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Title: Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems Abstract: Physics-Informed Neural Networks (PINNs) have received much attention recently due to their potential for high-performance computations for complex physical systems, including data-based computing, systems with unknown parameters, and others. However, applications of these methods to predict the long-term evolution of systems with little friction, such as many systems encountered in space exploration, oceanography/climate, and many other fields, are difficult as the errors tend to accumulate, and the results may quickly become unreliable. We provide a solution to the problem of data-based computation of Hamiltonian systems utilizing symmetry methods. Many Hamiltonian systems with symmetry can be written as a Lie-Poisson system, where the underlying symmetry with respect to a Lie group defines the Poisson bracket. For data-based computing of such systems, we design the Lie-Poisson neural networks (LPNets). By design, our method preserves all special integrals of the Lie-Poisson brackets (Casimirs) to machine precision. LPNets yield an efficient and promising computational method for many particular cases, such as rigid body (satellite) motion (the case of SO(3) group), Kirchhoff’s equations for an underwater vehicle (SE(3) group), and others. We also discuss symmetry-reduced data-based computations for cases of incomplete symmetry reduction, which is important for data-based computations of elasticity problems. This class of problems is defined as coupled dynamics on a product of Lie groups with appropriate symmetry. Our methods resist the curse of dimensionality by keeping acceptable accuracy and simplicity of networks and training when the number of dimensions increases. We also briefly discuss the application of LPNets to dissipative systems and control problems. W/Chris Eldred, Francois Gay-Balmaz, and Sofiia Huraka, partial support from an NSERC Discovery Grant.