非线性偏微分方程正反问题中学习引导的Kansa配置点法

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

February 8, 2026

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

cs.AI

摘要

偏微分方程在建模物理、生物及图形现象方面具有精确性，但数值方法仍面临维度灾难、计算成本高昂和领域特定离散化等挑战。本研究旨在系统探讨不同PDE求解器的优劣，并将其应用于具体科学模拟问题，包括正问题求解、反问题求解及方程发现。特别地，我们将近期提出的CNF（NeurIPS 2023）框架求解器扩展至多因变量与非线性场景，并开发下游应用。研究成果涵盖选定方法的实现、自适应调参技术、基准问题评估，以及对神经PDE求解器与科学模拟应用的全面综述。

English

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 multi-dependent-variable 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.