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

TL;DR

PINNs often struggle with Euler-Bernoulli beams under doubly-clamped boundaries due to ill-conditioned loss landscapes from boundary conditions and formulation choice. The authors conduct a Hessian-based analysis to compare strong-form and energy-based PINN losses, evaluate BC-handling strategies, and test optimizers (Adam vs $L$-BFGS) across cantilever, simply-supported, and doubly-clamped cases. They find the strong-form loss is highly sensitive to BC-induced conditioning, which is mitigated by a Dirichlet-accurate ansatz and Gauss-Newton-like Hessian tracking, especially when paired with $L$-BFGS or a hybrid optimizer; the energy-based loss can be inherently saddle-point dominated due to indefinite Hessians. Energy-based formulations exhibit saddle points that hinder convergence, while strong-form with BC-embedding and a hybrid optimizer achieves high accuracy and smooth strain fields across all boundary conditions. The results provide a diagnostic framework for developing robust physics-based surrogates for lattice-structured and complex beam systems.

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.

Figures (12)

• Figure 1: Euler-Bernoulli beam with distributed load and general boundary conditions.
• Figure 2: Schematic of the PINN architecture implementing Strong and Energy-based formulations.
• Figure 3: Snapshots of deflection predictions during training with Adam optimizer: (a) CV, (b) SS, and (c) CC.
• Figure 4: Evolution of individual loss components during training with Adam optimizer: (a) CV, (b) SS, and (c) CC.
• Figure 5: Evolution of the Hessian eigenvalues and condition number ($\kappa$) during training: (a) CV, (b) SS, and (c) CC.