Constrained Hamiltonian Systems And Physicsinformed Neural Networks Hamiltondirac Neural Networks Dimitrios A Kaltsas by Dimitrios A. Kaltsas instant download after payment.

(Received 27 January 2024; revised 31 December 2024; accepted 9 January 2025; published 4 February 2025)The effectiveness of physics-informed neural networks (PINNs) for learning the dynamics of constrainedHamiltonian systems is demonstrated using the Dirac theory of constraints for regular systems with holonomicconstraints and systems with nonstandard Lagrangians. By utilizing Dirac brackets, we derive the HamiltonDirac equations and minimize their residuals, incorporating also energy conservation and the Dirac constraints,using appropriate regularization terms in the loss function. The resulting PINNs, referred to as Hamilton-Diracneural networks (HDNNs), successfully learn constrained dynamics without deviating from the constraint manifold. Two examples with holonomic constraints are presented: the nonlinear pendulum in Cartesian coordinatesand a two-dimensional, elliptically restricted harmonic oscillator. In both cases, HDNNs exhibit superior performance in preserving energy and constraints compared to traditional explicit solvers. To demonstrate applicabilityin systems with singular Lagrangians, we computed the guiding center motion in a strong magnetic field startingfrom the guiding center Lagrangian. The imposition of energy conservation during the neural network trainingproved essential for accurately determining the orbits of the guiding center. The HDNN architecture enables thelearning of parametric dependencies in constrained dynamics by incorporating a problem-specific parameter asan input, in addition to the time variable. Additionally, an example of semisupervised, data-driven learning ofguiding center dynamics with parameter inference is presented.DOI: 10.1103/PhysRevE.111.025301