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Journal of Optical Communications

Solutions to the Schrödinger equation using deep neural networks for integrated photonics

Sourabh Kumar Dubey, Hibah Islahi, Raghvendra Singh

Published Sep 30, 2024• 0 citations
Publisher

Abstract

Abstract This paper introduces a novel method for solving the Schrödinger equation through the use of deep neural networks (DNNs), presenting a significant departure from traditional techniques. Conventional approaches to solving the Schrödinger equation, such as analytical methods and numerical algorithms, often face challenges when dealing with complex quantum systems due to their inherent limitations. These traditional methods can become cumbersome or even infeasible as the complexity of the systems increases. In contrast, our approach harnesses the capabilities of deep neural networks to approximate both the wavefunction and the energy eigenvalues of quantum systems. By leveraging the flexible and powerful nature of DNNs, we provide a new pathway to solving the Schrödinger equation that can potentially overcome the constraints of classical methods. To validate the effectiveness of our approach, we apply it to the particle in a box problem – a fundamental quantum mechanics model with well-established analytical solutions. This benchmark problem serves as a useful test case, allowing us to demonstrate that DNNs can not only replicate the known results accurately but also offer insights into how these networks can handle more intricate quantum systems. Our results reveal that DNNs are capable of accurately reproducing the analytical solutions for the particle in a box, illustrating their potential as a versatile tool for quantum mechanics.

