Large-scale Neural Solvers for Partial Differential Equations

Abstract

Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. However, recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. Scanning the parameters of the underlying model significantly increases the runtime as the simulations have to be cold-started for each parameter configuration. Machine Learning based surrogate models denote promising ways for learning complex relationship among input, parameter and solution. However, recent generative neural networks require lots of training data, i.e. full simulation runs making them costly. In contrast, we examine the applicability of continuous, mesh-free neural solvers for partial differential equations: physics-informed neural networks (PINNs) solely requiring initial/boundary values and validation points for training but no simulation data. The induced curse of dimensionality is approached by learning a domain decomposition that steers the number of neurons per unit volume and significantly improves runtime. Distributed training on large-scale cluster systems also promises great utilization of large quantities of GPUs which we assess by a comprehensive evaluation study. Finally, we discuss the accuracy of GatedPINN with respect to analytical solutions – as well as state-of-the-art numerical solvers.

Speaker

Nico Hoffmann, young investigator group leader. Nico earned his PhD in 2016 from Technische Universität Dresden in medical image analysis. He developed statistical machine learning methods for analysis of intraoperative neuroimaging data of the exposed human brain. He visited the Laboratory of Mathematics in Imaging of Harvard University from 2018 to 2019. During that time, he developed recurrent convolutional neural networks for reconstruction of nerve fibre bundles of the human brain. He is currently heading a Helmholtz AI Young Investigators Group at Helmholtz-Zentrum Dresden-Rossendorf “AI for Future Photon Sciences” researching Physics-guided Neural Networks for PDE learning as well as inverse problems.