Quantum Pendulum via Physics-Informed Neural Networks: A Topological Approach to Quantum Dynamics
TL;DR
This work bridges the gap between modern Scientific Machine Learning and canonical quantum mechanics, serving as a tutorial on embedding topological constraints into neural solvers.
Abstract
Abstract The numerical simulation of quantum systems with non-trivial topology typically requires specialized coordinate charts or symplectic integrators. Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative, but standard implementations struggle with periodic boundary conditions. We present a robust PINN architecture for the Quantum Pendulum (a particle on S 1) that enforces strict periodicity via a Trigonometric Coordinate Embedding layer. This “Hard Constraint” approach eliminates the need for unstable boundary penalty terms. We successfully reconstruct the unitary evolution of the Mathieu ground state under a non-linear cosine potential, achieving a norm conservation > 0.99 and initial condition error of �� ( 10 − 6 ). The method captures subtle quantum phenomena, including macroscopic tunneling tails and phase space delocalization. Validation is performed against the exact analytical solution via Mathieu functions. This work bridges the gap between modern Scientific Machine Learning and canonical quantum mechanics, serving as a tutorial on embedding topological constraints into neural solvers.