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Physics-Informed Neural Network for Quantum Systems with Time-Dependent Boundary Conditions and Equation Form Analysis

A. M. Kalitenko, P. I. Pronin

Published Jan 1, 2025• 0 citations
Publisher
DOI
10.2139/ssrn.5239468

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Physics-Informed Neural Networks (PINNs) offer an effective framework for solving partial differential equations by incorporating governing physical laws into the learning process. When applied to quantum mechanical differential equations such as the Schr¨odinger equation, this approach produces smooth wavefunctions suitable for computing both conventional observables and Bohmian quantities. In this work, a PINNs based framework is developed to solve the one-dimensional time-dependent Schr¨odinger equation for the Quantum Har monic Oscillator, a fundamental model with broad relevance in quantum and plasma physics. A fully connected neural network is used to represent the real and imaginary parts of the wavefunction, with the Schr¨odinger equation enforced through the loss function together with appropriate initial and boundary conditions. Automatic differentiation is employed to compute the required derivatives and to extract physical quantities from the learned solution. The method accurately reproduces the ground-state wavefunction and yields a ground-state energy in agreement with the analytical result. The Bohm quantum potential obtained from the network also matches the corresponding analytical expression, demon strating that the essential spatial structure of the quantum amplitude is well captured. These results demonstrate that PINNs constitute a reliable and conceptually transparent framework for solving the time-dependent Schr¨odinger equation. Within this differential equation based setting, PINNs also enable systematic exploration of Bohmian aspects of quantum systems, where Bohmian quantities such as the quantum potential provide physical insight into confinement effects and the emergence of effective potentials.
We investigate a time and energy minimization optimal control problem for open quantum systems, whose dynamics is governed through the Lindblad (or Gorini-Kossakowski-Sudarshan-Lindblad) master equation. The dissipation is Markovian time-independent, and the control is governed by the Hamiltonian of a quantum-mechanical system. We are specifically interested to study the purity in a dissipative system constrained by state and control inputs. We deal with the state constraints through Gamkrelidze revisited method, while handling control constraints through the idea of saturation functions and system extensions. This is the first time that quantum purity conservation is formulated in such framework. We obtain the necessary conditions of optimality through the Pontryagin Minimum Principle. Finally, the resulted boundary value problem is solved by a Physics-Informed Neural Network (PINN) approach, a technique that is also new in quantum control context. We show that these PINNs play an effective role in learning optimal control actions.
Abstract Classical Physics-informed neural networks (PINNs) approximate solutions to PDEs with the help of deep neural networks trained to satisfy the differential operator and the relevant boundary conditions. We revisit this idea in the quantum computing realm, using parameterised random quantum circuits as trial solutions. We further adapt recent PINN-based techniques to our quantum setting, in particular Gaussian smoothing. Our analysis concentrates on the Poisson, the Heat and the Hamilton–Jacobi–Bellman equations, which are ubiquitous in most areas of science. On the theoretical side, we develop a complexity analysis of this approach, and show numerically that random quantum networks can outperform more traditional quantum networks as well as random classical networks.