Solving the Elastic Wave Equation with Physics-Informed Neural Networks: A Robust and Critical Assessment

• URL: http://arxiv.org/abs/2609.07983v1
• Date: Mon, 07 Sep 2026 21:01:29 GMT
• Title: Solving the Elastic Wave Equation with Physics-Informed Neural Networks: A Robust and Critical Assessment
• Abstract summary: Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving Partial Differential Equations (PDEs)<n>We provide a robust and critical assessment of PINNs for solving the elastic wave equation in seismology.
• Score: 0.42970700836450476
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Physics-Informed Neural Networks (PINNs) have recently emerged as a promising approach for solving Partial Differential Equations (PDEs), offering a meshfree alternative that integrates physical principles into the learning process. This presents a new paradigm compared to traditional discretization methods and purely data-driven machine learning techniques. While promising, PINNs are not a panacea; they inherit challenges such as spectral bias and unstable convergence. Moreover, their potential in seismology remains largely unexplored. In this work, we provide a robust and critical assessment of PINNs for solving the elastic wave equation in seismology. We investigate the performance of PINNs on problems with varying degrees of complexity across various seismic sources and parameter models, from constant to highly heterogeneous settings. A pivotal aspect of our work involves investigating whether embedding physical principles directly into the network architecture enhances convergence and accuracy. We test an extensive range of neural architecture designs, from unrestricted, uninformed PINNs to highly specialized ones. We find that integrating an understanding of wave physics into the network design significantly improves accuracy. For instance, introducing a custom wavelet or plane wave layer, coupled with encoder and decoder layers, consistently yields a relative $L_2$ error approximately half that of the standard PINN, as evidenced across numerous experiments. We further demonstrate that this novel architecture enhances accuracy when applied to the acoustic wave equation, underlying the versatility of our network. Another key contribution of our research is the successful conditioning of PINNs on seismic source locations. This signifies a considerable advancement towards rapid seismic hazard detection and seismic analysis.

Related papers

• Learning Metamaterial Eigenmodes with Wavelet-Encoded Fourier Neural Operators [57.64362886310329]
  We show that a machine learning surrogate based on neural operators can learn and predict multiple eigenmodes of the elastic wave equation.<n>For metamaterial design, the resulting surrogate accelerates the simulation stage of the design cycle by three orders of magnitude relative to finite element analysis on a consumer-grade CPU.<n>These results also carry broader implications for designing input encodings in other multi-mode PDE solvers based on spectral neural operators.
  arXiv  Detail & Related papers  (2026-09-08T01:21:46Z)
• Physics-Informed Neural Networks with Architectural Physics Embedding for Large-Scale Wave Field Reconstruction [8.953088273349499]
  Physics-informed neural networks (PINNs) integrate physical principles into machine learning models.<n>PINNs embed physical principles only in loss functions, leading to slow convergence, optimization instability, and spectral bias.<n>This work introduces architecture physics embedded (PE)-PINN, which integrates additional physical guidance directly into the neural network architecture.
  arXiv  Detail & Related papers  (2026-02-13T04:28:23Z)
• Physics-Informed Neural Network Frameworks for the Analysis of Engineering and Biological Dynamical Systems Governed by Ordinary Differential Equations [0.0]
  PINNs offer a powerful approach for handling challenging numerical scenarios.<n> PINNs can achieve superior results by embedding physical laws directly into the learning process.
  arXiv  Detail & Related papers  (2025-10-28T04:00:59Z)
• ProPINN: Demystifying Propagation Failures in Physics-Informed Neural Networks [71.02216400133858]
  Physics-informed neural networks (PINNs) have earned high expectations in solving partial differential equations (PDEs)<n>Previous research observed the propagation failure phenomenon of PINNs.<n>This paper provides a formal and in-depth study of propagation failure and its root cause.
  arXiv  Detail & Related papers  (2025-02-02T13:56:38Z)
• Towards Model Discovery Using Domain Decomposition and PINNs [44.99833362998488]
  The study evaluates the performance of two approaches, namely Physics-Informed Neural Networks (PINNs) and Finite Basis Physics-Informed Neural Networks (FBPINNs) We find a better performance for the FBPINN approach compared to the vanilla PINN approach, even in cases with data from only a quasi-stationary time domain with few dynamics.
  arXiv  Detail & Related papers  (2024-10-02T14:38:37Z)
• An efficient wavelet-based physics-informed neural networks for singularly perturbed problems [0.0]
  Physics-informed neural networks (PINNs) are a class of deep learning models that utilize physics in the form of differential equations to address complex problems.<n>We present a wavelet-based PINNs model to tackle solutions of differential equations with rapid oscillations, steep gradients, or singular behavior.<n>The proposed model significantly improves with traditional PINNs, recently developed wavelet-based PINNs, and other state-of-the-art methods.
  arXiv  Detail & Related papers  (2024-09-18T10:01:37Z)
• Point Neuron Learning: A New Physics-Informed Neural Network Architecture [8.545030794905584]
  This paper proposes a new physics-informed neural network architecture. It embeds the fundamental solution of the wave equation into the network architecture, enabling the learned model to strictly satisfy the wave equation. Compared to other PINN methods, our approach directly processes complex numbers and offers better interpretability and generalizability.
  arXiv  Detail & Related papers  (2024-08-30T02:07:13Z)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations [64.78260098263489]
  In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs) We show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.
  arXiv  Detail & Related papers  (2022-10-14T15:01:32Z)
• Auto-PINN: Understanding and Optimizing Physics-Informed Neural Architecture [77.59766598165551]
  Physics-informed neural networks (PINNs) are revolutionizing science and engineering practice by bringing together the power of deep learning to bear on scientific computation. Here, we propose Auto-PINN, which employs Neural Architecture Search (NAS) techniques to PINN design. A comprehensive set of pre-experiments using standard PDE benchmarks allows us to probe the structure-performance relationship in PINNs.
  arXiv  Detail & Related papers  (2022-05-27T03:24:31Z)
• Characterizing possible failure modes in physics-informed neural networks [55.83255669840384]
  Recent work in scientific machine learning has developed so-called physics-informed neural network (PINN) models. We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena even for simple PDEs. We show that these possible failure modes are not due to the lack of expressivity in the NN architecture, but that the PINN's setup makes the loss landscape very hard to optimize.
  arXiv  Detail & Related papers  (2021-09-02T16:06:45Z)
• On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks [0.0]
  We show that neural networks (PINNs) struggle in cases where the target functions to be approximated exhibit high-frequency or multi-scale features. We construct novel architectures that employ multi-scale random observational features and justify how such coordinate embedding layers can lead to robust and accurate PINN models.
  arXiv  Detail & Related papers  (2020-12-18T04:19:30Z)
• A deep learning framework for solution and discovery in solid mechanics [1.4699455652461721]
  We present the application of a class of deep learning, known as Physics Informed Neural Networks (PINN), to learning and discovery in solid mechanics. We explain how to incorporate the momentum balance and elasticity relations into PINN, and explore in detail the application to linear elasticity.
  arXiv  Detail & Related papers  (2020-02-14T08:24:53Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.