A Comparative Study of Stiffness-Matrix-Based Loss Functions for Displacement Prediction with PINNs

In recent years, Physics-Informed Machine Learning (PIML) has made significant strides in diverse fields. However, in solid mechanics, conventional strong-form physics-informed neural networks often involve multiple loss terms and high-order spatial derivatives, which may lead to training instability and convergence difficulties. In this work, we present a comparative study of stiffness-matrix-informed loss functions for training neural networks to approximate displacement fields without labeled displacement data. The objective of the study is to investigate how different discrete physics-based loss formulations affect training accuracy, convergence behavior, conditioning, and computational efficiency. The structural domain is first discretized using finite elements, and the neural network maps spatial coordinates to displacement components. Four stiffness-matrix-based loss formulations are considered: the minimum-potential-energy-based loss, a Cholesky-preconditioned equilibrium residual loss, a Schur-complement-reduced residual loss, and a Schur-complement-reduced energy loss. By imposing physical constraints through assembled stiffness matrices and load vectors, the proposed formulations avoid high-order derivatives with respect to spatial coordinates and reduce the dependence on automatic differentiation used in conventional PINN formulations. By evaluating these formulations across 2D and 3D solid mechanics benchmark problems, we analyze their respective impacts on training stability and final approximation accuracy. This study provides practical guidance for designing stiffness-matrix-based neural-network loss functions for solid mechanics applications.