DOI
10.1515/joc-2024-0193

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Abstract This paper introduces a novel method for solving the Schrödinger equation through the use of deep neural networks (DNNs), presenting a significant departure from traditional techniques. Conventional approaches to solving the Schrödinger equation, such as analytical methods and numerical algorithms, often face challenges when dealing with complex quantum systems due to their inherent limitations. These traditional methods can become cumbersome or even infeasible as the complexity of the systems increases. In contrast, our approach harnesses the capabilities of deep neural networks to approximate both the wavefunction and the energy eigenvalues of quantum systems. By leveraging the flexible and powerful nature of DNNs, we provide a new pathway to solving the Schrödinger equation that can potentially overcome the constraints of classical methods. To validate the effectiveness of our approach, we apply it to the particle in a box problem – a fundamental quantum mechanics model with well-established analytical solutions. This benchmark problem serves as a useful test case, allowing us to demonstrate that DNNs can not only replicate the known results accurately but also offer insights into how these networks can handle more intricate quantum systems. Our results reveal that DNNs are capable of accurately reproducing the analytical solutions for the particle in a box, illustrating their potential as a versatile tool for quantum mechanics.
This paper presents a novel approach to quantum state simulation utilizing multi-layer neural networks. Traditional methods of quantum simulation, particularly those based on solving the Schrödinger equation, face significant computational challenges, especially for complex quantum systems with numerous qubits. This research proposes a framework where a multi-layer neural network is trained to approximate the dynamics of a quantum system, effectively mimicking the behavior of a quantum system without explicitly solving the underlying equations. The core of this method lies in the neural network's inherent ability to learn complex, non-linear mappings, allowing it to represent the time evolution dictated by the Schrödinger equation. We explore the architecture of the network, focusing on its ability to capture the intricate correlations within quantum states. The results demonstrate the potential for significantly faster and more efficient quantum state simulation compared to conventional approaches, offering a viable path toward simulating larger and more complex quantum systems.
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The solution of Schrödinger equation generates the quantisation properties of the system, which can completely describe the quantum behaviour of microscopic particles in a physical system. However, solving the Schrödinger equation is a non-deterministic polynomial time (NP)-hard problem. Numerical methods for solving the Schrödinger equation require careful selection of basis configurations, and the complete basis set has high accuracy but its complexity increases exponentially in the number of particles. Therefore, it is crucial to find an effective numerical method. To solve this problem, this paper present a deep learning architecture, VMCNet, using the powerful computational efficiency of neural networks to improve the speed of numerical computation. Moreover, this paper proposes a suitable wavefunction ansatz, witch determines the accuracy of neural networks, that achieves more accurate energy solutions of the many-electron Schrödinger equation compared to the conventional single-determinant variational molecular Monte Carlo. We introduce non-local and local blocks to represent quantum mechanical interactions, which contain more physical information about the particle system. The experimental results show that VMCNet can cover more than 93% of the correlation energy of the main group elements (Be–Ne) for the monoatomic system and more than 90% of the correlation energy for the diatomic system.
Quantum many-body systems are of great interest for many research areas, including physics, biology, and chemistry. However, their simulation is extremely challenging, due to the exponential growth of the Hilbert space with system size, making it exceedingly difficult to parameterize the wave functions of large systems by using exact methods. Neural networks and machine learning, in general, are a way to face this challenge. For instance, methods like tensor networks and neural quantum states are being investigated as promising tools to obtain the wave function of a quantum mechanical system. In this tutorial, we focus on a particularly promising class of deep learning algorithms. We explain how to construct a Physics-Informed Neural Network (PINN) able to solve the Schrödinger equation for a given potential, by finding its eigenvalues and eigenfunctions. This technique is unsupervised, and utilizes a novel computational method in a manner that is barely explored. PINNs are a deep learning method that exploit automatic differentiation to solve integro-differential equations in a mesh-free way. We show how to find both the ground and the excited states. The method discovers the states progressively by starting from the ground state. We explain how to introduce inductive biases in the loss to exploit further knowledge of the physical system. Such additional constraints allow for a faster and more accurate convergence. This technique can then be enhanced by a smart choice of collocation points in order to take advantage of the mesh-free nature of the PINN. The methods are made explicit by applying them to the infinite potential well and the particle in a ring, a challenging problem to be learned by an artificial intelligence agent due to the presence of complex-valued eigenfunctions and degenerate states
Quantum theory has been remarkably successful in providing an understanding of physical systems at foundational scales. Solving the Schrödinger equation provides full knowledge of all dynamical quantities of the physical system. However closed form solutions to this equation are only available for a few systems and approximation methods are typically used to find solutions. In this paper we address the problem of solving the time independent Schrödinger equation for the ground state solution of physical systems. We propose using end-to-end deep learning approach in a variational optimization scheme for approximating the ground state energies and wave functions of these systems. A neural network realizes a universal trial wave function and is trained in an unsupervised learning framework by optimizing the expectation value of the Hamiltonian of a physical system. The proposed approach is evaluated on physical systems consisting of a particle in a box with and without a perturbation. We demonstrate that our approach obtains approximations of ground state energies and wave functions that are highly accurate, which makes it a potentially plausible candidate for solving more complex physical systems for which analytical solutions are beyond reach.
Traditional linear approximation of quantum mechanical wave functions are not practically applicable for systems with more than 3 degrees of freedom due to the “the curse of dimensionality”. Indeed,the number of parameters required to describe a wave function in high-dimensional space grows exponentially with the number of degrees of freedom. Inevitably, strong model assumptions should be used when studying such systems numerically. There are, however, estimates of the complexity of a function reproduced by a deep neural network (DNN) that demonstrate the same exponential growth with respect to the number of the network layers. The number of parameters for DNN grows only linearly with the number of layers. This gives us a hope that application of DNN as an approximant for a wave function in high-dimensional space might moderate the computational requirements for reproducing such systems and make 4- or higher- dimensional systems feasible for direct numerical modeling. We present a study of DNN approximation properties for a multi-dimensional quantum harmonic oscillator. We demonstrate that the computational resources required to reproduce the wave function depend on the dimensionality of the problem and the quantum numbers of the state. Increasing the number of hidden layers in a fully-connected feed-forward DNN we can reproduce some excited states of a multidimensional system with computational resources comparable to low-dimensional cases. Using the DNN as an approximant for a wave function paves a way to developing a new class of computational schemes for solving the Schroedinger equation for high-dimensional systems.