Analytic continuation via domain knowledge free machine learning

Abstract

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 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 stochastic gradient descent optimiser. Machinelearning 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 -10 5 times faster than maximum entropy method.

Applications of deep learning in high energy nuclear physics

“…������ 图 14 利用神经网络重建势函数的自动微分框架 [82] 。 Figure 14 Flow chart of the potential reconstruction scheme [82] . [83] 。这个问题难以克服却又极为重 要， 因为从这些重建的谱出发， 我们可以得到许多重 要的物理信息，除粒子结构外还包括非平衡演化或 者输运行为中的动力学系数等。 同时， 重建谱函数时 面临的困难也并非强相互作用多体系统所独有，在 量子液体和超导研究中也有类似的问题 [84] 。 在过去 的二十年里，尝试解决这种反问题中最常见的方法 是贝叶斯推断。这是一种经典的统计学习方法，它 将包含有物理规律的知识表示为明显的先验，引入 推断过程从而使的重建任务得到良好的正规化 [83] 。 在我们最近的工作中 [85] ， 物理驱动的自动微分 框架成为了有效重建谱函数的新工具。如图 15 所 示，我们利用关联函数作为观测量，对物理的谱进 行重构。它们两者之间的关系由 Källén-Lehmann 谱表示联系起来， [87] 。 还有几项研究已经尝试用神经网络表征映 射或反映射来重建谱函数 [88][89][90][91] 。例如，在监督学习 框架中，先验知识被编码在大量的训练数据中，从 数据出发可以训练得到从关联函数到谱函数的反映 射 [88][89][90] 。而为了减轻对冗余训练数据的依赖，也有 一些研究中采用了径向基函数、高斯过程或变分自 编码器做重建，均取得了不错的效果 [92][93][94] Deep learning methods are exquisitely tailored to uncover the structure in complex data and efficiently describe it with a finite number of parameters. This data-driven method has recently emerged in high-energy nuclear physics.…”

Automatic differentiation approach for reconstructing spectral functions with neural networks

“…For example, as two of axioms, the scale invariance and proper constant form prior are both embedded into the Bayesian approach with Shannon-Jaynes entropy, which is termed as maximum entropy method (MEM) [4,6]. Besides, recent several studies have investigated reconstructing spectral functions through a neural network [7][8][9][10][11]. In a supervised learning framework, the prior knowledge is encoded in amounts of training data and the inverse transformation is explicitly unfolded through a training process [7][8][9].…”

Bryan’s Maximum Entropy Method—Diagnosis of a Flawed Argument and Its Remedy

“…Together with the SOM method presented in [17], these stochastic methods have for example, been deployed in the study of nuclear matter at high temperatures in [18]. Recently, the community has seen heightened activity in exploring the use of neural networks for the solution of inverse problems, e.g., in [19][20][21][22].…”