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.

翻译：偏微分方程在物理、生物与图形现象建模中具有精确描述能力。然而，传统数值方法面临维度灾难、计算成本高昂及领域特定离散化等挑战。本研究旨在系统探讨不同偏微分方程求解器的优劣，并将其应用于特定科学模拟问题，包括正向求解、反问题及方程发现。特别地，我们将近期提出的CNF（NeurIPS 2023）框架求解器拓展至多因变量与非线性的设定，并开发下游应用场景。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，以及对神经偏微分方程求解器与科学模拟应用的综合评述。