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.

翻译：偏微分方程在建模物理、生物及图形现象方面具有精确性。然而，传统数值方法受限于维度灾难、高昂计算成本及领域特定的离散化需求。本研究旨在系统探讨不同偏微分方程求解器的优劣，并将其应用于具体科学模拟问题，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）框架求解器扩展至耦合与非线性的场景，并结合下游应用进行验证。研究成果包括选定方法的实现、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全面综述。