将神经微分方程求解器拓展至非线性与耦合场景，提升科学模拟的通用性。
Learning-guided Kansa collocation for forward and inverse PDEs beyond linearity
- 基于Kansa配点法，融合学习引导机制求解非线性偏微分方程
- 在多个基准测试中实现高精度正向求解与逆问题重建
- 适用于需要高维、复杂物理建模的科研人员
偏微分方程在物理、生物和图形现象建模中具有精确性。然而，传统数值方法面临维度灾难、计算成本高及领域依赖离散化等问题。本文旨在探索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任务，包括正向求解、反问题及方程发现。特别地，将近期提出的CNF（NeurIPS 2023）框架求解器扩展至耦合与非线性情形，并配套下游应用。成果包括选定方法的实现、自调参技术、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。
原文摘要 · Abstract (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 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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