Dynamical systems describe the evolution of natural phenomena over time and space through mathematical frameworks, often using differential equations. Accurate predictions in these systems are crucial for various applications, yet traditional methods face challenges due to rigidity and complex dynamic behaviors.

Existing models often oversimplify these systems, leading to biases and errors.

Purely data-driven approaches and physics-guided machine learning methods have emerged to address these issues. However, there remains a need for more effective methods to accurately learn and predict the behavior of dynamical systems under rigid conditions. Due to these challenges, an in-depth study was necessary.

The new research, conducted by researchers from Qingdao University, China, introduces the multiscale differential-algebraic neural network (MDANN) method. Published in the International Journal of Mechanical System Dynamics on 20 March, 2024, this study aims to improve the learning of dynamical systems, especially those characterized by rigid conditions.

By incorporating Lagrangian mechanics and multiscale information, the MDANN method enhances the accuracy and efficiency of predictions in complex systems.

The MDANN method features two key components: the Lagrangian mechanics module and the multiscale module. The Lagrangian mechanics module simplifies the learning process by embedding the system in Cartesian coordinates, utilizing a differential-algebraic equation format, and explicitly imposing constraints through Lagrange multipliers.

The multiscale module effectively converts high-frequency components into low-frequency components via radial scaling, enabling the system to learn subprocesses with varying velocities. Experimental validations on a coupled pendulum system, a double pendulum system, and a scissor-type deployable mast system showcased the MDANN method's superior performance.

The method achieved a mean squared error (MSE) of 3.214e-2 for position and 2.590e-3 for energy in the coupled pendulum system. In the double pendulum system, it recorded an MSE of 9.638e-02 for position and 5.091e-01 for energy. Additionally, it optimized control forces in the scissor-type deployable mast system, ensuring uniform motion.

These results underscore MDANN's capability to manage the complexities of rigid systems, significantly enhancing prediction accuracy and computational efficiency.

Prof. Jieyu Ding, one of the lead researchers, stated, "The development of the MDANN method marks a significant advancement in the field of dynamical system learning. By integrating Lagrangian mechanics and multiscale information, we have addressed the long-standing challenges associated with rigid systems. This approach not only enhances prediction accuracy but also offers practical solutions for complex engineering applications."

The MDANN method's application is set to transform fields relying on precise dynamical system modeling, such as aerospace and robotics. Its ability to refine control strategies and ensure operational safety heralds a new era of efficiency and reliability in system predictions and optimizations.

More information: Yin Huang et al, A multiscale differential‐algebraic neural network‐based method for learning dynamical systems, International Journal of Mechanical System Dynamics (2024). DOI: 10.1002/msd2.12102

Provided by Qingdao University