On learning Hamiltonian systems using machine learning with focus on system symmetry preservation

Symmetries are fundamental properties of many natural phenomena and structures. In physics, they enable the formulation of conservation laws such as the conservation of energy, momentum, and angular momentum, which simplify the mathematical description and modeling of physical systems. In the field of machine learning, particularly in neural networks, symmetries can be utilized to improve the efficiency and accuracy of models. By incorporating symmetries into the architecture and training of neural networks, these models can become more robust and generalizable. While neural networks are widely used in areas such as image analysis, speech recognition, and text processing, the learning of nonlinear dynamical systems that consider physical laws is less explored. To address this gap, Hamiltonian Neural Networks (HNNs) have been developed, specifically designed to learn dynamical systems while preserving the Hamiltonian structure. This approach ensures that the symplecticity of the system is maintained during data-driven modeling. However, preserving additional symmetry properties requires extra attention. This work proposes two methods to improve HNNs by considering system symmetries. The first part integrates known symmetry information during training through the introduction of symmetry-preserving extensions to the Hamiltonian network architecture. Discrete symmetries, such as periodicity, and continuous symmetries, like translational or rotational invariance, are discussed. The second part focuses on identifying system symmetries alongside learning the Hamiltonian function. The proposed method extends the structure of HNNs with a Lie algebra framework to recognize and embed symmetries into the neural network. This allows for the simultaneous learning of the symmetry group and the total energy of the system. This work examines the results from simulations of various physical systems to demonstrate the effectiveness of the model approaches. Illustrative examples include the simple pendulum, the cart-pendulum system, and the two-body problem in astrodynamics. The results show that incorporating symmetry into the learning process results in more robust and accurate predictions.