Date: 04 October 2023
Time: 12.45 Hours
Room: RA1501 & online
Speaker: Dr. Alexander Heinlein (TUDelft)
Title: “Decomposing physics-informed neural networks"
Abstract:
Scientific machine learning (SciML) is a rapidly evolving field of research that combines techniques from scientific computing and machine learning. A major branch of SciML is the approximation of the from scientific computing and machine learning. A major branch of SciML is the approximation of the solutions of partial differential equations (PDEs) using neural networks. In classical physics-informed neural networks (PINNs) [4], simple feed-forward neural networks are employed to discretize a PDE. The loss function may include a combination of data (e.g., initial, boundary, and/or measurement data) and the residual of the PDE. Challenging applications, such as multiscale problems, require neural networks with high capacity, and the training is often not robust and may take large numbers of iterations.
In this talk, domain decomposition-based network architectures for PINNs using the finite basis physics-informed neural network (FBPINN) approach [3, 1] will be discussed. In particular, the global network function is constructed as a combination of local network functions defined on an overlapping domain decomposition. Similar to classical domain decomposition methods, the one-level method generally lacks scalability, but scalability can be achieved by introducing a multi-level hierarchy of overlapping domain decompositions. The performance of the multi-level FBPINN [2] method will be investigated based on numerical results for several model problems, showing robust convergence for up to 64 subdomains on the finest level and challenging multi-frequency problems.
This talk is based on joint work with Victorita Dolean (University of Strathclyde, Côte d’Azur University), Siddhartha Mishra, and Ben Moseley (ETH Zürich).
References
[1] Victorita Dolean, Alexander Heinlein, Siddhartha Mishra, and Ben Moseley. Finite basis physics-informed neural networks as a Schwarz domain decomposition method, November 2022. arXiv:2211.05560 [physics].
[2] Victorita Dolean, Alexander Heinlein, Siddhartha Mishra, and Ben Moseley. Multilevel domain decomposition-based architectures for physics-informed neural networks, June 2023. arXiv:2306.05486 [cs, math].
[3] Ben Moseley, Andrew Markham, and Tarje Nissen-Meyer. Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations. Advances in Computational Mathematics, 49(4):62, July 2023.
[4] M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686–707, 2019.