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

Zheyuan Hu, Weitao Chen, Cengiz Öztireli, Chenliang Zhou, Fangcheng Zhong

TL;DR

This work extends the CNF-based learning framework to multi-variable and nonlinear PDEs using Kansa collocation with Gaussian RBFs, enabling forward, inverse, and discovery tasks in a mesh-free setting. It introduces coupled-field and nonlinear operator extensions, various time-stepping strategies, and auto-tuning of hyperparameters, complemented by a comprehensive empirical benchmark against classical and neural PDE solvers across Advection, Lotka-Volterra, Maxwell, and Burgers problems. The results show that learning-guided Kansa solvers can achieve competitive accuracy and efficiency, with distinct trade-offs in stability and training cost, and demonstrate robust inverse parameter recovery. The study provides practical insights into solver performance, scalability, and integration potential with differentiable pipelines in scientific computing contexts.

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.

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

TL;DR

This work extends the CNF-based learning framework to multi-variable and nonlinear PDEs using Kansa collocation with Gaussian RBFs, enabling forward, inverse, and discovery tasks in a mesh-free setting. It introduces coupled-field and nonlinear operator extensions, various time-stepping strategies, and auto-tuning of hyperparameters, complemented by a comprehensive empirical benchmark against classical and neural PDE solvers across Advection, Lotka-Volterra, Maxwell, and Burgers problems. The results show that learning-guided Kansa solvers can achieve competitive accuracy and efficiency, with distinct trade-offs in stability and training cost, and demonstrate robust inverse parameter recovery. The study provides practical insights into solver performance, scalability, and integration potential with differentiable pipelines in scientific computing contexts.

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.
Paper Structure (44 sections, 106 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 44 sections, 106 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Methodology
  4. General form of PDEs
  5. Kansa collocation
  6. Linear operator case
  7. Extension 1: coupled solution fields of PDEs
  8. Extension 2: nonlinear operator case
  9. Time-stepping approach with linear system.
  10. Auto-tuning of Kansa hyperparameters
  11. Solutions of inverse PDE problems
  12. Inverse PDE problems.
  13. Evaluation
  14. Performance metrics
  15. Evaluation of solvers for the Advection equation
  16. ...and 29 more sections

Figures (4)

  • Figure 1: Advection equation solution visualization in 1D, 2D and 3D.
  • Figure 2: Lotka-Volterra predator-prey model solution and phase space.
  • Figure 3: Maxwell's equations solution visualization in 1D, 2D and 3D.
  • Figure 4: Burgers' equation solution visualization in 1D, 2D and 3D.