Forward and Inverse Problems with the Elastic Wave Equation: A Comparison of Traditional Numerical and Machine-Learning Methods

Tamanna Saini, & Brittany A. Erickson

Submitted September 7, 2025, SCEC Contribution #14653, 2025 SCEC Annual Meeting Poster #TBD

The seismic wave equation, a fundamental tool in modeling earthquakes, presents unique computational challenges, especially in accurately capturing wave propagation dynamics. Traditional numerical approaches, including finite difference methods (FDM), have long been used to solve the forward problem, providing well-established, theoretically robust frameworks. However, physics-informed neural networks offer a promising alternative, integrating the governing physical laws directly into the learning process and potentially improving efficiency and adaptability in complex scenarios without mesh generation. We perform a detailed comparative analysis of traditional numerical methods and physics-informed neural networks (PINNs) to solve both forward and inverse problems related to the elastic wave equation. By analyzing the computational benefits and limitations of these two approaches, we hope to identify the comparative strengths of PINNs versus traditional methodologies.
For the forward problem, we systematically benchmark three numerical strategies for the elastic wave equation—FDM, soft-enforced PINNs, and hard-enforced PINNs—under two optimization schemes (Adam and L-BFGS). FDM delivers the lowest errors and shortest runtimes when grid resolution is sufficient, but hard-enforced PINNs narrow the gap dramatically. With hard enforcement, L-BFGS training reaches near-FDM accuracy. Soft enforcement proves sensitive to sharp localized features, whereas hard enforcement stabilizes convergence and preserves accuracy. Across all trials, L-BFGS consistently outperforms Adam on both speed and fidelity, highlighting optimizer choice as a key design factor. Overall, the results position hard-enforced, L-BFGS-optimized PINNs as an attractive mesh-free alternative when exact constraint satisfaction is possible and grid-based methods become cumbersome or expensive. Next, we consider a classical problem of inferring elastic moduli given limited surface observations. Given the material density, we set up the PINN to solve for the material displacement as well as the shear modulus. Two networks are defined to approximate both the displacement field and the shear modulus. The composite loss function couples the PDE residual, initial/boundary conditions, surface observations, and an optional regularizer on the surface shear modulus. A warm-up phase with Adam optimizes both nets on mini-batches, followed by full-batch L-BFGS to reach machine precision.

Key Words
Physics Informed Neural Network, Finite Difference Methods, Traditional Numerical Methods, Inverse Problem

Citation
Saini, T., & Erickson, B. A. (2025, 09). Forward and Inverse Problems with the Elastic Wave Equation: A Comparison of Traditional Numerical and Machine-Learning Methods. Poster Presentation at 2025 SCEC Annual Meeting.

Related Projects & Working Groups
Research Computing (RC)