Automatic differentiation approach for reconstructing spectral functions with neural networks

Abstract

Reconstructing spectral functions from Euclidean Green's functions is an important inverse problem in physics. The prior knowledge for specific physical systems routinely offers essential regularization schemes for solving the ill-posed problem approximately. Aiming at this point, we propose an automatic differentiation framework as a generic tool for the reconstruction from observable data. We represent the spectra by neural networks and set chi-square as loss function to optimize the parameters with backward automatic differentiation unsupervisedly. In the training process, there is no explicit physical prior embedding into neural networks except the positive-definite form. The reconstruction accuracy is assessed through Kullback-Leibler(KL) divergence and mean square error(MSE) at multiple noise levels. It should be noted that the automatic differential framework and the freedom of introducing regularization are inherent advantages of the present approach and may lead to improvements of solving inverse problem in the future.

High-energy nuclear physics meets machine learning

“…In Refs. [114,118,119], the authors developed an unsupervised approach based on DNN representation for the spectral function together with automatic differentiation (AD) to reconstruct the spectral function, which does not need training data preparation for supervision (a similar DNN-based inverse problem solving strategy within the AD framework was used for reconstructing the neutron-star…”

“…In addition to Gaussian-like and Lorentzian-like spectral reconstruction tests, the newly devised framework presented in Refs. [114,118] was validated through two physicsmotivated tests. One was for non-positive definite spectral reconstruction, which is beyond the scope of classical MEM applicability but is often encountered for spectral functions related to confinement phenomenon of, e.g., gluons and ghosts, or thermal excitations with long-range correlation in strongly coupled systems.…”

Applications of Machine Learning to Lattice Quantum Field Theory

“…Analysis -Physically interpretable results are extracted from observable measurements. ML applications thus far include cross-observable regression [82,83], action parameter regression [67,84], and new methods for ill-posed inverse problems [85][86][87][88][89][90]. As discussed further in Sec.…”

Neural network approach to reconstructing spectral functions and complex poles of confined particles

“…Examples are Fournier et al [22] and Yoon et al [23] which both trained neural networks to perform analytic continuation on normalized sums of Gaussian distributions. Kades et al [24] and Wang et al [25] used linear combinations of unnormalized Breit-Wigner peaks as their mock spectral functions. In comparison, we propose a more intricate spectral function and reconstruct not only the spectral function, but also pairs of complex poles and an IR cutoff, generalizing this approach and as such considerably extending its potential applicability.…”