A multiscale differential-algebraic neural network-based method for learning dynamical systems

The objective of dynamical system learning tasks is to forecast the future behavior of a system by leveraging observed data. However, such systems can sometimes exhibit rigidity due to significant variations in component parameters or the presence of slow and fast variables, leading to challenges in learning. To overcome this limitation, we propose a multiscale differential-algebraic neural network (MDANN) method that utilizes Lagrangian mechanics and incorporates multiscale information for dynamical system learning. The MDANN method consists of two main components: the Lagrangian mechanics module and the multiscale module. The Lagrangian mechanics module embeds the system in Cartesian coordinates, adopts a differential-algebraic equation format, and uses Lagrange multipliers to impose constraints explicitly, simplifying the learning problem. The multiscale module converts high-frequency components into low-frequency components using radial scaling to learn subprocesses with large differences in velocity. Experimental results demonstrate that the proposed MDANN method effectively improves the learning of dynamical systems under rigid conditions.