Discovering symmetry invariants and conserved quantities by interpreting siamese neural networks

Wetzel, Sebastian Johann; Melko, Roger G.; Scott, Joseph; Panju, Maysum; Ganesh, Vijay

doi:10.1103/physrevresearch.2.033499

Cited by 68 publications

(63 citation statements)

References 34 publications

(37 reference statements)

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“…One important issue of interpretability is to characterize the importance of features, which will assist in explaining the main factors that affect the results and implementing feature extractions, to name but a few. In the literature, some methods are proposed to improve the intepretability of machine learning [100][101][102][103][104][105], although there still exist various limitations. Our work explicitly shows how entanglement properties of MPS can be used to characterize the importance of features.…”

Section: Discussionmentioning

confidence: 99%

Entanglement-Based Feature Extraction by Tensor Network Machine Learning

Liu

Zhang

et al. 2021

Front. Appl. Math. Stat.

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It is a hot topic how entanglement, a quantity from quantum information theory, can assist machine learning. In this work, we implement numerical experiments to classify patterns/images by representing the classifiers as matrix product states (MPS). We show how entanglement can interpret machine learning by characterizing the importance of data and propose a feature extraction algorithm. We show on the MNIST dataset that when reducing the number of the retained pixels to 1/10 of the original number, the decrease of the ten-class testing accuracy is only O (10–3), which significantly improves the efficiency of the MPS machine learning. Our work improves machine learning’s interpretability and efficiency under the MPS representation by using the properties of MPS representing entanglement.

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Section: Discussionmentioning

confidence: 99%

Entanglement-Based Feature Extraction by Tensor Network Machine Learning

Liu

Zhang

et al. 2021

Front. Appl. Math. Stat.

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“…More recently, Liu and Tegmark [28] discover conservation laws of Hamiltonian systems by performing local Monte Carlo sampling followed by linear dimensionality estimation to predict the number of conserved quantities from an explained ratio diagram. Wetzel et al [55] learn a Siamese Network that learns to differentiate between trajectories. Contrary to both, Noether Networks do not need segmentations into trajectories with different conserved quantities and can deal with raw pixel inputs and outputs.…”

Section: Related Workmentioning

confidence: 99%

Noether Networks: Meta-Learning Useful Conserved Quantities

Alet¹,

Doblar²,

Zhou³

et al. 2021

Preprint

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Progress in machine learning (ML) stems from a combination of data availability, computational resources, and an appropriate encoding of inductive biases. Useful biases often exploit symmetries in the prediction problem, such as convolutional networks relying on translation equivariance. Automatically discovering these useful symmetries holds the potential to greatly improve the performance of ML systems, but still remains a challenge. In this work, we focus on sequential prediction problems and take inspiration from Noether's theorem to reduce the problem of finding inductive biases to meta-learning useful conserved quantities. We propose Noether Networks: a new type of architecture where a meta-learned conservation loss is optimized inside the prediction function. We show, theoretically and experimentally, that Noether Networks improve prediction quality, providing a general framework for discovering inductive biases in sequential problems.

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“…Refs. [7][8][9][10]. These examples employed a computer-assisted search to either discover new families of trajectories or to rediscover previously known invariants of motion in integrable systems.…”

Section: Introductionmentioning

confidence: 99%

Machine-assisted discovery of integrable symplectic mappings

Zolkin,

Kharkov,

Nagaitsev

2022

Preprint

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I. Introduction II. 2D symplectic mappings and integrability: historical overview III. 1-piece maps A. Polygon maps with integer coefficients B. Fundamental polygons of the first kind IV. 2-piece maps A. Linear mappings 1. Polygon maps with integer coefficients 2. Fundamental polygons of the second kind 3. Ingredients of mappings with polygon invariant B. Nonlinear mappings 1. Zoo mappings 2. Nonlinear integrable maps with polygon invariants V. Search algorithm VI. 3-piece maps A. Isolated d B. Discrete d, (D) C. Continuous d, (C) VII. 4-piece maps A. Isolated d and r B. Isolated d, discrete r C. Isolated d, continuous r D. Additional cases 1. Continuous and semi-continuous d with unity r. 2. Non-unity values of r. VIII. Applications of mappings with polygon invariants A. Connection to Painlevé equations and Suris mappings B. Near-integrable systems C. Discrete perturbation theory IX. Summary X. Acknowledgments A. Calculation of action-angle variables for polygon mappings 1. Action 2. Rotation number and angle variable a. Method I b. Method II B. Pseudo-code for the search algorithm C. Action-angle variables for 4-piece mappings D. Properties of invariant polygons for Zoo maps References

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Discovering symmetry invariants and conserved quantities by interpreting siamese neural networks

Cited by 68 publications

References 34 publications

Entanglement-Based Feature Extraction by Tensor Network Machine Learning

Entanglement-Based Feature Extraction by Tensor Network Machine Learning

Noether Networks: Meta-Learning Useful Conserved Quantities

Machine-assisted discovery of integrable symplectic mappings

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