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arXiv:1612.04895

Projected Regression Methods for Inverting Fredholm Integrals: Formalism and Application to Analytical Continuation

Louis-François Arsenault, Richard Neuberg, Lauren A. Hannah, Andrew J. Millis

Published Dec 15, 2016• 9 citations
arXivHTMLPDF

Abstract

We present a machine learning approach to the inversion of Fredholm integrals of the first kind. The approach provides a natural regularization in cases where the inverse of the Fredholm kernel is ill-conditioned. It also provides an efficient and stable treatment of constraints. The key observation is that the stability of the forward problem permits the construction of a large database of outputs for physically meaningful inputs. We apply machine learning to this database to generate a regression function of controlled complexity, which returns approximate solutions for previously unseen inputs; the approximate solutions are then projected onto the subspace of functions satisfying relevant constraints. We also derive and present uncertainty estimates. We illustrate the approach by applying it to the analytical continuation problem of quantum many-body physics, which involves reconstructing the frequency dependence of physical excitation spectra from data obtained at specific points in the complex frequency plane. Under standard error metrics the method performs as well or better than the Maximum Entropy method for low input noise and is substantially more robust to increased input noise. We expect the methodology to be similarly effective for any problem involving a formally ill-conditioned inversion, provided that the forward problem can be efficiently solved.

DOI
10.48550/arxiv.1612.04895

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We present a machine learning approach to the inversion of Fredholm integrals of the first kind. The approach provides a natural regularization in cases where the inverse of the Fredholm kernel is ill-conditioned. It also provides an efficient and stable treatment of constraints. The key observation is that the stability of the forward problem permits the construction of a large database of outputs for physically meaningful inputs. We apply machine learning to this database to generate a regression function of controlled complexity, which returns approximate solutions for previously unseen inputs; the approximate solutions are then projected onto the subspace of functions satisfying relevant constraints. We also derive and present uncertainty estimates. We illustrate the approach by applying it to the analytical continuation problem of quantum many-body physics, which involves reconstructing the frequency dependence of physical excitation spectra from data obtained at specific points in the complex frequency plane. Under standard error metrics the method performs as well or better than the Maximum Entropy method for low input noise and is substantially more robust to increased input noise. We expect the methodology to be similarly effective for any problem involving a formally ill-conditioned inversion, provided that the forward problem can be efficiently solved.
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.
A data-science approach to solving the ill-conditioned inverse problem for analytical continuation is proposed. The root of the problem lies in the fact that even tiny noise of imaginary-time input data has a serious impact on the inferred real-frequency spectra. By means of a modern regularization technique, we eliminate redundant degrees of freedom that essentially carry the noise, leaving only relevant information unaffected by the noise. The resultant spectrum is represented with minimal bases and thus a stable analytical continuation is achieved. This framework further provides a tool for analyzing to what extent the Monte Carlo data need to be accurate to resolve details of an expected spectral function.
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 104–105{10}^{4}\text{--}{10}^{5}104–105 times faster than the maximum entropy method.
Many fields of physics use quantum Monte Carlo (QMC) simulations to simulate quantum systems in imaginary-time τττ and estimate imaginary-time correlation functions (ITCF). However, extracting dynamic ωωω-dependent quantities from ITCFs is a notoriously difficult task, known as analytic continuation (AC), that amounts to solving an exponentially ill-conditioned inverse problem. Within the AC literature, there are competing stochastic and regularized approaches, as well as an emerging collection of works using parameterized models like neural networks. Here we transcend the traditional divides between the communities, introducing the regularized stochastic optimization method (RSOM). This method reformulates AC as a dictionary learning problem, discovering a sparse dictionary to represent the solution. Our approach is motivated by the astounding results dictionary learning has produced in many scientific fields. Remarkably, RSOM discovers a sparse dictionary that maps an ill-conditioned inverse problem to a low-dimensional problem that is well-conditioned. We demonstrate that the method yields competitive results for common synthetic test problems as well as for authentic QMC data from the finite temperature electron gas. This work exposes that a dictionary exists within all stochastic and regularized methods and that dictionary learning provides a new angle of attack for future AC methods.
Path integral Monte Carlo (PIMC) simulations are a cornerstone method for studying quantum many-body systems, such as warm dense matter and ultracold atoms. The analytic continuation needed to estimate dynamic quantities from these simulations amounts to an inverse Laplace transform, which is an ill-conditioned problem. If this challenging problem were surmounted, dynamical observables such as the dynamic structure factor (DSF) S ( q , ω ) —a key property e.g. in x-ray and neutron scattering experiments—could be extracted from the imaginary-time correlation functions F ( q , τ ) estimates. Although of fundamental importance, the analytic continuation problem remains challenging due to its ill-posedness, and state-of-the-art techniques continue to deliver unsatisfactory results. To address this challenge, we express the DSF as a linear combination of kernel functions with known Laplace transforms that have been tailored to satisfy its physical constraints, e.g. , detailed balance. Then we employ least-squares optimization regularized with a Bayesian prior estimate to determine the coefficients of this linear combination. We explore various regularization term, such as the commonly used entropic regularizer, as well as the uncommon L 2 -distance and CDF L 2 -distance. We also explore techniques for setting the regularization weight. A key outcome and contribution is the open-source package PyLIT ( Py thon L aplace I nverse T ransform), which leverages Numba for C-level performance, unifying the presented formulations. PyLIT’s core functionality is kernel construction and optimization. In our applications, we find that PyLIT’s DSF estimates share many qualitative features with other more established methods. Drawing from our insights, we identify three key findings. Firstly, independent of the regularization choice, utilizing non-uniform grid point distributions reduced the number of unknowns and thus reduced our space of possible solutions. Secondly, the L 2 -distance and CDF L 2 -distance, previously unexplored regularizers, benefit from their linear gradients and perform about as well as entriopic regularization. Thirdly, future work can meaningfully combine regularized and stochastic optimization.
We address the problem of analytic continuation of imaginary-frequency Green's functions, which is crucial in many-body physics, using machine learning based on a multilevel residual neural network. We specifically address potential biases that can be introduced due to the use of artificially created spectral functions that are employed to train the neural network. We also implement an uncertainty estimation of the predicted spectral function, based on Monte Carlo dropout, which allows us to identify frequency regions where the prediction might not be accurate, and we study the effect of noise, in particular also for situations where the noise level during training is different from that in the actual data. Our analysis demonstrates that this method can indeed achieve a high quality of prediction, comparable to or better than the widely used maximum entropy method, but that further improvement is currently limited by the lack of true data that can be used for training. We also benchmark our approach by applying it to the case of SrVO3{\mathrm{SrVO}}_{3}SrVO3​, where an accurate spectral function has been obtained from dynamical mean-field theory using a solver that works directly on the real frequency axis.