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：偏微分方程在建模物理、生物与图形现象方面具有精确性。然而，数值方法面临维度灾难、高计算成本及领域特定离散化的局限。本研究旨在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CNF（NeurIPS 2023）框架求解器扩展至多因变量及非线性场景，并结合下游应用。研究成果包括选定方法的实现、自调优技术、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全面综述。