Rethinking the ill-posedness of the spectral function reconstruction -- why is it fundamentally hard and how Artificial Neural Networks can help

Abstract

Reconstructing hadron spectral functions through Euclidean correlation functions are of the important missions in lattice QCD calculations. However, in a Källen-Lehmann(KL) spectral representation, the reconstruction is observed to be ill-posed in practice. It is usually ascribed to the fewer observation points compared to the number of points in the spectral function. In this paper, by solving the eigenvalue problem of continuous KL convolution, we show analytically that the ill-posedness of the inversion is fundamental and it exists even for continuous correlation functions. We discussed how to introduce regulators to alleviate the predicament, in which include the Artificial Neural Networks(ANNs) representations recently proposed by the Authors in [arXiv:2111.14760]. The uniqueness of solutions using ANNs representations is manifested analytically and validated numerically. Reconstructed spectral functions using different regularization schemes are also demonstrated, together with their eigen-mode decomposition. We observe that components with large eigenvalue can be reliably reconstructed by all methods, whereas those with low eigenvalues need to be constrained by regulators.

Bayesian inference of real-time dynamics from lattice QCD

“…Indeed, each of the tiny eigenvalues of the kernel is associated with a mode along frequencies, which can be added to the spectral function without significantly changing the correlator. Reference [44] has recently investigated this fact in detail analytically for the bosonic finite temperature kernel relevant in transport coefficient computations.…”

“…Over the past years interest in machine learning approaches to spectral function reconstruction has increased markedly (see also [128]). Several groups have put forward pioneering studies that explore how established machine learning strategies, such as supervised kernel ridge regression [129,130], artificial neural networks [44,45,[131][132][133][134][135] or Gaussian processes [136,137] can be used to tackle the inverse problem in the context of extracting spectral functions from Euclidean lattice correlators. The machine learning mindset has already lead to new developments in the spectral reconstruction community, by providing new impulses to regularization of the ill-posed problem.…”

“…[45] and recently applied to the study of finite temperature spectra in Ref. [44] this approach allows to infuse the reconstruction with additional information about the analytic properties of ρ. Traditionally one would choose a specific parametrization apriori such as rational functions (Padé) or SVD basis functions (Bryan) and vary their parameters.…”

Sampling QCD field configurations with gauge-equivariant flow models

“…The rapid development of machine learning, and in particular deep learning, over the last decade has spawned a new wave of algorithms for computational physics [1][2][3]. In lattice quantum field theory, custom machine learning tools are being developed to accelerate almost every step of the computational workflow [4], from configuration generation to calculation or design of observables [5,18,[47][48][49][50][51][52][53][54][55][56][57][58][59] and various aspects of analysis [60][61][62][63][64][65][66][67][68][69][70][71], while maintaining rigorous guarantees of exactness.…”

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.…”