Constrained Hamiltonian Systems and Physics-Informed Neural Networks: Hamilton-Dirac Neural Networks
The effectiveness of the Physics Informed Neural Networks (PINNs) for learning the dynamics of constrained Hamiltonian systems is demonstrated using the Dirac theory of constraints for regular systems with holonomic constraints and systems with non-standard Lagrangians. By utilizing Dirac brackets, we derive the Hamilton-Dirac 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-Dirac Neural Networks (HDNNs), successfully learn constrained dynamics without deviating from the constraint manifold. Two examples with holonomic constraints are presented: the nonlinear pendulum in Cartesian coordinates and 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 applicability in systems with singular Lagrangians, we computed the guiding center motion in a strong magnetic field starting from the guiding center Lagrangian. The imposition of energy conservation during the neural network training proved essential for accurately determining the orbits of the guiding center. The HDNN architecture enables the learning of parametric dependencies in constrained dynamics by incorporating a problem-specific parameter as an input, in addition to the time variable. Additionally, an example of semi-supervised, data-driven learning of guiding center dynamics with parameter inference is presented.