The R-mAtrIx Net

• URL: http://arxiv.org/abs/2304.07247v1
• Date: Fri, 14 Apr 2023 16:50:42 GMT
• Title: The R-mAtrIx Net
• Authors: Shailesh Lal, Suvajit Majumder, Evgeny Sobko
• Abstract summary: We provide a novel Neural Network architecture that can output R-matrix for a given quantum integrable spin chain. We also explore the space of Hamiltonians around already learned models and reconstruct the family of integrable spin chains which they belong to.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We provide a novel Neural Network architecture that can: i) output R-matrix for a given quantum integrable spin chain, ii) search for an integrable Hamiltonian and the corresponding R-matrix under assumptions of certain symmetries or other restrictions, iii) explore the space of Hamiltonians around already learned models and reconstruct the family of integrable spin chains which they belong to. The neural network training is done by minimizing loss functions encoding Yang-Baxter equation, regularity and other model-specific restrictions such as hermiticity. Holomorphy is implemented via the choice of activation functions. We demonstrate the work of our Neural Network on the two-dimensional spin chains of difference form. In particular, we reconstruct the R-matrices for all 14 classes. We also demonstrate its utility as an \textit{Explorer}, scanning a certain subspace of Hamiltonians and identifying integrable classes after clusterisation. The last strategy can be used in future to carve out the map of integrable spin chains in higher dimensions and in more general settings where no analytical methods are available.

Related papers

• Deep Learning based discovery of Integrable Systems [0.0]
  We introduce a novel machine learning based framework for discovering integrable models. Our approach first employs a synchronized ensemble of neural networks to find high-precision numerical solution to the Yang-Baxter equation.
  arXiv  Detail & Related papers  (2025-03-13T15:42:37Z)
• Neural Control Variates with Automatic Integration [49.91408797261987]
  This paper proposes a novel approach to construct learnable parametric control variates functions from arbitrary neural network architectures. We use the network to approximate the anti-derivative of the integrand. We apply our method to solve partial differential equations using the Walk-on-sphere algorithm.
  arXiv  Detail & Related papers  (2024-09-23T06:04:28Z)
• Symplectic Neural Networks Based on Dynamical Systems [0.0]
  We present and analyze a framework for Symplectic neural networks (SympNets) based on geometric for Hamiltonian differential equations. The SympNets are universal approximators in the space of Hamiltonian diffeomorphisms, interpretable and have a non-vanishing property.
  arXiv  Detail & Related papers  (2024-08-19T09:18:28Z)
• A Unified Algebraic Perspective on Lipschitz Neural Networks [88.14073994459586]
  This paper introduces a novel perspective unifying various types of 1-Lipschitz neural networks. We show that many existing techniques can be derived and generalized via finding analytical solutions of a common semidefinite programming (SDP) condition. Our approach, called SDP-based Lipschitz Layers (SLL), allows us to design non-trivial yet efficient generalization of convex potential layers.
  arXiv  Detail & Related papers  (2023-03-06T14:31:09Z)
• Gradient Descent in Neural Networks as Sequential Learning in RKBS [63.011641517977644]
  We construct an exact power-series representation of the neural network in a finite neighborhood of the initial weights. We prove that, regardless of width, the training sequence produced by gradient descent can be exactly replicated by regularized sequential learning.
  arXiv  Detail & Related papers  (2023-02-01T03:18:07Z)
• 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)
• Symplectically Integrated Symbolic Regression of Hamiltonian Dynamical Systems [11.39873640706974]
  Symplectically Integrated Symbolic Regression (SISR) is a novel technique for learning physical governing equations from data. SISR employs a deep symbolic regression approach, using a multi-layer LSTM-RNN with mutation to probabilistically sample Hamiltonian symbolic expressions.
  arXiv  Detail & Related papers  (2022-09-04T03:17:40Z)
• On Rademacher Complexity-based Generalization Bounds for Deep Learning [18.601449856300984]
  We show that the Rademacher complexity-based approach can generate non-vacuous generalisation bounds on Convolutional Neural Networks (CNNs) Our results show that the Rademacher complexity does not depend on the network length for CNNs with some special types of activation functions such as ReLU, Leaky ReLU, Parametric Rectifier Linear Unit, Sigmoid, and Tanh.
  arXiv  Detail & Related papers  (2022-08-08T17:24:04Z)
• NeuralEF: Deconstructing Kernels by Deep Neural Networks [47.54733625351363]
  Traditional nonparametric solutions based on the Nystr"om formula suffer from scalability issues. Recent work has resorted to a parametric approach, i.e., training neural networks to approximate the eigenfunctions. We show that these problems can be fixed by using a new series of objective functions that generalizes to space of supervised and unsupervised learning problems.
  arXiv  Detail & Related papers  (2022-04-30T05:31:07Z)
• Representation Theorem for Matrix Product States [1.7894377200944511]
  We investigate the universal representation capacity of the Matrix Product States (MPS) from the perspective of functions and continuous functions. We show that MPS can accurately realize arbitrary functions by providing a construction method of the corresponding MPS structure for an arbitrarily given gate. We study the relation between MPS and neural networks and show that the MPS with a scale-invariant sigmoidal function is equivalent to a one-hidden-layer neural network.
  arXiv  Detail & Related papers  (2021-03-15T11:06:54Z)
• 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)

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.