Geometric Mechanics and Its Application to Neural Networks with Evaluation

Abstract

In recent years, deep learning designed to adhere to physical laws, such as neural network models based on Hamiltonian and Lagrangian systems, has received significant attention. However, there have been few developments addressing cases with constraints. In this study, we consider a neural network learning model based on a physical system with existing holonomic constraints. We propose a novel deep learning method based on a Lagrange-Dirac system (an implicit Lagrangian system), whose geometric structure is defined by a Dirac structure on the cotangent bundle over a configuration manifold. It is noteworthy that such a Lagrange-Dirac system is applicable to physical systems with constraints. Our approach allows to consider mechanical systems with holonomic constraints, in which given training data comprising coordinates and velocities, the Lagrangian can be obtained the output by using the constraint force, momentum and its time derivative as teaching data. We demonstrate its validity through numerical experiments using the example of a double pendulum. The proposed deep learning approach shows that training data improves the accuracy of learning physical laws for holonomic mechanical systems.