Physics-preserved graph learning of differential equations for structural dynamics

Abstract

Spatio-temporal differential equations are fundamental to understanding the world, describing the dynamic behavior of a structure/system under external stimuli. The equations are typically solved with numerical methods, such as the finite element method, which is computationally inefficient for complex structures, especially under multi-load case analysis requiring repeated calls to slow numerical solvers. Meanwhile, the emerging data-driven deep learning approaches heavily rely on extensive labeled datasets. Here we propose a physics-preserved neural network that seamlessly integrates physical knowledge towards accurate and rapid computation of dynamic characteristics for complex systems without relying on labeled data. A graph convolutional network is created for modal computation in space domain, where physical laws and constraints are inherently encoded within the network architecture (termed ‘hard-embedding’). A physics-informed neural network is then adopted for the dynamic response computation in time domain. This hard-embedding approach remarkably improves computational accuracy compared to the state-of-the-art soft-constraint methods based on loss functions. The proposed model also realizes end-to-end generalization computations under different loading and initial conditions, thereby improving computational speed by hundreds of times compared to the finite element method. This characteristic renders our approach a promising alternative for realizing real-time structural dynamic computations.