A Machine Learning Approach to Trapped Many-Fermion Systems - Variational Neural Network 与 VQMC 连接

Aug 3, 2026

A variational neural-network Ansatz is applied to spin-1/2 fermions confined in a harmonic trap with short-range contact interactions. The study frames variational quantum Monte Carlo within an unsupervised machine-learning analogy, where neural-network parameters define a family of wave functions and the energy functions as a cost. Results show fast convergence to the ground state for weak coupling, and an “interaction-strength during training” strategy to efficiently extend performance to stronger couplings. The work emphasizes transfer learning to reuse knowledge across coupling regimes and improve efficiency while clarifying scaling behavior.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages8

Reading Time20 min

AccessFree to download

Introduction

Machine learning and VQMC analogy

Training as cost minimization

Motivation and scope

Model

Harmonic trap and contact interaction

Hamiltonian and assumptions

A Machine Learning Approach to Trapped Many-Fermion Systems Paulo F. Bedaque, 1, ∗ Hersh Kumar, 1,† and Andy Sheng 1,‡ 1 Department of Physics, University of Maryland, College Park, MD 20742 We apply a variational Ansatz based on neural networks to the problem of spin-~~1~~2 fermions in a harmonic trap interacting through a short distance potential. We showed that standard machine learning techniques lead to a quick convergence to the ground state, especially in weakly coupled cases. Higher couplings can be handled efficiently by increasing the strength of interactions during“training”. arXiv :2410 . 17383v1 [nucl-th] 22 Oct 2024 I. INTRODUCTION In recent years, there has been a noteworthy and promising convergence between the methods of machine learning and quantum many-body physics. The central idea has been the recognition of a natural analogy between the training of neural networks and Variational Quantum Monte Carlo methods (VQMC) . This connection has been explored both in the condensed matter/quantum chemistry context where the long-range Coulomb force is relevant (for a small sample see references [1–4]) and in the nuclear/atomic trap case where short-range forces are of interest [5–13] . Common uses of machine learning techniques are based on creating an artificial neural network with some specific architecture and choosing the parameters of this network in order to accomplish some task of interest. In the unsupervised learning approach, the network parameters are tuned in order to minimize a “cost function” — a function of the network parameters whose value correlates with the skill in that particular task. By numerically minimizing the cost function by varying the network parameters (“training” the network) one arrives at a network with some amount of skill in that task. Similarly, in the VQMC method, one parameterizes the ground state of a system and minimizes the energy in relation to these parameters. The wave function | Unsupervised Machine learning | Variational
Monte Carlo | | --- | --- | | cost function | energy | | --- | --- | | neural network | wave function | | weights and biases | wave function parameters | TABLE I: Analogy between unsupervised machine learning and Variational Monte Carlo. ∗ [bedaque@umd.edu](bedaque@umd.edu)[ ](bedaque@umd.edu)† [hekumar@umd.edu](hekumar@umd.edu)[ ](hekumar@umd.edu)‡ [asheng@umd.edu](asheng@umd.edu) with the smallest energy within that parameterized family is the best estimate of the ground state (it is actually a rigorous upper bound on the exact value) . The analogy then is that the energy plays the role of cost function and the network is the family of parameterized wave functions (see Table I) . The “Monte Carlo” part of VQMC refers to the fact that the energies and their gradients cannot be computed analytically for many-particle systems with complicated wave functions. Instead, they are estimated stochastically by Monte Carlo methods. The stochastic noise present in the final evaluation of the energy is kept to a minimum by using a long Monte Carlo chain; the noise in the gradient does not need to be very small. In fact, some stochastic noise is useful in order to avoid getting trapped in local minima of the energy during training [14] . The great advantage of framing VQMC in the machine learning language is to use a number of tricks and techniques used in that field so that training large systems, with wave functions parameterized by many (sometimes millions of) parameters, can be readily accomplished on standard laptops. In this work, we focus on a class of problems permeating several fields of Physics: systems comprised of multiple fermions interacting through short-range potentials. These systems, while conceptually simple, pose significant computational challenges due to the exponential growth of the Hilbert space with particle number and a serious sign problem in some Monte Carlo approaches. Our method leverages the aforementioned

FAQs

How does the paper connect machine learning training with variational quantum Monte Carlo?

It establishes an analogy where neural-network parameters define a variational family of wave functions, and the energy plays the role of the cost function minimized during training.

Why do weakly coupled cases converge quickly in the proposed approach?

The variational neural-network method yields rapid convergence to the ground state when interactions are weak, as demonstrated by the reported behavior of standard machine-learning techniques.

How are higher coupling strengths handled efficiently?

Higher couplings are addressed by increasing the interaction strength during training, combined with a transfer learning strategy that carries knowledge from lower-coupling regimes to accelerate calculations at stronger coupling.

Aug 3, 2026

Upload by Valentina