Seminários de métodos numéricos

Resumo: Restricting Hamiltonian dynamics to satisfy algebraic constraints is often challenging for traditional numerical algorithms. By leveraging Dirac’s theory of constraints [1,2], we introduce Hamilton–Dirac Neural Networks (HDNNs) [3], a new class of physics-informed neural networks [4] designed to learn constrained Hamiltonian dynamics. This parameter- informed machine learning model incorporates the Hamilton–Dirac equations, energy conservation, and Dirac constraints through regularization terms in the loss function. This formulation enables HDNNs to accurately predict constrained dynamics while confining system trajectories to the constraint manifold, even in cases where traditional solvers fail. Their effectiveness is demonstrated on systems with holonomic constraints, such as the nonlinear pendulum and a restricted two-dimensional harmonic oscillator, where they outperform conventional solvers in preserving energy and constraints. Additionally, HDNNs are applied to systems with singular Lagrangians, modeling the guiding-center motion of a charged particle in a strong magnetic field.

Keywords: Physics-Informed Neural Networks, Dirac Constraints, Dirac Bracket, Hamilton-Dirac Neural Networks

References

[1] P. A. M. Dirac, Proc. R. Soc. Lond. Ser. A 246, 326 (1958).

[2] P. A. M. Dirac, Can. J. Math. 2, 129 (1950).

[3] D. A. Kaltsas, Phys. Rev. E 111, 025301 (2025).

[4] M. Raissi, P. Perdikaris and G. E. Karniadakis, J. Comput. Phys. 378, 686 (2019).