TY  - JOUR
AU  - Ferrer-Sanchez, A.
AU  - Martin-Guerrero, J.
AU  - Ruiz de Austri, R.
AU  - Torres-Forne, A.
AU  - Font, J. A.
PY  - 2024
DA  - 2024//
TI  - Gradient-annihilated PINNs for solving Riemann problems: Application to relativistic hydrodynamics
T2  - Comput. Meth. Appl. Mech. Eng.
JO  - Computer Methods in Applied Mechanics and Engineering
SP  - 116906
EP  - 18pp
VL  - 424
PB  - Elsevier Science Sa
KW  - Riemann problem
KW  - Euler equations
KW  - Machine learning
KW  - Neural networks
KW  - Relativistic hydrodynamics
AB  - We present a novel methodology based on Physics-Informed Neural Networks (PINNs) for solving systems of partial differential equations admitting discontinuous solutions. Our method, called Gradient-Annihilated PINNs (GA-PINNs), introduces a modified loss function that forces the model to partially ignore high-gradients in the physical variables, achieved by introducing a suitable weighting function. The method relies on a set of hyperparameters that control how gradients are treated in the physical loss. The performance of our methodology is demonstrated by solving Riemann problems in special relativistic hydrodynamics, extending earlier studies with PINNs in the context of the classical Euler equations. The solutions obtained with the GA-PINN model correctly describe the propagation speeds of discontinuities and sharply capture the associated jumps. We use the relative l(2) error to compare our results with the exact solution of special relativistic Riemann problems, used as the reference ''ground truth'', and with the corresponding error obtained with a second-order, central, shock-capturing scheme. In all problems investigated, the accuracy reached by the GA-PINN model is comparable to that obtained with a shock-capturing scheme, achieving a performance superior to that of the baseline PINN algorithm in general. An additional benefit worth stressing is that our PINN-based approach sidesteps the costly recovery of the primitive variables from the state vector of conserved variables, a well-known drawback of grid-based solutions of the relativistic hydrodynamics equations. Due to its inherent generality and its ability to handle steep gradients, the GA-PINN methodology discussed in this paper could be a valuable tool to model relativistic flows in astrophysics and particle physics, characterized by the prevalence of discontinuous solutions.
SN  - 0045-7825
UR  - https://arxiv.org/abs/2305.08448
UR  - https://doi.org/10.1016/j.cma.2024.116906
DO  - 10.1016/j.cma.2024.116906
LA  - English
N1  - WOS:001221797400001
ID  - Ferrer-Sanchez_etal2024
ER  -