Neural networks and PDEs

Research at our chair

• Expressivity of neural networks for parametric PDEs
  A theoretical analysis of deep neural networks and parametric PDEs (https://arxiv.org/pdf/1904.00377.pdf)
• Identifiability of PDEs from the solution
  Symbolic Recovery of Differential Equations: The Identifiability Problem. (https://arxiv.org/abs/2210.08342)
• Continuous optimization for Symbolic Regression
  ParFam -- (Neural Guided) Symbolic Regression Based on Continuous Global Optimization. (https://arxiv.org/abs/2310.05537)

General References

• Solving PDEs with deep learning (PINNs)
  Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
  (https://arxiv.org/abs/1711.10561)
  DGM: A deep learning algorithm for solving partial differential equations (https://arxiv.org/pdf/1708.07469.pdf)
  Solving high-dimensional partial differential equations using deep learning (https://www.pnas.org/content/pnas/115/34/8505.full.pdf)
• Solving PDEs with deep learning (neural operators)
  Neural Operator: Learning Maps Between Function Spaces
  (https://arxiv.org/abs/2108.08481)
  DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
  (https://arxiv.org/abs/1910.03193)
  Fourier Neural Operator for Parametric Partial Differential Equations
  (https://arxiv.org/abs/2010.08895)
• Symbolic Regression
  Contemporary Symbolic Regression Methods and their Relative Performance (https://arxiv.org/abs/2107.14351)
  Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl (https://arxiv.org/abs/2305.01582)
  End-to-end symbolic regression with transformers (https://arxiv.org/abs/2204.10532)

Contact

Do you have questions about our research in this area?

Please do not hesitate to contact us directly. Feel free to write an e-mail to Phillip Scholl, one of our PhD students in the field of neural networks and PDEs.

Inquiries from students are very welcome!