Artificial Neural Network Approach to the Analytic Continuation Problem

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 (ANN) that solves this task with a supervised learning approach. Application of the ANN approach to quantum Monte Carlo calculations and simulated Green’s function data demonstrates its high accuracy. 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. The computational cost of the proposed neural network approach is reduced by almost three orders of magnitude compared to the maximum entropy method.

Figure 1

(a) Examples of imaginary-time correlation functions present in the validation set and (b) their corresponding spectral density functions. (c) Illustration of the rapid decrease of the explained variance ratio (EVR) for the first five principal components. (d) Projection of data on the first two principal components.

Figure 2

(a) Example of architecture with $m=3$ and $k=8$. The last layers have 1024 nodes. (b) Validation MAE at the end of the training phase for different training set sizes $N$ and units per layer $k$. (c) Same quantity for different number of hidden layers ($m+1$) for $N=100000$ and $k=256$.

Figure 3

Analytic continuation of the QMC data performed using MaxEnt and the proposed ANN approach for different training set sizes $N$.

Figure 4

(a) Comparison of the starting spectral functions $A(\omega )$ (solid lines) with the predicted $\widehat{A}(\omega )$ calculated using the MaxEnt approach and the proposed ANN model at different noise levels $\eta$ for three examples of spectral density functions not present in the training dataset. (b) MAE distributions for the two methods and different noise levels $\eta$. (c) Comparison of average MAE for the two methods and different noise levels $\eta$.