Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems

By Ameya Dilip Jagtap, E. Kharazmi, G. Em Karniadakis. 2020

Abstract 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.

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Citations in the corpus

Physics informed deep learning for computational elastodynamics without labeled data (2020, methodology): “performance, the PINN framework has also been extended via incorporating variational/energy formulations of the residual physics loss function [37{40], distributed learning using domain decomposition [41,42], and uncertainty quantication via variational/Bayesian inference [26,43{45]. It is important”
Analyses of internal structures and defects in materials using physics-informed neural networks (2022, methodology): “4 of 12 can be directly estimated using the standard formulation of PINNs for inverse problems (12, 18, 41, 42).”
Physics‐Informed Neural Networks (PINNs) for Wave Propagation and Full Waveform Inversions (2021, methodology): “The interested reader is directed to Jagtap et al., 2020a&b for a discussion of adaptive activation functions and their effect on convergence.”
Physics-informed neural networks for inverse problems in supersonic flows (2022, methodology): “In recent years physics-informed neural networks (PINNs) [1] emerged as an alternative simple method to solve many problems in computational science and engineering, see, for example [2, 3, 4, 5, 6, 7, 8, 9, 10, 4, 11, 12, 13, 14, 15].”
Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems (2021, methodology): “First, they suffer from high training cost since the point-wise formulation requires huge amount of AD computations on vast collocation points in a high-dimensional spatiotemporal (and parameter) domain [16,17].”
Physics-Informed Neural Network for Modelling the Thermochemical Curing Process of Composite-Tool Systems During Manufacture (2020, methodology): “Due to these advantages, many variations of the PINN approach have been successfully used to solve PDEs in a wide range of engineering applications [24, 29–37].”
Extended Physics-informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition based Deep Learning Framework for Nonlinear Partial Differential Equations (2020, methodology): “This issue was first addressed in the conservative PINN (cPINN) method for conservation laws [46] by employing the domain decomposition approach in PINN framework.”
Parallel Physics-Informed Neural Networks via Domain Decomposition (2021, methodology): “To reduce the computational cost, Jagtap et al. [16] introduced a domain decomposition-based PINN for conservation laws, namely conservative PINN (cPINN), where continuity of the states as well as their ﬂuxes across subdomain interfaces is used to stitch the subdomains together.”
Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs (2020, background): “There has been an explosive growth of papers that present PINNs for different applications and a very incomplete list of references includes [15,27,29,39,44,51,54,61].”
Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, methodology): “As done in Jagtap et al [71], cPINN splits the domain into a number of small subdomains in which multiple neural networks with different architectures (known as sub-PINN networks) can be used to solve the same underlying PDE.”
Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks (2020, methodology): “One of the main limitation of PINN algorithm is its high computational cost for high-dimensional optimization problem, which is addressed in [13] by employing the domain decomposition approach.”
Physics-informed neural networks (PINNs) for fluid mechanics: a review (2021, methodology): “In addition, the variational hp-VPINN considered domain decomposition, and similar pointwise versions were also studied in CPINN [26] and XPINN [27].”