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Non-diffusive neural network method for hyperbolic conservation laws

Emmanuel Lorin, Arian Novruzi

TL;DR

This work introduces a non-diffusive neural network (NDNN) framework for solving hyperbolic conservation laws by explicitly tracking discontinuity lines (DLs) and solving smooth problems in subdomains separated by RH-based DLs. DLs are learned as time-dependent neural networks while local solutions are captured by dedicated subdomain networks, with a nonlinear loss that enforces PDE residuals, initial/boundary conditions, and Rankine-Hugoniot jump conditions. The approach extends to systems and supports shock generation, interactions, and efficient initial wave decomposition, complemented by a domain-decomposition strategy (DDM) that enables embarrassingly parallel training. Numerical experiments across scalar and system cases demonstrate diffusion-free shock capturing, accurate entropy-satisfying behavior, and scalability through SWR-like decomposition. The method offers a practical, differentiable alternative to traditional diffusion-augmented PINNs for sharp shock tracking in hyperbolic conservation laws.

Abstract

In this paper we develop a non-diffusive neural network (NDNN) algorithm for accurately solving weak solutions to hyperbolic conservation laws. The principle is to construct these weak solutions by computing smooth local solutions in subdomains bounded by discontinuity lines (DLs), the latter defined from the Rankine-Hugoniot jump conditions. The proposed approach allows to efficiently consider an arbitrary number of entropic shock waves, shock wave generation, as well as wave interactions. Some numerical experiments are presented to illustrate the strengths and properties of the algorithms.

Non-diffusive neural network method for hyperbolic conservation laws

TL;DR

This work introduces a non-diffusive neural network (NDNN) framework for solving hyperbolic conservation laws by explicitly tracking discontinuity lines (DLs) and solving smooth problems in subdomains separated by RH-based DLs. DLs are learned as time-dependent neural networks while local solutions are captured by dedicated subdomain networks, with a nonlinear loss that enforces PDE residuals, initial/boundary conditions, and Rankine-Hugoniot jump conditions. The approach extends to systems and supports shock generation, interactions, and efficient initial wave decomposition, complemented by a domain-decomposition strategy (DDM) that enables embarrassingly parallel training. Numerical experiments across scalar and system cases demonstrate diffusion-free shock capturing, accurate entropy-satisfying behavior, and scalability through SWR-like decomposition. The method offers a practical, differentiable alternative to traditional diffusion-augmented PINNs for sharp shock tracking in hyperbolic conservation laws.

Abstract

In this paper we develop a non-diffusive neural network (NDNN) algorithm for accurately solving weak solutions to hyperbolic conservation laws. The principle is to construct these weak solutions by computing smooth local solutions in subdomains bounded by discontinuity lines (DLs), the latter defined from the Rankine-Hugoniot jump conditions. The proposed approach allows to efficiently consider an arbitrary number of entropic shock waves, shock wave generation, as well as wave interactions. Some numerical experiments are presented to illustrate the strengths and properties of the algorithms.
Paper Structure (36 sections, 59 equations, 17 figures, 2 algorithms)

This paper contains 36 sections, 59 equations, 17 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Introductory remarks
  3. Basics of neural networks
  4. About the entropy of direct PINN methods
  5. Organization of the paper
  6. Non-diffusive neural network solver for one dimensional scalar HCLs
  7. One shock wave
  8. Arbitrary number of shock waves
  9. Shock wave generation
  10. Shock wave interaction
  11. Non-diffusive neural network solver for one dimensional systems of CLs
  12. Efficient initial wave decomposition
  13. Initial wave decomposition for arbitrary $m$
  14. Initial wave decomposition for 2 equation systems $(m=2,\, \mu=2)$
  15. Initial wave decomposition for 3 equation systems $(m=3,\, \mu=3$)
  16. ...and 21 more sections

Figures (17)

  • Figure 1: Experiment 1. (Left) Neural network solution. (Middle) Solution of reference. (Right) Direct PINN solution.
  • Figure 2: Experiment 1. Loss function.
  • Figure 3: Experiment 2.(Left) Loss function. (Right) Space-time solution.
  • Figure 4: Experiment 2. (Left) Godunov scheme solution at CFL=0.9 and neural network solution at time $T=0.5$.
  • Figure 5: Experiment 3. (Left) Space-time solution (number of neurons is equal to $5$ with learning nodes is $1800$). (Right) Loss function.
  • ...and 12 more figures