We develop a constructive approach to generate artificial neural networks
representing the exact ground states of a large class of many-body lattice
Hamiltonians. It is based on the deep Boltzmann machine architecture, in which
two layers of hidden neurons mediate quantum correlations among physical
degrees of freedom in the visible layer. The approach reproduces the exact
imaginary-time Hamiltonian evolution, and is completely deterministic. In turn,
compact and exact network representations for the ground states are obtained
without stochastic optimization of the network parameters. The number of
neurons grows linearly with the system size and total imaginary time,
respectively. Physical quantities can be measured by sampling configurations of
both physical and neuron degrees of freedom. We provide specific examples for
the transverse-field Ising and Heisenberg models by implementing efficient
sampling. As a compact, classical representation for many-body quantum systems,
our approach is an alternative to the standard path integral, and it is
potentially useful also to systematically improve on numerical approaches based
on the restricted Boltzmann machine architecture.

我们开发了一种构造性方法来生成人工神经网络，代表大量多体格子哈密顿量的准确基态。它基于深层玻尔兹曼机架构，在其中，隐藏层的两层中介绍了可见层中的物理自由度之间的量子相关性。通过对物理和神经元自由度的配置进行采样，可以测量物理量，并获得基态的紧凑、准确的网络表示，而且完全是确定性的。有了我们的方法，作为多体量子系统的紧凑、经典表示，它是标准路径积分的一种替代方案，也有可能对基于受限玻尔兹曼机架构的数值方法进行系统改进。

使用深度神经网络构建量子多体系统的精确表示

Constructing exact representations of quantum many-body systems with  deep neural networks

The curse of dimensionality is commonly encountered in numerical partial
differential equations (PDE), especially when uncertainties have to be modeled
into the equations as random coefficients. However, very often the variability
of physical quantities derived from a PDE can be captured by a few features on
the space of the coefficient fields. Based on such an observation, we propose
using a neural-network (NN) based method to parameterize the physical quantity
of interest as a function of input coefficients. The representability of such
quantity using a neural-network can be justified by viewing the neural-network
as performing time evolution to find the solutions to the PDE. We further
demonstrate the simplicity and accuracy of the approach through notable
examples of PDEs in engineering and physics.

该研究论文展示了如何使用基于神经网络的方法用于数值偏微分方程中处理随机系数的问题，通过神经网络解决 PDE 问题从而简单和精确地表示物理量。