Deep learning methods for approximating (stochastic) partial differential equations
Contact person: Salvador Ortiz-Latorre
Keywords: (S)PDEs, physics informed neutral networks, generative adversarial networks, regularity of solutions, climate modelling.
Research group: Risk and Stochastics
Department of Mathematics
Stochastic partial differential equations ((S)PDEs) represent the dynamic tools for the modelling of time-space dynamics, ranging from finance in e.g. interest rate theory, to climate modelling as for wind or turbulent flows. Climate models are often represented by SPDEs and the main equations appearing in stochastic filtering and data assimilation are of this type. Explicit solution formulas are only available for very specific classes of (S)PDEs. Hence, numerical simulations are of crucial importance for most practical applications. The entailed challenges are in the pursuit of both methodological rigour and computational efficiency, while mitigating the “curse of dimensionality” (exponential increase in computational effort when increasing the dimension of the problem) for high dimensional problems.
We are interested in research proposals that are directed towards the theoretical and the computational aspects of the approximation of possibly high dimensional (S)PDEs
The research proposals may span several methodological approaches within stochastic control, optimisation, and computational methods, as well as a number of application domains.
Methodological research topics:
- Physics Informed Neural Networks (PINNs) for the approximation of partial differential equations
- Generative adversarial networks (GANs) for the learning of probability distributions from data
- Stochastic filtering, data assimilation and uncertainty quantification.
- Impact of the regularity of solutions of SPDEs in approximations by PINNs
Application domains:
- Climate modelling
- Energy systems
- Engineering
- Sustainability
Mentoring and internship will be offered by a relevant external partner.
