ABSTRACT
It is widely hoped that artificial intelligence will boost data-driven surrogate models in science and engineering. However, fundamental spatial aspects of AI surrogate models remain under-studied. We investigate the ability of neural-network surrogate models to predict solutions to PDEs under variable boundary values. We do not wish to retrain the model when the boundary values change but to make them inputs to the model and infer the solution of the PDE under those boundary conditions. Such a capability is essential to making AI-based surrogate models practically useful. While simple feedforward networks are used for one-dimensional (1D) Poisson equation, an encoder-decoder architecture with a tensor-product layer is developed for the two-dimensional Poisson equation posed on a rectangular domain. We show that it is indeed possible to infer solutions to PDEs from variable boundary data using neural networks in this relatively simple setting, and point to future directions. 1
- Daniel Arndt, Wolfgang Bangerth, Maximilian Bergbauer, Marco Feder, Marc Fehling, Johannes Heinz, Timo Heister, Luca Heltai, Martin Kronbichler, Matthias Maier, Peter Munch, Jean-Paul Pelteret, Bruno Turcksin, David Wells, and Stefano Zampini. 2023, accepted for publication. The deal.II Library, Version 9.5. Journal of Numerical Mathematics (2023, accepted for publication). https://dealii.org/deal95-preprint.pdfGoogle Scholar
- Steven L. Brunton, Bernd R. Noack, and Petros Koumoutsakos. 2020. Machine Learning for Fluid Mechanics. Annual Review of Fluid Mechanics 52, 1 (2020), 477–508. https://doi.org/10.1146/annurev-fluid-010719-060214Google ScholarCross Ref
- Yuyao Chen, Lu Lu, George Em Karniadakis, and Luca Dal Negro. 2020. Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Optics Express 28, 8 (April 2020), 11618–11633. https://doi.org/10.1364/OE.384875 Publisher: Optica Publishing Group.Google ScholarCross Ref
- Kai Fukami, Kazuto Hasegawa, Taichi Nakamura, Masaki Morimoto, and Koji Fukagata. 2021. Model order reduction with neural networks: application to laminar and turbulent flows. SN Computer Science 2, 467 (2021). https://doi.org/10.1007/s42979-021-00867-3Google ScholarDigital Library
- Viktor Grimm, Alexander Heinlein, and Axel Klawonn. 2023. Learning the solution operator of two-dimensional incompressible Navier-Stokes equations using physics-aware convolutional neural networks. arxiv:2308.02137 [math.NA]Google Scholar
- Stephen Hudson, Jeffrey Larson, John-Luke Navarro, and Stefan M. Wild. 2022. libEnsemble: A Library to Coordinate the Concurrent Evaluation of Dynamic Ensembles of Calculations. IEEE Transactions on Parallel and Distributed Systems 33, 4 (2022), 977–988. https://doi.org/10.1109/tpds.2021.3082815Google ScholarCross Ref
- Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. 2022. Neural Operator: Learning maps between function spaces. https://doi.org/10.48550/arXiv.2108.08481 arXiv:2108.08481 [cs, math].Google ScholarCross Ref
- Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. 2020. Neural Operator: Graph Kernel Network for partial differential equations. https://doi.org/10.48550/arXiv.2003.03485 arXiv:2003.03485 [cs, math, stat].Google ScholarCross Ref
- Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. 2021. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature Machine Intelligence 3, 3 (mar 2021), 218–229. https://doi.org/10.1038/s42256-021-00302-5Google ScholarCross Ref
- Christopher Rackauckas, Yingbo Ma, Julius Martensen, Collin Warner, Kirill Zubov, Rohit Supekar, Dominic Skinner, Ali Ramadhan, and Adam Edelman. 2021. Universal differential equations for scientific machine learning. https://doi.org/10.48550/arXiv.2001.04385 arXiv:2001.04385 [cs].Google ScholarCross Ref
- M. Raissi, P. Perdikaris, and G. E. Karniadakis. 2019. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378 (Feb. 2019), 686–707. https://doi.org/10.1016/j.jcp.2018.10.045Google ScholarCross Ref
- Ricardo Vinuesa and Steven L. Brunton. 2022. Enhancing computational fluid dynamics with machine learning. Nature Computational Science 2, 6 (jun 2022), 358–366. https://doi.org/10.1038/s43588-022-00264-7Google ScholarCross Ref
- Tingtao Zhou, Xuan Wan, Daniel Zhengyu Huang, Zongyi Li, Zhiwei Peng, Anima Anandkumar, John F. Brady, Paul W. Sternberg, and Chiara Daraio. 2023. AI-aided Geometric Design of Anti-infection Catheters. arxiv:2304.14554 [physics.med-ph]Google Scholar
Index Terms
- Tencoder: tensor-product encoder-decoder architecture for predicting solutions of PDEs with variable boundary data
Recommendations
Spatially Dispersionless, Unconditionally Stable FC---AD Solvers for Variable-Coefficient PDEs
We present fast, spatially dispersionless and unconditionally stable high-order solvers for partial differential equations (PDEs) with variable coefficients in general smooth domains. Our solvers, which are based on (i) A certain "Fourier continuation" (...
Multigrid Methods for Implicit Runge--Kutta and Boundary Value Method Discretizations of Parabolic PDEs
Advanced time discretization schemes for stiff systems of ordinary differential equations (ODEs), such as implicit Runge--Kutta and boundary value methods, have many appealing properties. However, the resulting systems of equations can be quite large and ...
Comments