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

Source: arXiv - 2602.13811v1

Overview

This paper explores how Physics‑Informed Neural Networks (PINNs) can be used as a mesh‑free solver for a one‑dimensional, coupled electro‑ and elastodynamic wave problem—essentially the equations that describe piezoelectric materials. By embedding the governing PDEs directly into the loss function, the authors demonstrate that a standard feed‑forward network can predict both mechanical displacement and electric potential with only a few percent error, opening the door to fast, differentiable simulations for multi‑physics systems.

Key Contributions

• Unified PINN formulation for the coupled elastodynamic‑electrodynamic system in stress‑charge form.
• Three‑stage loss optimization strategy that balances PDE residuals, boundary/initial conditions, and a physics‑based regularization term to improve training stability.
• Quantitative benchmark showing global relative L₂ errors of 2.34 % (displacement) and 4.87 % (electric potential) on a representative 1‑D piezoelectric wave propagation problem.
• Analysis of error accumulation and stiffness issues specific to coupled eigenvalue problems, providing practical guidance for future PINN deployments in multi‑physics contexts.

Methodology

1. Problem definition – The authors model a linear piezoelectric rod using the stress‑charge formulation, which couples the elastodynamic equation (Newton’s second law for displacement) with the electrodynamic equation (Gauss’s law for electric potential).

2. Network architecture – A simple fully‑connected feed‑forward network takes space‑time coordinates ((x, t)) as input and outputs two fields:

   • Mechanical displacement (u(x,t))
   • Electric potential (\phi(x,t))

3. Physics‑informed loss – The loss function is built from three components:

   • PDE residuals (the differential equations evaluated with automatic differentiation).
   • Boundary & initial condition penalties (enforcing known values at the rod ends and at (t=0)).
   • Stage‑specific regularization that gradually shifts emphasis from boundary conditions to PDE residuals, helping the optimizer avoid getting stuck in stiff regions.

4. Training – The network is trained with Adam followed by L‑BFGS (a quasi‑Newton method) to fine‑tune the solution, a common practice in PINN literature to achieve higher accuracy.

5. Evaluation – After training, the predicted fields are compared against a high‑fidelity finite‑difference reference solution, and global relative L₂ errors are reported.

Results & Findings

• The error remains uniform across the domain, indicating that the PINN captures both wave propagation speed and coupling effects without needing a mesh.
• Three‑stage loss scheduling proved crucial: early stages focus on satisfying boundary conditions, later stages tighten the PDE residuals, reducing error drift over time.
• The authors observed error accumulation for longer simulation times, a known challenge when dealing with wave‑type PDEs, and noted that the coupled eigenvalue nature of the system can cause stiffness that slows convergence.

Practical Implications

• Rapid prototyping of multi‑physics models – Engineers can replace traditional FEM/FD solvers with a PINN that only requires a dataset of collocation points, dramatically cutting setup time for new geometries or material parameters.
• Differentiable simulations – Because the solution is represented by a neural network, gradients with respect to material constants, geometry, or control inputs are readily available, enabling inverse design, parameter estimation, and real‑time control loops.
• Mesh‑free scalability – For problems where mesh generation is costly (e.g., complex micro‑structured piezoelectric composites), PINNs provide a straightforward alternative that works directly in the continuous domain.
• Integration with existing ML pipelines – The feed‑forward architecture can be combined with other data‑driven components (e.g., sensor fusion, reinforcement learning) to build end‑to‑end digital twins of smart structures.

Limitations & Future Work

• Error growth over long time horizons suggests that additional stabilization techniques (e.g., adaptive collocation, curriculum learning, or physics‑based preconditioners) are needed for sustained simulations.
• Stiffness in coupled eigenvalue systems limited training speed; exploring implicit PINN formulations or operator‑learning approaches could mitigate this.
• The study is confined to a 1‑D benchmark; extending the framework to 2‑D/3‑D geometries, heterogeneous material distributions, and nonlinear piezoelectric behavior remains an open research direction.

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Bottom line: This work shows that a well‑designed PINN can serve as a practical, mesh‑free solver for coupled electro‑elastic wave propagation, delivering accuracy competitive with traditional numerical methods while unlocking differentiable, data‑centric workflows that are highly attractive for modern engineering and AI‑driven product development.*

Authors

• Suhas Suresh Bharadwaj
• Reuben Thomas Thovelil

Paper Information

• arXiv ID: 2602.13811v1
• Categories: cs.NE, cs.LG, physics.comp-ph
• Published: February 14, 2026
• PDF: Download PDF