Table of Links
Abstract and 1. Introduction
1.1. Introductory remarks
1.2. Basics of neural networks
1.3. About the entropy of direct PINN methods
1.4. Organization of the paper
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Non-diffusive neural network solver for one dimensional scalar HCLs
2.1. One shock wave
2.2. Arbitrary number of shock waves
2.3. Shock wave generation
2.4. Shock wave interaction
2.5. Non-diffusive neural network solver for one dimensional systems of CLs
2.6. Efficient initial wave decomposition
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Gradient descent algorithm and efficient implementation
3.1. Classical gradient descent algorithm for HCLs
3.2. Gradient descent and domain decomposition methods
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Numerics
4.1. Practical implementations
4.2. Basic tests and convergence for 1 and 2 shock wave problems
4.3. Shock wave generation
4.4. Shock-Shock interaction
4.5. Entropy solution
4.6. Domain decomposition
4.7. Nonlinear systems
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Conclusion and References
5. Conclusion
In this paper, we have proposed an original method for solving hyperbolic conservation laws using a non-diffusive neural network method. The principle of the method is to track
the DLs and solve the conservation laws in the subdomains the DLs define, and where the solution is smooth. We have used neural networks to approximate the DLs and the solution of conservation laws in each of the subdomains. The networks are trained by minimizing a loss functional that measures the (norm of the) residues of conservation laws, boundary and initial conditions and Rankine-Hugoniot conditions. This approach allows for a computation of shock waves without using a weak formulation of HCL. Other functions could be used to approximate the solution and the DLs, but neural networks provide interesting features. Indeed, they allow for automatic differentiation, which avoids the approximation errors for the derivatives, facilitates the implementation of the algorithm, and allows for an accurate/diffusion-free computation of shock waves, solutions to nonlinear hyperbolic conservation laws.
When the global loss functional approach is slow to converge, a rapidly convergent and embarrassingly parallel (Schwarz) domain decomposition method can be used. The latter allows for a decoupling of the optimization procedure thanks to the optimization of local neural networks approximating from which the global approximate solution is reconstructed, as shown in [23].
In a future work, we plan to apply to this methodology to higher dimensional problems.
CRediT authorship contribution statement
The authors have contributed equally.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Data availability
No data was used for the research described in this article.
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Authors:
(1) Emmanuel LORIN, School of Mathematics and Statistics, Carleton University, Ottawa, Canada, K1S 5B6 and Centre de Recherches Mathematiques, Universit´e de Montr´eal, Montreal, Canada, H3T 1J4 ([email protected]);
(2) Arian NOVRUZI, a Corresponding Author from Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5, Canada ([email protected]).
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This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license.
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