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 solutions of partial differential equations (PDEs) using neural networks. The network models be can trained in a data-driven and/or physics-informed way, that is, using reference data (from simulations or measurements) or a loss function based on the PDE, respectively.

In physics-informed neural networks (PINNs) [4], simple feedforward neural networks are employed to discretize the PDEs, and a single network is trained to approximate the solution of one specific boundary value problem. The loss function may include a combination of 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 iteration counts. Therefore, in the first part of the talk, domain decomposition-based network architectures improving the training performance using the finite basis physics-informed neural network (FBPINN) approach [3, 1] will be discussed. It is based on joint work with Victorita Dolean (University of Strathclyde, Côte d’Azur University), Siddhartha Mishra, and Ben Moseley (ETH Zürich).

In the second part of the talk, surrogate models for computational fluid dynamics (CFD) simulations based on convolutional neural networks (CNNs) [2] will be discussed. In particular, the network is trained to approximate a solution operator, taking a representation of the geometry as input and the solution field(s) as output. In contrast to the classical PINN approach, a single network is trained to approximate a variety of boundary value problems. This makes the approach potentially very efficient. As in the PINN approach, data as well as the residual of the PDE may be used in the loss function for training the network. The second part of the talk is based on joint work with Matthias Eichinger, Viktor Grimm, and Axel Klawonn (University of Cologne).

[1] V. Dolean, A. Heinlein, S. Mishra, and B. Moseley. Finite basis physics-informed neural networks as a Schwarz domain decomposition method, November 2022. arXiv:2211.05560.

[2] M. Eichinger, A. Heinlein, and A. Klawonn. Surrogate convolutional neural network models for steady computational fluid dynamics simulations. Electronic Transactions on Numerical Analysis, 56:235–255, 2022.

[3] B. Moseley, A. Markham, and T. Nissen-Meyer. Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations, July 2021. arXiv:2107.07871.

[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.