We present a machine-learning approach to a long-standing issue in quantum many-body physics, namely, analytic continuation. This notorious ill-conditioned problem of obtaining spectral function from an imaginary time Green's function has been a focus of new method developments for past decades. Here we demonstrate the usefulness of modern machine-learning techniques including convolutional neural networks and the variants of a stochastic gradient descent optimizer. The machine-learning continuation kernel is successfully realized without any “domain knowledge,” which means that any physical “prior” is not utilized in the kernel construction and the neural networks “learn” the knowledge solely from “training.” The outstanding performance is achieved for both insulating and metallic band structure. Our machine-learning-based approach not only provides the more accurate spectrum than the conventional methods in terms of peak positions and heights, but is also more robust against the noise which is the required key feature for any continuation technique to be successful. Furthermore, its computation speed is ${10}^4\text{–}{10}^5$ times faster than the maximum entropy method.

• Received 9 May 2018
• Revised 17 October 2018

DOI:https://doi.org/10.1103/PhysRevB.98.245101