Effective uncertainty quantification and ANN dynamics via amplitude equations

We consider multiple scaling problems for which effective asymptotic models do exist. These regular limit systems very often can be obtained via singular perturbation techniques. These approaches allow for an effective numerical simulation and are less expensive than direct simulations of the original sys- tems. These tools have been applied successfully to various deterministic and stochastic multi-physics problems.

There are three goals of this project.

1. We would like to use the regular limit systems for an effective simulation of stochastic multiple scaling multi-physics problems. In order to do so the approach for stochastic systems has to be extended from toy problems to real world applications.
2. We would like to use the regular limit systems for an effective uncertainty quantification. Beside the control of noise in pattern forming systems we plan to quantify uncertainty for highly oscillatory systems via averaging or normal form techniques. We would like to extract stable objects such as modulating fronts from an uncertain background.

3. We would like to continue our research about the effective learning of PDE Dynamics through ar- tificial neural networks from deterministic multiple scaling problems to stochastic multiple scaling problems.