Physics-informed neural networks (PINNs) for inverse analysis: An illustration

Abstract

This paper investigates the application of physics-informed neural networks (PINNs) to solve inverse problems in solid mechanics. Traditional numerical methods, such as finite element method (FEM) and isogeometric analysis (IGA), often require significant computational time, hindering their applicability in scenarios requiring rapid calculations. To overcome these challenges, PINNs offer a robust alternative by directly incorporating physical information—such as governing equations and boundary conditions—into the neural network's loss function, reducing the reliance on purely experimental data. This study specifically focuses on a two-dimensional linear elasticity plane-strain problem defined on a unit square domain. The primary objective is to accurately determine the spatial distribution of Young's modulus, alongside displacement and stress components, based on established governing equations, boundary conditions, and sparsely measured data. To achieve this, the proposed framework employs two distinct neural networks: the first (N1) is designed to predict displacements, while the second (N2) estimates Young's modulus. Both network architectures consist of four hidden layers containing 40 neurons each. The models were trained over 20,000 epochs, utilizing a dataset comprising displacements measured at 400 points and Young's modulus measured at 9 points randomly distributed across the computational domain. The results demonstrate a strong agreement between the PINNs predictions and the exact analytical solutions. Although slight relative errors were observed at the boundaries where exact values approach zero, the overall deviation remained acceptably small, at only around 10-5. These findings validate that PINNs provide a highly effective tool that balances computational efficiency and error. While the study highlights a current lack of neural network architecture optimization, systematically addressing this in future research will further enhance the model's speed and predictive accuracy for complex mechanical systems.