A Unified Physics-Informed Neural Network for Modeling Coupled Electro- and Elastodynamic Wave Propagation Using Three-Stage Loss Optimization

Physics-informed neural networks (PINNs) provide a promising framework for solving partial differential equations while embedding the underlying physical laws directly into the learning process. This study presents a PINN-based framework for modeling transient elastodynamic wave propagation in bimaterial systems governed by the axisymmetric equations of linear elasticity. A steel-aluminum specimen representative of a Split Hopkinson Pressure Bar configuration is considered, and the governing elastodynamic equations, together with the corresponding initial, boundary, and interface conditions, are incorporated directly into the network through a physics-informed loss function. High-fidelity finite-element simulations performed using ANSYS Workbench Explicit Dynamics are used for validation and as supplementary data constraints during training. The proposed framework accurately predicts wave transmission and reflection across the bimaterial interface and reproduces axial and radial displacement histories, face-averaged responses, and the dominant stress and strain evolution with close agreement to the finite-element solutions. The trained network further demonstrates the ability to predict wave responses at previously unseen time instants and for modified material properties without requiring additional finite-element simulations, providing a continuous surrogate model for elastodynamic analysis. Mesh-sensitivity studies confirm numerical robustness, while additional material combinations demonstrate the generality of the proposed methodology. The results show that integrating physics-informed neural networks with explicit finite-element analysis provides an accurate and computationally efficient framework for elastodynamic wave propagation in heterogeneous solids, offering an effective surrogate modeling approach for high-rate solid mechanics and impact engineering applications.

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.

Physics-informed neural networks (PINNs) for coupled multiphysics systems suffer systematic accuracy degradation as inter-equation coupling strengthens. We provide a theoretical explanation through neural tangent kernel (NTK) analysis: for linearly coupled systems, the standard NTK's spectral radius grows as $Ω(γ^2)$ with coupling strength $γ$, shrinking the stable learning rate, while block-diagonal Gauss--Newton (GN) preconditioning yields a preconditioned NTK $K_P = JH^{+}J^\top$ whose spectral radius is bounded by $S$ (number of networks), independent of $γ$. Adam's diagonal preconditioning destroys this projector structure -- inflating $λ_{\max}$ far above $S$ for any coupling type -- and its residual-dynamics kernel grows as $Θ(γ)$, placing its stable learning rate strictly between gradient descent and GN. For one-way coupling the limitation is class-wide: no diagonal preconditioner, fixed or adaptive, halves the driving residual in fewer than $Ω(γ)$ iterations ($Ω(γ^2)$ if fixed), whereas block-diagonal GN requires $O(1)$. We verify $Ω(γ^2)$ growth across linearly coupled benchmarks and confirm $λ_{\max}(K_P) = S$ in all three 1D systems, including nonlinearly coupled NP+P. Combining the Kronecker-preconditioned optimizer SOAP with inverse-gradient-norm loss balancing (SOAP+GradNorm) yields coupling-robust accuracy: across 222 experiments spanning three 1D systems and a 2D electroosmotic flow benchmark, SOAP+GradNorm maintains final-epoch $L_2$ accuracy across coupling strengths, with $\leq 2.3\times$ degradation in nonlinear NP+P while Adam+GradNorm fails ($L_2 > 0.1$). SOAP+GradNorm further scales to a 2D, 6-PDE electroosmotic flow at EDL-resolved conditions down to $\varepsilon = 0.01$ -- a regime all prior PINN electrokinetics studies have avoided -- where Adam+GradNorm fails entirely ($L_2 > 0.3$).