Visualizing a neural network that develops quantum perturbation theory

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Visualizing a neural network that develops quantum perturbation theory

Yadong Wu, Pengfei Zhang, Huitao Shen, and Hui Zhai

Phys. Rev. A 98, 010701(R) – Published 30 July 2018

Abstract

Motivated by the question whether the empirical fitting of data by neural networks can yield the same structure of physical laws, we apply neural networks to a quantum-mechanical two-body scattering problem with short-range potentials—a problem that by itself plays an important role in many branches of physics. After training, the neural network can accurately predict $s$ -wave scattering length, which governs the low-energy scattering physics. By visualizing the neural network, we show that it develops perturbation theory order by order when the potential depth increases, without solving the Schrödinger equation or obtaining the wave function explicitly. The result provides an important benchmark to the machine-assisted physics research or even automated machine learning physics laws.

Received 14 March 2018

DOI:https://doi.org/10.1103/PhysRevA.98.010701

Physics Subject Headings (PhySH)

Few-body systems Machine learning Scattering theory

Artificial neural networks

Perturbation theory

NetworksGeneral PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Yadong Wu¹, Pengfei Zhang¹, Huitao Shen², and Hui Zhai^1,3

¹Institute for Advanced Study, Tsinghua University, Beijing 100084, China
²Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
³Collaborative Innovation Center of Quantum Matter, Beijing 100084, China

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Issue

Vol. 98, Iss. 1 — July 2018

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Images

Figure 1
(a) Schematic of the fully connected neural network. (b) Schematic of the input data, which is a finite-range potential discretized at $N_{0}$ different points. (c) The error $ε$ of the neural network after training as a function of maximum scattering length ${\tilde{a}}_{s}$ , with different number of neurons $N_{1}$ in the first hidden layer. See main text for the definition of $ε$ and ${\tilde{a}}_{s}$ .
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Figure 2
Visualization of the matrix $W$ [(a) and (b)] and a typical $N_{0}$ -dimensional vector $ζ_{α} = {W_{α i}, i = 1, \dots, N_{0}}$ as a function of $r / R = i / N_{0}$ [(c) and (d)], for ${\tilde{a}}_{s} / R = - 0.1$ [(a) and (c)] and for ${\tilde{a}}_{s} / R = - 0.5$ [(b) and (d)]. The solid line in (c) is a fitting of $ζ_{α = 46}$ . The orange (lower) and green (upper) solid lines in (d) are fittings of $ζ_{α_{1} = 46}$ and $ζ_{α_{2} = 54}$ as linear combinations of $ξ$ and $ζ$ .
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Figure 3
Neural network with the same architecture as that in Fig. 2, but trained with $a_{s}^{(2)}$ as the output instead of $a_{s}$ . (a) Visualization of the matrix $W$ . (b) A typical vector $ζ_{α = 32}$ as a function of $r / R = i / N_{0}$ for ${\tilde{a}}_{s} / R = - 0.5$ . It is compared with the eigenvector of $K$ matrix with the largest eigenvalue $\sim ξ$ shown by the solid line. The inset of (b) is the eigenvectors associated with first two largest eigenvalues. For $N_{0} = 64$ , the blue solid curve shows $\sim ξ_{1}$ with the largest eigenvalue $Λ_{1} = 29.5$ , and the orange dashed curve shows $\sim ξ_{2}$ with the second largest eigenvalue $Λ_{2} = 1.8$ .
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Figure 4
(a) When the first layer only contains one neuron, $W$ becomes an $N_{0}$ -dimensional vector. Here we compare $W$ and different linear combinations of $ζ$ and $ξ$ . (b) The relation between the actual scattering length $a_{s}$ and $a_{s}^{(1)}$ from the first-order Born approximation. The black line is the empirical formula, Eq. (13), and the orange (light-gray) dots are scattering lengths computed from first two orders of Born approximation.
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Figure 5
Random square well potential generated by two kinds of Gaussian distributions.
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Figure 6
Visualization of the matrix $W$ [(a) and (b)] and a typical $N_{0}$ -dimensional vector $ζ_{α}$ as a function of $r / R = i / N_{0}$ [(c) and (d)], for $p$ -wave [(a) and (c)] and for $d$ -wave scattering [(b) and (d)] in the first Born approximation regime. The solid line in (c) is $ζ_{α = 38}$ , fitted by $r^{4}$ and the solid line in (d) is $ζ_{α = 64}$ fitted by $r^{6}$ . For $p$ -wave training set, the scattering volumes $v$ distribute uniformly within $v / R^{3} \in (- 0.02, 0)$ . For $d$ -wave training set, the scattering supervolumes $D$ distribute uniformly with $D / R^{5} \in (- 0.01, 0)$ .
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