Artificial Neural Network Approach to the Analytic Continuation Problem

Abstract

Inverse problems are encountered in many domains of physics, with analytic continuation of the imaginary Green's function into the real frequency domain being a particularly important example. However, the analytic continuation problem is ill-defined and currently no analytic transformation for solving it is known. We present a general framework for building an artificial neural network that efficiently solves this task with a supervised learning approach, provided that the forward problem is sufficiently stable for creating a large training dataset of physically relevant examples. By comparing with the commonly used maximum entropy approach, we show that our method can reach the same level of accuracy for low-noise input data, while performing significantly better when the noise strength increases. We also demonstrate that adding an unsupervised denoising step significantly improves the accuracy of the maximum entropy predictions for noisy inputs. The computational cost of the proposed neural network appoach is reduced by almost three orders of magnitude compared to the maximum entropy method.Numerical simulations are playing an extensive role in a growing number of scientific disciplines. Most commonly, numerical simulations approximate a process or a field on a discretized map, taking as input an equation describing the model as well as initial and boundary conditions [1]. Problems falling under this definition are known as direct or forward problems. However, in a number of situations it is required to reconstruct an approximation of the input data or the model that generated it given the observable data. One particularly important example, known as the Fredholm integral equation of the first kind, takes the following form:where g(s, t) is the available quantity, f (s) is the quantity of interest and k(t, s) is the kernel. Such problem definitions are known as inverse problems and are mostly illposed [2]. Noise affecting the data may lead to arbitrarily large errors in the solutions. Formally speaking, one is interested in operators k• that are ill-conditioned or degenerated. Therefore, the formal inversion f = k −1 • g is not a stable operation [1]. One popular approach for solving such problems is to construct regularizing algorithms that converge to a "pseudo-solution" [3]. In this work, we will focus on a particularly important case of the analytic continuation problem in quantum many-body physics. The analytic continuation problem seeks to recover the electron single-particle spectral density in the real frequency domain A(ω) from the fermionic Green's function G(τ ) in imaginary time domain. These quantities are related by the following equation:G(τ ) = − e −ωτ 1 + e − ωβ A(ω)dω.(2) *

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“…the popular (standard) Maximum Entropy Method [18,19,1]. Alternative methods include Padé rational function approximation [20] or machine learning-based methodologies [21], of which it still needs to be established if these also perform well for unphysical Green functions. Besides these numerical approaches, analytical estimates of spectral functions can also be made.…”

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“…Fortunately, there has been extensive research in the last several years using artificial neural networks and other machine learning algorithms to attack such inverse problems [23][24][25]. The setting of such approaches are data driven -meaning that a large number of forward problems are first solved in order to produce a set of training data.…”

“…In this paper we use a powerful artificial neural network scheme, adopted from previous applications to inverse problems in physics [25] and slightly modified for our particular application -see Fig. (2). The neural network takes as input a set of normalized unique (target) pairwise interactions, which we callĴ ij , defined as,…”

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“…Together with the SOM method presented in [17], these stochastic methods have for example, been deployed in the study of nuclear matter at high temperatures in [18]. Recently, the community has seen heightened activity in exploring the use of neural networks for the solution of inverse problems, e.g., in [19][20][21][22].…”