Learning-guided Kansa collocation for forward and inverse PDEs beyond linearity
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Abstract (arXiv)
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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References · 9
- 2015Automatic differentiation in machine learning: a surveycited by 846
- 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
- 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
- 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 396
- 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 279
- 2018Optimal approximation of continuous functions by very deep ReLU networkscited by 165
- 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
- 2023Differentiable Visual Computing for Inverse Problems and Machine Learningcited by 7
- 2023Neural Fields with Hard Constraints of Arbitrary Differential Ordercited by 5
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