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 4.3. Shock wave generation In this section, we demonstrate the potential of our algorithms to handle shock wave generation, as described in Subsection 2.3. One of the strengths of the proposed algorithm is that it does not require to know the initial position&time of birth, in order to accurately track the DLs. Recall that the principle is to assume that in a given (sub)domain and from a smooth function a shock wave will eventually be generated. Hence we decompose the corresponding (sub)domain in two subdomains and consider three neural networks: two neural networks will approximate the solution in each subdomain, and one neural network will approximate the DL. As long as the shock wave is not generated (say for t < t∗ ), the global solution remains smooth and the Rankine-Hugoniot condition is trivially satisfied (null jump); hence the DL for t < t∗ does not have any meaning. Experiment 4. We again consider the inviscid Burgers’ equation, Ω × [0, T] = (−1, 2) × [0, 0.5] and the initial condition 4.4. Shock-Shock interaction In this subsection, we are proposing a test involving the interaction of two shock waves merging to generate a third shock wave. As explained in Subsection 2.4, in this case it is necessary re-decompose the full domain once the two shock waves have interacted. 4.5. Entropy solution We propose here an experiment dedicated to the computation of the viscous shock profiles and rarefaction waves and illustrating the discussion from Subsection 1.3. In this example, a regularized non-entropic shock is shown to be “destabilized” into rarefaction wave by the direct PINN method. 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). 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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Shocks, Collisions, and Entropy—Neural Networks Handle It All

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