Machine learning determination of dynamical parameters: The Ising model case

Cossu, Guido; Debbio, Luigi Del; Giani, Tommaso; Khamseh, Ava; Wilson, M. G.

doi:10.1103/physrevb.100.064304

Cited by 31 publications

(67 citation statements)

References 14 publications

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“…In Figure 13 the plot for L = 150 indicates that the transition occurs right at the critical temperature. One can fit the points of the latent variable which behave linearly to a constant as a function of the temperature and can restrict that the collapse of the two states located at 1 and −1 occurs at T = 2.288 (21). This is in good agreement with the theoretical prediction.…”

Section: Results For the Anti-ferromagnetic Ising Modelsupporting

confidence: 75%

“…Recent advances in the implementation of Artificial Intelligence (AI) for physical systems, especially, on those which can be formulated on a lattice, appear to be suitable for observing the corresponding underlying phase structure [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. So far methods such as the Principal Component Analysis (PCA) [6,10,12,18,19], Supervised Machine Learning (ML) [2,15,20], Restricted Boltzmann Machines (RBMs) [21,22], as well as autoencoders [6,13] appear to successfully identify different phase regions of classical statistical systems, such as the 2-dimensional (2D) Ising model that describes the (anti)ferromagneticparamagnetic transition. These techniques were also applied on quantum statistical systems, such as the Hubbard model [4] that describes the transition between conducting and insulating systems.…”

Section: Introductionmentioning

confidence: 99%

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The critical temperature of the 2D-Ising model through deep learning autoencoders

et al. 2020

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We investigate deep learning autoencoders for the unsupervised recognition of phase transitions in physical systems formulated on a lattice. We focus our investigation on the 2-dimensional ferromagnetic Ising model and then test the application of the autoencoder on the anti-ferromagnetic Ising model. We use spin configurations produced for the 2-dimensional ferromagnetic and anti-ferromagnetic Ising model in zero external magnetic field. For the ferromagnetic Ising model, we study numerically the relation between one latent variable extracted from the autoencoder to the critical temperature Tc. The proposed autoencoder reveals the two phases, one for which the spins are ordered and the other for which spins are disordered, reflecting the restoration of the ℤ2 symmetry as the temperature increases. We provide a finite volume analysis for a sequence of increasing lattice sizes. For the largest volume studied, the transition between the two phases occurs very close to the theoretically extracted critical temperature. We define as a quasi-order parameter the absolute average latent variable z̃, which enables us to predict the critical temperature. One can define a latent susceptibility and use it to quantify the value of the critical temperature Tc(L) at different lattice sizes and that these values suffer from only small finite scaling effects. We demonstrate that Tc(L) extrapolates to the known theoretical value as L →∞ suggesting that the autoencoder can also be used to extract the critical temperature of the phase transition to an adequate precision. Subsequently, we test the application of the autoencoder on the anti-ferromagnetic Ising model, demonstrating that the proposed network can detect the phase transition successfully in a similar way. Graphical abstract

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Section: Results For the Anti-ferromagnetic Ising Modelsupporting

confidence: 75%

Section: Introductionmentioning

confidence: 99%

The critical temperature of the 2D-Ising model through deep learning autoencoders

et al. 2020

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“…In parallel, tools from Statistical Physics have been applied to analyze the learning ability of RBMs (Decelle et al , 2018; Huang, 2017b), characterizing the sparsity of the weights, the effective temperature, the non-linearities in the activation functions of hidden units, and the adaptation of fields maintaining the activity in the visible layer (Tubiana and Monasson, 2017). Spin glass theory motivated a deterministic framework for the training, evaluation, and use of RBMs (Tramel et al , 2017); it was demonstrated that the training process in RBMs itself exhibits phase transitions (Barra et al , 2016, 2017); learning in RBMs was studied in the context of equilibrium (Cossu et al , 2018; Funai and Giataganas, 2018) and nonequilibrium (Salazar, 2017) thermodynamics, and spectral dynamics (Decelle et al , 2017); mean-field theory found application in analyzing DBMs (Huang, 2017a). Another interesting direction of research is the use of generative models to improve Monte Carlo algorithms (Cristoforetti et al , 2017; Nagai et al , 2017; Tanaka and Tomiya, 2017b; Wang, 2017).…”

