{
  "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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  "affiliations": [
    "Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Peshawar, Khyber Pakhtunkhwa, Pakistan",
    "Abdus Salam School of Mathematical Sciences, GC University Lahore, Lahore, Pakistan",
    "Department of Mathematics, University of Hafr Al Batin, Hafar Al-Batin, Saudi Arabia"
  ],
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      "name": "Muhammad Israr",
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      "name": "Shahbaz Ahmad",
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    {
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      "name": "F. D. Zaman",
      "orcid": "0000-0002-6498-0664"
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    {
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        2
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      "name": "Iftikhar Ali"
    }
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  "doi": "10.1371/journal.pone.0360636",
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  "published": "2026-10-27",
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  "title": "Causality-enhanced and symmetry-preserving PINNs for damped elastodynamic PDEs",
  "title_text": "Causality-enhanced and symmetry-preserving PINNs for damped elastodynamic PDEs",
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  "updated_at": "2026-10-31T12:23:13Z",
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