Investigation of PINN Stability and Robustness for the Euler-Bernoulli Beam Problem

TL;DR

This paper investigates the stability issues of Physics-Informed Neural Networks (PINNs) applied to Euler-Bernoulli beam problems, identifying formulation-dependent failure modes and proposing diagnostic insights for improved robustness.

Contribution

It provides a detailed analysis of PINN loss landscape challenges for beam problems, comparing strong and energy-based formulations and offering guidance for enhancing PINN robustness.

Findings

Strong formulation suffers from ill-conditioning due to boundary conditions.

Energy-based formulation faces saddle point issues despite lower-order derivatives.

Best performance achieved with strong formulation, BC handling, and L-BFGS optimizer.

Abstract

Physics-Informed Neural Networks (PINNs) encounter significant training difficulties when applied to doubly-clamped beam problems, and the underlying causes are not fully understood. This study investigates the PINN loss landscape to identify the failure mechanisms of two primary formulations: the high-order strong formulation and the energy-based formulation. The results demonstrate that the Strong Formulation suffers from landscape ill-conditioning driven by the boundary conditions (BCs), leading to convergence issues in the doubly-clamped case. Conversely, while the energy-based formulation requires only lower-order derivatives, its loss functional can become indefinite, causing optimization difficulties near saddle points. Based on strain field benchmarks against Finite Element Method (FEM), it is found that the strong formulation, combined with a BC handling method and the L-BFGS optimizer, yields the best performance across three classical boundary condition cases. These findings clarify distinct, formulation-dependent failure modes, offering a diagnostic foundation for developing robust physics-based surrogate models for complex beam systems.