Section: Generative Models In Physicsmentioning

confidence: 99%

A high-bias, low-variance introduction to Machine Learning for physicists

et al. 2019

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Machine Learning (ML) is one of the most exciting and dynamic areas of modern research and application. The purpose of this review is to provide an introduction to the core concepts and tools of machine learning in a manner easily understood and intuitive to physicists. The review begins by covering fundamental concepts in ML and modern statistics such as the bias-variance tradeoff, overfitting, regularization, generalization, and gradient descent before moving on to more advanced topics in both supervised and unsupervised learning. Topics covered in the review include ensemble models, deep learning and neural networks, clustering and data visualization, energy-based models (including MaxEnt models and Restricted Boltzmann Machines), and variational methods. Throughout, we emphasize the many natural connections between ML and statistical physics. A notable aspect of the review is the use of Python Jupyter notebooks to introduce modern ML/statistical packages to readers using physics-inspired datasets (the Ising Model and Monte-Carlo simulations of supersymmetric decays of proton-proton collisions). We conclude with an extended outlook discussing possible uses of machine learning for furthering our understanding of the physical world as well as open problems in ML where physicists may be able to contribute.

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“…ML models are able to automatically capture statistical characteristics by identifying and learning hidden patterns in data (Pathak et al 2018), making them ideally suited to detecting warning signals. Indeed, ML has been used to classify phases of matter, study phase behavior, detect phase transitions, and predict chaotic dynamics (Scandolo 2019, Van Nieuwenburg et al 2017, Canabarro et al 2019, Zhao and Fu 2019, Pathak et al 2018, whilst supervised learning algorithms such as artificial neural networks have been used to study second-order phase transitions, especially the Ising model (Morningstar and Melko 2017, Cossu et al 2019, Ni et al 2019, Giannetti et al 2019. However, thus far machine learning tools have not been used to classify the most common transitions seen in ecological, financial, and climatic systems -catastrophic (i.e., firstorder or discontinuous) and non-catastrophic (i.e., second-order or continuous) transitions (Martín et al 2015).…”

Section: Introductionmentioning

confidence: 99%

Machine learning methods trained on simple models can predict critical transitions in complex natural systems

Deb

Sidheekh

Cf³

et al. 2021

Preprint

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Sudden transitions from one stable state to a contrasting state occur in complex systems ranging from the collapse of ecological populations to climatic change, with much recent work seeking to develop methods to predict these unexpected transitions from signals in time series data. However, previously developed methods vary widely in their reliability, and fail to classify whether an approaching collapse might be catastrophic (and hard to reverse) or non-catastrophic (easier to reverse) with significant implications for how such systems are managed. Here we develop a novel detection method, using simulated outcomes from a range of simple mathematical models with varying nonlinearity to train a deep neural network to detect critical transitions - the Early Warning Signal Network (EWSNet). We demonstrate that this neural network (EWSNet), trained on simulated data with minimal assumptions about the underlying structure of the system, can predict with high reliability observed real-world transitions in ecological and climatological data. Importantly, our model appears to capture latent properties in time series missed by previous warning signals approaches, allowing us to not only detect if a transition is approaching but critically whether the collapse will be catastrophic or non-catastrophic. The EWSNet can flag a critical transition with unprecedented accuracy, overcoming some of the major limitations of traditional methods based on phenomena such as Critical Slowing Down. These novel properties mean EWSNet has the potential to serve as a universal indicator of transitions across a broad spectrum of complex systems, without requiring information on the structure of the system being monitored. Our work highlights the practicality of deep learning for addressing further questions pertaining to ecosystem collapse and have much broader management implications.

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Machine learning determination of dynamical parameters: The Ising model case

Cited by 31 publications

References 14 publications

The critical temperature of the 2D-Ising model through deep learning autoencoders

The critical temperature of the 2D-Ising model through deep learning autoencoders

A high-bias, low-variance introduction to Machine Learning for physicists

Machine learning methods trained on simple models can predict critical transitions in complex natural systems

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