on benchmark problems and a comprehensive survey of neural PDE solvers and scientific simulation applications.

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