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

Zheyuan Hu, Weitao Chen, Cengiz Öztireli, Chenliang Zhou, Fangcheng Zhong
2/8/2026
cs.CEcs.AIcs.LG

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Abstract

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.

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Cite this paper

@article{hu2026learningguided,
  title  = {Learning-guided Kansa collocation for forward and inverse PDEs beyond linearity},
  author = {Zheyuan Hu and Weitao Chen and Cengiz Öztireli and Chenliang Zhou and Fangcheng Zhong},
  year   = {2026},
  eprint = {2602.07970},
  archivePrefix = {arXiv},
  url    = {http://arxiv.org/abs/2602.07970v1}
}

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