Physics-Informed Neural Networks for Nonlinear Structural Dynamics via Mixed Lagrangian Formalism

Physics-informed Neural Networks (PINNs) have recently emerged as a promising approach for solving structural dynamics problems. Conventional PINNs, which are typically formulated based on the strong form of the equation of motion or its corresponding state-space representation, often encounter challenges associated with the calculation of numerical stiffness, poor energy consistency, and instability issues when applied to highly nonlinear or non-smooth behaviours commonly encountered in civil engineering structures, such as impact, contact, and material nonlinearity. To address these challenges, we propose integrating a Mixed Lagrangian Formulation (MLF) into the PINN framework, resulting in a Mixed Lagrangian Physics-Informed Neural Network (MLF-PINN). MLF-PINN provides enhanced numerical stability and accuracy for non-smooth dynamic systems by treating displacement and internal forces as independent field variables. In the proposed framework, the loss function is constructed from the variational (weak) form mixed Lagrangian functional rather than the residual of the strong form equations of motion. This formulation preserves both energy balance and local compatibility, reducing the need for high-order derivatives, and improving the numerical robustness in the estimation of strongly nonlinear or discontinuous responses. Case studies demonstrate that the proposed approach offers improved accuracy and stability on predicting non-smooth behaviour compared to conventional PINNs, providing a physically consistent, structure-preserving learning framework for structural dynamic simulation.