Tensor network renormalization: application to dynamic correlation functions and non-hermitian systems

• URL: http://arxiv.org/abs/2311.18785v1
• Date: Thu, 30 Nov 2023 18:34:32 GMT
• Title: Tensor network renormalization: application to dynamic correlation functions and non-hermitian systems
• Authors: Ying-Jie Wei and Zheng-Cheng Gu
• Abstract summary: We present the implementation of the Loop-TNR algorithm, which allows for the computation of dynamical correlation functions. We highlight that the Loop-TNR algorithm can also be applied to investigate critical properties of non-Hermitian systems.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In recent years, tensor network renormalization (TNR) has emerged as an efficient and accurate method for studying (1+1)D quantum systems or 2D classical systems using real-space renormalization group (RG) techniques. One notable application of TNR is its ability to extract central charge and conformal scaling dimensions for critical systems. In this paper, we present the implementation of the Loop-TNR algorithm, which allows for the computation of dynamical correlation functions. Our algorithm goes beyond traditional approaches by not only calculating correlations in the spatial direction, where the separation is an integer, but also in the temporal direction, where the time difference can contain decimal values. Our algorithm is designed to handle both imaginary-time and real-time correlations, utilizing a tensor network representation constructed from a path-integral formalism. Additionally, we highlight that the Loop-TNR algorithm can also be applied to investigate critical properties of non-Hermitian systems, an area that was previously inaccessible using density matrix renormalization group(DMRG) and matrix product state(MPS) based algorithms.

Related papers

• Interval Reachability of Nonlinear Dynamical Systems with Neural Network Controllers [5.543220407902113]
  This paper proposes a computationally efficient framework, based on interval analysis, for rigorous verification of nonlinear continuous-time dynamical systems with neural network controllers. Inspired by mixed monotone theory, we embed the closed-loop dynamics into a larger system using an inclusion function of the neural network and a decomposition function of the open-loop system. We show that one can efficiently compute hyper-rectangular over-approximations of the reachable sets using a single trajectory of the embedding system.
  arXiv  Detail & Related papers  (2023-01-19T06:46:36Z)
• A Recursively Recurrent Neural Network (R2N2) Architecture for Learning Iterative Algorithms [64.3064050603721]
  We generalize Runge-Kutta neural network to a recurrent neural network (R2N2) superstructure for the design of customized iterative algorithms. We demonstrate that regular training of the weight parameters inside the proposed superstructure on input/output data of various computational problem classes yields similar iterations to Krylov solvers for linear equation systems, Newton-Krylov solvers for nonlinear equation systems, and Runge-Kutta solvers for ordinary differential equations.
  arXiv  Detail & Related papers  (2022-11-22T16:30:33Z)
• Verification of Neural-Network Control Systems by Integrating Taylor Models and Zonotopes [0.0]
  We study the verification problem for closed-loop dynamical systems with neural-network controllers (NNCS) We present an algorithm to chain approaches based on Taylor models and zonotopes, yielding a precise reachability algorithm for NNCS.
  arXiv  Detail & Related papers  (2021-12-16T20:46:39Z)
• Revisiting Recursive Least Squares for Training Deep Neural Networks [10.44340837533087]
  Recursive least squares (RLS) algorithms were once widely used for training small-scale neural networks, due to their fast convergence. Previous RLS algorithms are unsuitable for training deep neural networks (DNNs), since they have high computational complexity and too many preconditions. We propose three novel RLS optimization algorithms for training feedforward neural networks, convolutional neural networks and recurrent neural networks.
  arXiv  Detail & Related papers  (2021-09-07T17:43:51Z)
• Fractal Structure and Generalization Properties of Stochastic Optimization Algorithms [71.62575565990502]
  We prove that the generalization error of an optimization algorithm can be bounded on the complexity' of the fractal structure that underlies its generalization measure. We further specialize our results to specific problems (e.g., linear/logistic regression, one hidden/layered neural networks) and algorithms.
  arXiv  Detail & Related papers  (2021-06-09T08:05:36Z)
• Metric Entropy Limits on Recurrent Neural Network Learning of Linear Dynamical Systems [0.0]
  We show that RNNs can optimally learn - or identify in system-theory parlance - stable LTI systems. For LTI systems whose input-output relation is characterized through a difference equation, this means that RNNs can learn the difference equation from input-output traces in a metric-entropy optimal manner.
  arXiv  Detail & Related papers  (2021-05-06T10:12:30Z)
• Connecting Weighted Automata, Tensor Networks and Recurrent Neural Networks through Spectral Learning [58.14930566993063]
  We present connections between three models used in different research fields: weighted finite automata(WFA) from formal languages and linguistics, recurrent neural networks used in machine learning, and tensor networks. We introduce the first provable learning algorithm for linear 2-RNN defined over sequences of continuous vectors input.
  arXiv  Detail & Related papers  (2020-10-19T15:28:00Z)
• Activation Relaxation: A Local Dynamical Approximation to Backpropagation in the Brain [62.997667081978825]
  Activation Relaxation (AR) is motivated by constructing the backpropagation gradient as the equilibrium point of a dynamical system. Our algorithm converges rapidly and robustly to the correct backpropagation gradients, requires only a single type of computational unit, and can operate on arbitrary computation graphs.
  arXiv  Detail & Related papers  (2020-09-11T11:56:34Z)
• Hierarchical Deep Learning of Multiscale Differential Equation Time-Steppers [5.6385744392820465]
  We develop a hierarchy of deep neural network time-steppers to approximate the flow map of the dynamical system over a disparate range of time-scales. The resulting model is purely data-driven and leverages features of the multiscale dynamics. We benchmark our algorithm against state-of-the-art methods, such as LSTM, reservoir computing, and clockwork RNN.
  arXiv  Detail & Related papers  (2020-08-22T07:16:53Z)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)
• Iterative Algorithm Induced Deep-Unfolding Neural Networks: Precoding Design for Multiuser MIMO Systems [59.804810122136345]
  We propose a framework for deep-unfolding, where a general form of iterative algorithm induced deep-unfolding neural network (IAIDNN) is developed. An efficient IAIDNN based on the structure of the classic weighted minimum mean-square error (WMMSE) iterative algorithm is developed. We show that the proposed IAIDNN efficiently achieves the performance of the iterative WMMSE algorithm with reduced computational complexity.
  arXiv  Detail & Related papers  (2020-06-15T02:57:57Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.