We propose a conservative physics-informed neural network (cPINN) on discrete domains for nonlinear conservation laws. Here, the term discrete domain represents the discrete sub-domains obtained after division of the computational domain, where PINN is applied and the conservation property of cPINN is obtained by enforcing the flux continuity in the strong form along the sub-domain interfaces. In case of hyperbolic conservation laws, the convective flux contributes at the interfaces, whereas in case of viscous conservation laws, both convective and diffusive fluxes contribute. Apart from the flux continuity condition, an average solution (given by two different neural networks) is also enforced at the common interface between two sub-domains. One can also employ a deep neural network in the domain, where the solution may have complex structure, whereas a shallow neural network can be used in the sub-domains with relatively simple and smooth solutions. Another advantage of the proposed method is the additional freedom it gives in terms of the choice of optimization algorithm and the various training parameters like residual points, activation function, width and depth of the network etc. Various forms of errors involved in cPINN such as optimization, generalization and approximation errors and their sources are discussed briefly. In cPINN, locally adaptive activation functions are used, hence training the model faster compared to its fixed counterparts. Both, forward and inverse problems are solved using the proposed method. Various test cases ranging from scalar nonlinear conservation laws like Burgers, Korteweg–de Vries (KdV) equations to systems of conservation laws, like compressible Euler equations are solved. The lid-driven cavity test case governed by incompressible Navier–Stokes equation is also solved and the results are compared against a benchmark solution. The proposed method enjoys the property of domain decomposition with separate neural networks in each sub-domain, and it efficiently lends itself to parallelized computation, where each sub-domain can be assigned to a different computational node.
If you make use of the code or the idea/algorithm in your work, please cite our papers
References: For Domain Decomposition based PINN framework
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A.D.Jagtap, G.E.Karniadakis, Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations, Commun. Comput. Phys., Vol.28, No.5, 2002-2041, 2020. (https://doi.org/10.4208/cicp.OA-2020-0164)
@article{jagtap2020extended, title={Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations}, author={Jagtap, Ameya D and Karniadakis, George Em}, journal={Communications in Computational Physics}, volume={28}, number={5}, pages={2002--2041}, year={2020} } -
A.D.Jagtap, E. Kharazmi, G.E.Karniadakis, Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems, Computer Methods in Applied Mechanics and Engineering, 365, 113028 (2020). (https://doi.org/10.1016/j.cma.2020.113028)
@article{jagtap2020conservative, title={Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems}, author={Jagtap, Ameya D and Kharazmi, Ehsan and Karniadakis, George Em}, journal={Computer Methods in Applied Mechanics and Engineering}, volume={365}, pages={113028}, year={2020}, publisher={Elsevier} } -
K. Shukla, A.D. Jagtap, G.E. Karniadakis, Parallel Physics-Informed Neural Networks via Domain Decomposition, Journal of Computational Physics 447, 110683, (2021).
@article{shukla2021parallel, title={Parallel Physics-Informed Neural Networks via Domain Decomposition}, author={Shukla, Khemraj and Jagtap, Ameya D and Karniadakis, George Em}, journal={Journal of Computational Physics}, volume={447}, pages={110683}, year={2021}, publisher={Elsevier} }
References: For adaptive activation functions
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A.D. Jagtap, K.Kawaguchi, G.E.Karniadakis, Adaptive activation functions accelerate convergence in deep and physics-informed neural networks, Journal of Computational Physics, 404 (2020) 109136. (https://doi.org/10.1016/j.jcp.2019.109136)
@article{jagtap2020adaptive, title={Adaptive activation functions accelerate convergence in deep and physics-informed neural networks}, author={Jagtap, Ameya D and Kawaguchi, Kenji and Karniadakis, George Em}, journal={Journal of Computational Physics}, volume={404}, pages={109136}, year={2020}, publisher={Elsevier} } -
A.D.Jagtap, K.Kawaguchi, G.E.Karniadakis, Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 20200334, 2020. (http://dx.doi.org/10.1098/rspa.2020.0334).
@article{jagtap2020locally, title={Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks}, author={Jagtap, Ameya D and Kawaguchi, Kenji and Em Karniadakis, George}, journal={Proceedings of the Royal Society A}, volume={476}, number={2239}, pages={20200334}, year={2020}, publisher={The Royal Society} } -
A.D. Jagtap, Y. Shin, K. Kawaguchi, G.E. Karniadakis, Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functions, Neurocomputing, 468, 165-180, 2022. (https://www.sciencedirect.com/science/article/pii/S0925231221015162)
@article{jagtap2022deep, title={Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functions}, author={Jagtap, Ameya D and Shin, Yeonjong and Kawaguchi, Kenji and Karniadakis, George Em}, journal={Neurocomputing}, volume={468}, pages={165--180}, year={2022}, publisher={Elsevier} }
Recommended software versions: TensorFlow 1.14, Python 3.6, Latex (for plotting figures)
Example details: Conservative PINN code with 4 spatial subdomains for the one-dimensional Burgers equations.
For any queries regarding the cPINN code, feel free to contact me : ameya_jagtap@brown.edu, ameyadjagtap@gmail.com