Type
Master thesis / Bachelor thesis
Prerequisites
- Sound knowledge of machine learning
- (Preferred) Proficiency in Python and deep learning frameworks (PyTorch or Tensorflow)
- Familiarity with partial differential equations
Description
Deep learning methods have also recently had an exciting impact on the numerical analysis of PDEs. It is well-known that PDEs can model many complex processes. Their numerical solution constitutes one of the biggest challenges in scientific computing. Classical numerical representations are not expressive enough to accurately represent complicated high-dimensional structures such as wave functions with long-range interactions. An emerging body of work shows that Neural Networks can potentially overcome such shortcomings and enjoy superior expressivity compared to standard numerical representations. Such results include (linear and semi-linear) parabolic evolution equations, stationary elliptic PDEs, nonlinear Hamilton Jacobi Bellman equations, or parametric PDEs. In all these cases, the absence of the curse of dimensionality (in terms of the theoretical approximation power of neural networks) was rigorously established.
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)