Bavarian AI Chair for Mathematical Foundations of Artificial Intelligence

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Deep Neural Networks and PDEs


Master's thesis / Bachelor's thesis


  • Strong machine learning knowledge
  • (Preferred) Proficiency with python and deep learning frameworks (PyTorch or Tensorflow)
  • Familiarity with partial differential equations


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.