Causality-enhanced and symmetry-preserving PINNs for damped elastodynamic PDEs
- 1Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Peshawar, Khyber Pakhtunkhwa, Pakistan
- 2Abdus Salam School of Mathematical Sciences, GC University Lahore, Lahore, Pakistan
- 3Department of Mathematics, University of Hafr Al Batin, Hafar Al-Batin, Saudi Arabia
Abstract
This study develops a causality-enhanced and symmetry-preserving Physics-Informed Neural Network (S-PINN) framework for nonlinear damped elastodynamic partial differential equations. The governing nonlinear elastic wave equation with viscous damping is reduced through Lie symmetry analysis into four invariant ordinary differential equation forms, including scaling, translation, damped, and generalized damped symmetries. The proposed framework combines symmetry-aware solution reconstruction, composite residual and symmetry losses, and causality-based temporal weighting to improve the training stability and physical consistency of PINNs. Numerical experiments are performed using a fully connected neural network with four hidden layers, 128 neurons per layer, the tanh activation function, and the L-BFGS optimizer. The S-PINN solutions are compared with conventional PINNs and exact analytical references for all four symmetry-reduced cases. The results show that the proposed S-PINN framework provides faster convergence, lower loss, and improved solution accuracy. In Case IV, S-PINNs achieve a 35% faster convergence rate during the first 500 epochs and reduce the final loss from 3.8 for PINNs to 2.4. The S-PINN loss stabilizes after approximately 1000 epochs, whereas the PINN loss continues to oscillate with fluctuations up to 15%. The gradient norm remains within 0.5–1.5 for S-PINNs, compared with 0.2–2.5 for PINNs. At α=0.4, the S-PINN solution attains 98% agreement with the exact solution, and its maximum absolute error remains below 0.05, compared with 0.18 for the standard PINN. Contour comparisons further show less than 5% distortion for S-PINNs, while PINNs exhibit distortions up to 18%. These findings demonstrate that embedding Lie symmetries and causal temporal learning into PINNs constrains the solution space to physically admissible manifolds and improves the reliability of neural solvers for nonlinear wave propagation in computational mechanics.
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BibTeX
@article{israr2026causalityenhanced,
title = {{Causality-enhanced and symmetry-preserving PINNs for damped elastodynamic PDEs}},
author = {Muhammad Israr and Shahbaz Ahmad and F. D. Zaman and Iftikhar Ali},
journal = {PLOS One},
year = {2026},
month = oct,
doi = {10.1371/journal.pone.0360636},
url = {https://doi.org/10.1371/journal.pone.0360636},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Causality-enhanced and symmetry-preserving PINNs for damped elastodynamic PDEs
AU - Muhammad Israr
AU - Shahbaz Ahmad
AU - F. D. Zaman
AU - Iftikhar Ali
PY - 2026
DA - 2026/10/27
JO - PLOS One
DO - 10.1371/journal.pone.0360636
UR - https://doi.org/10.1371/journal.pone.0360636
AB - This study develops a causality-enhanced and symmetry-preserving Physics-Informed Neural Network (S-PINN) framework for nonlinear damped elastodynamic partial differential equations. The governing nonlinear elastic wave equation with viscous damping is reduced through Lie symmetry analysis into four invariant ordinary differential equation forms, including scaling, translation, damped, and generalized damped symmetries. The proposed framework combines symmetry-aware solution reconstruction, composite residual and symmetry losses, and causality-based temporal weighting to improve the training stability and physical consistency of PINNs. Numerical experiments are performed using a fully connected neural network with four hidden layers, 128 neurons per layer, the tanh activation function, and the L-BFGS optimizer. The S-PINN solutions are compared with conventional PINNs and exact analytical references for all four symmetry-reduced cases. The results show that the proposed S-PINN framework provides faster convergence, lower loss, and improved solution accuracy. In Case IV, S-PINNs achieve a 35% faster convergence rate during the first 500 epochs and reduce the final loss from 3.8 for PINNs to 2.4. The S-PINN loss stabilizes after approximately 1000 epochs, whereas the PINN loss continues to oscillate with fluctuations up to 15%. The gradient norm remains within 0.5–1.5 for S-PINNs, compared with 0.2–2.5 for PINNs. At α=0.4, the S-PINN solution attains 98% agreement with the exact solution, and its maximum absolute error remains below 0.05, compared with 0.18 for the standard PINN. Contour comparisons further show less than 5% distortion for S-PINNs, while PINNs exhibit distortions up to 18%. These findings demonstrate that embedding Lie symmetries and causal temporal learning into PINNs constrains the solution space to physically admissible manifolds and improves the reliability of neural solvers for nonlinear wave propagation in computational mechanics.
SN - 1932-6203
ER -