Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Abstract and 1. Introduction 1.1. Introductory remarks 1.1. Introductory remarks 1.2. Basics of neural networks 1.2. Basics of neural networks 1.3. About the entropy of direct PINN methods 1.3. About the entropy of direct PINN methods 1.4. Organization of the paper 1.4. Organization of the paper 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


Gradient descent algorithm and efficient implementation
3.1. Classical gradient descent algorithm for HCLs
3.2. Gradient descent and domain decomposition methods


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


Conclusion and References 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 Non-diffusive neural network solver for one dimensional scalar HCLs Non-diffusive neural network solver for one dimensional scalar HCLs Non-diffusive neural network solver for one dimensional scalar HCLs 2.1. One shock wave 2.1. One shock wave 2.2. Arbitrary number of shock waves 2.2. Arbitrary number of shock waves 2.3. Shock wave generation 2.3. Shock wave generation 2.4. Shock wave interaction 2.4. Shock wave interaction 2.5. Non-diffusive neural network solver for one dimensional systems of CLs 2.5. Non-diffusive neural network solver for one dimensional systems of CLs 2.6. Efficient initial wave decomposition 2.6. Efficient initial wave decomposition Gradient descent algorithm and efficient implementation
3.1. Classical gradient descent algorithm for HCLs
3.2. Gradient descent and domain decomposition methods Gradient descent algorithm and efficient implementation Gradient descent algorithm and efficient implementation 3.1. Classical gradient descent algorithm for HCLs 3.1. Classical gradient descent algorithm for HCLs 3.2. Gradient descent and domain decomposition methods 3.2. Gradient descent and domain decomposition methods 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 Numerics Numerics 4.1. Practical implementations 4.1. Practical implementations 4.2. Basic tests and convergence for 1 and 2 shock wave problems 4.2. Basic tests and convergence for 1 and 2 shock wave problems 4.3. Shock wave generation 4.3. Shock wave generation 4.4. Shock-Shock interaction 4.4. Shock-Shock interaction 4.5. Entropy solution 4.5. Entropy solution 4.6. Domain decomposition 4.6. Domain decomposition 4.7. Nonlinear systems 4.7. Nonlinear systems Conclusion and References Conclusion and References Conclusion and References 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. 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(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 (elorin@math.carleton.ca);
(2) Arian NOVRUZI, a Corresponding Author from Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5, Canada (novruzi@uottawa.ca). Authors: 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 (elorin@math.carleton.ca); (2) Arian NOVRUZI, a Corresponding Author from Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5, Canada (novruzi@uottawa.ca). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

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A Non-Diffusive Neural Network Method for Solving Hyperbolic Conservation Laws

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