By combining results from Time-X with state-of-the-art machine learning techniques for the solution of differential equations, our Inno4Scale project NeuralPint will develop new, more efficient parallel-in-time algorithms.
Parallel-across-the-steps methods like Parareal, MGRIT or PFASST require a coarse level solver to handle the unavoidable serial information transport in time. However, devising good coarse level models is hard and there is still only limited theoretical insight into how to design them. The key challenge is that since the coarse level constitutes a computational bottleneck, it needs to be computationally very fast. At the same time, it must be accurate enough to ensure rapid convergence of the iteration.
Within Time-X, we published a demonstrator study that showed that ML-based techniques like physics-informed neural networks (PINNs) or physics-informed neural operators (PINO) are promising approaches to build coarse models. They are generic, do no require much training data since they use the PDE in the loss function, provide reasonable accuracy and are very fast to evaluate once trained.
The NeuralPinT project will build on these achievements and further develop the combination of parallel-in-time and machine learning techniques. Jülich Supercomputing Centre and TU Hamburg will jointly develop distributed Neural Operators that work efficiently alongside PDE-solvers using large-scale space-time parallelism. Furthermore, we will scale up the complexity of our benchmarks by modelling turbulent, three-dimensional Rayleigh-Benard convection, a challenging benchmark problem with relevance for many scientific domains. All algorithms developed within NeuralPinT will be integrated into the pySDC package, which already saw significant development as part of the Time-X project.
Publications
- Ibrahim, A. Q., Götschel, S. & Ruprecht, D. Space-time parallel scaling of Parareal with a Fourier Neural Operator as coarse propagator arXiv:2404.02521v1 [math.NA]. 2024.
- Ibrahim, A. Q., Götschel, S. & Ruprecht, D. Parareal with a Physics-Informed Neural Network as Coarse Propagator in Euro-Par 2023: Parallel Processing (Springer Nature Switzerland, 2023), 649–663.