Deep Neural Networks and PDEs
Type
Master's thesis / Bachelor's thesis
Prerequisites
- Strong machine learning knowledge
- (Preferred) Proficiency with 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 high-dimensional partial differential equations using deep learning (https://www.pnas.org/content/pnas/115/34/8505.full.pdf)
- A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations (https://arxiv.org/pdf/1809.02362.pdf)
- Deep neural network approximation theory (https://arxiv.org/pdf/1901.02220.pdf)
- A theoretical analysis of deep neural networks and parametric PDEs (https://arxiv.org/pdf/1904.00377.pdf)
- DGM: A deep learning algorithm for solving partial differential equations (https://arxiv.org/pdf/1708.07469.pdf)