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

Suhas Suresh Bharadwaj, Reuben Thomas Thovelil

TL;DR

This work evaluates Physics-Informed Neural Networks (PINNs) for a 1D coupled electro-elastodynamic system modeling linear piezoelectricity in stress–charge form. By mapping space-time coordinates to displacement and electric potential and enforcing boundary/initial conditions with hard constraints, a three-stage loss optimization (Adam, AdamW, L-BFGS) demonstrates feasible, mesh-free learning of a time-dependent multiphysics problem. Displacement is captured with approximately 2% relative error while electric potential converges to about 5% relative error, with boundary errors largely suppressed by constraint enforcement; the results illuminate how coupling and derivative terms amplify errors in the electric field. The study provides practical insights into the strengths and limitations of PINNs for coupled wave propagation and outlines directions such as temporal domain decomposition and adaptive collocation to improve long-time accuracy and robustness.

Abstract

Physics-Informed Neural Networks present a novel approach in SciML that integrates physical laws in the form of partial differential equations directly into the NN through soft constraints in the loss function. This work studies the application of PINNs to solve a one dimensional coupled electro-elastodynamic system modeling linear piezoelectricity in stress-charge form, governed by elastodynamic and electrodynamic equations. Our simulation employs a feedforward architecture, mapping space-time coordinates to mechanical displacement and electric potential. Our PINN model achieved global relative L2 errors of 2.34 and 4.87 percent for displacement and electric potential respectively. The results validate PINNs as effective mesh free solvers for coupled time-dependent PDE systems, though challenges remain regarding error accumulation and stiffness in coupled eigenvalue systems.

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

TL;DR

This work evaluates Physics-Informed Neural Networks (PINNs) for a 1D coupled electro-elastodynamic system modeling linear piezoelectricity in stress–charge form. By mapping space-time coordinates to displacement and electric potential and enforcing boundary/initial conditions with hard constraints, a three-stage loss optimization (Adam, AdamW, L-BFGS) demonstrates feasible, mesh-free learning of a time-dependent multiphysics problem. Displacement is captured with approximately 2% relative error while electric potential converges to about 5% relative error, with boundary errors largely suppressed by constraint enforcement; the results illuminate how coupling and derivative terms amplify errors in the electric field. The study provides practical insights into the strengths and limitations of PINNs for coupled wave propagation and outlines directions such as temporal domain decomposition and adaptive collocation to improve long-time accuracy and robustness.

Abstract

Physics-Informed Neural Networks present a novel approach in SciML that integrates physical laws in the form of partial differential equations directly into the NN through soft constraints in the loss function. This work studies the application of PINNs to solve a one dimensional coupled electro-elastodynamic system modeling linear piezoelectricity in stress-charge form, governed by elastodynamic and electrodynamic equations. Our simulation employs a feedforward architecture, mapping space-time coordinates to mechanical displacement and electric potential. Our PINN model achieved global relative L2 errors of 2.34 and 4.87 percent for displacement and electric potential respectively. The results validate PINNs as effective mesh free solvers for coupled time-dependent PDE systems, though challenges remain regarding error accumulation and stiffness in coupled eigenvalue systems.
Paper Structure (17 sections, 20 equations, 2 figures)

This paper contains 17 sections, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Simulation and Methodology
  3. Mathematical Model of 1D Linear Piezoelectricity
  4. Physics-Informed Neural Network Architecture
  5. Loss Function and Physics Residuals
  6. Training Data and Collocation Sampling
  7. Three-Stage Loss Optimization Strategy
  8. Computational Infrastructure and Implementation Details
  9. Validation and Error Metrics
  10. Theoretical Foundations
  11. Results and Discussion
  12. Model Results
  13. Physical Interpretation
  14. Comparison with Baseline Approaches
  15. Boundary Condition Enforcement
  16. ...and 2 more sections

Figures (2)

  • Figure 1: (a) Elastic wave displacement $\mathbf{u}(x)$ with respect to position; (b) Electric potential $\mathbf{E}(x)$ with respect to position
  • Figure 2: (a) Absolute displacement error; (b) Absolute Electric potential error