Session EE01: V: Statistical and Nonlinear Physics II

Large and small fluctuations in oscillator networks from heterogeneous and correlated noise

Oscillatory networks subjected to noise are broadly used to model physical and technological systems. Due to their nonlinear coupling, such networks typically have multiple stable and unstable states that a network might visit due to noise. In this talk, we focus on the assessment of fluctuations resulting from heterogeneous and correlated noise inputs on Kuramoto model networks. We evaluate the typical, small fluctuations near synchronized states and connect the network variance to the overlap between stable modes of synchronization and the input noise covariance. Going beyond small to large fluctuations, we introduce the indicator mode approximation, that projects the dynamics onto a single amplitude dimension. Such an approximation allows for estimating rates of fluctuations to saddle instabilities, resulting in phase slips between connected oscillators. Statistics for both regimes are quantified in terms of effective noise amplitudes that are compared and contrasted for several noise models. Bridging the gap between small and large fluctuations, we show that a larger network variance does not necessarily lead to higher rates of large fluctuations.   

Statistical physics of regression with quadratic models

A central model in machine learning for theory and practice is the linear model, where the predictor is a linear function of the learnable parameters. Such models arise naturally in a common limit of infinitely-wide deep neural networks and have aided in the understanding of the dynamics of learning and generalization. However, linear models also have limitations and do not capture the richness of "feature learning" that arises in deep neural networks. We consider quadratic models – predictors which allow a quadratic dependence on parameters – as a class of models for studying the effects of feature learning. We theoretically investigate the generalization scaling with sample size and learning dynamics in these models via replica methods from statistical physics and a dynamical mean field theory.   

Engineered Ordinary Differential Equations as Classification Algorithm (EODECA): a Bridge between Dynamical Systems and Machine Learning

In a world increasingly reliant on machine and deep learning, the interpretability of these models remains a substantial challenge, with many equating their functionality to an enigmatic black box. This study seeks to bridge the domains of machine learning and dynamical systems. Recognizing the deep parallels between dense neural networks and dynamical systems, particularly in the light of non-linearities and successive transformations, this talk introduces the Engineered Ordinary Differential Equations as Classification Algorithms (EODECAs). Uniquely designed as neural networks underpinned by continuous ordinary differential equations (ODEs), EODECAs aim to capitalize on the well-established toolkit of dynamical systems. Unlike traditional deep learning models, which often suffer from opacity and non-invertibility, EODECAs promise both high classification performance and intrinsic interpretability. They are naturally invertible, granting them an edge in understanding and transparency over their counterparts. By bridging these domains, we hope to usher in a new era of machine learning models where genuine comprehension of data processes complements predictive prowess. Drawing inspiration from Sir Winston Churchill, this research might signify the end of the beginning for opaque machine learning models, emphasizing the imperative of interpretability in design.    

Phase classification and finite-size analysis with supervised machine learning

We analyze the problem of supervised learning of ferromagnetic phase transitions from a statistical physics perspective [1]. We consider two systems in two universality classes, a two-dimensional Ising model and a two-dimensional Baxter-Wu model, and perform a thorough finite-dimensional analysis of the supervised learning phase results of each model. We found that the variance of the neural network (NN) output function (VOF) as a function of temperature peaks in the critical region. Qualitatively, VOF is related to the degree of NN classification. We find that the VOF peak width displays a finite size scaling determined by the correlation length metric of the model's universality class. We test this conclusion using several neural network architectures—a fully connected neural network, a convolutional neural network, and several members of the ResNet family—and discuss the accuracy of the extracted critical metrics.

We used this approach to investigate how the anisotropy of the model being tested affects the accuracy of the critical temperature and critical index estimates by conducting a training and validation test run on an isotropic model. In other words, we would like to understand how stable the predictions are on a test set with unknown isotropic properties. We simulate a two-dimensional Ising model with two types of anisotropy using precise knowledge of the orthogonal anisotropy of the Onsager solution and the Huthappel solution of the diagonal anisotropy. The result is interesting in that the universal property of the Ising model, expressed in the correlation length exponent, is well restored as a result of supervised machine learning, while the non-universal property - the critical temperature value - is shifted in some regular way. This is true even for the antiferromagnetic sector of the Hautapell model.

[1] V. Chertenkov, E. Burovski, and L. Shchur, PRE 108, L032102 (2023)   

Thermodynamics of deterministic finite automata operating locally and periodically

Real-world computers have operational constraints that cause nonzero entropy production (EP). In particular, almost all real-world computers are `"periodic'', iteratively undergoing the same physical process; and "local", in that subsystems evolve whilst physically decoupled from the rest of the computer. These constraints are so universal because decomposing a complex computation into small, iterative calculations is what makes computers so powerful. We first derive the nonzero EP caused by the locality and periodicity constraints for deterministic finite automata (DFA), a foundational system of computer science theory. We then relate this minimal EP to the computational characteristics of the DFA. We thus divide the languages recognised by DFA into two classes: those that can be recognised with zero EP, and those that necessarily have non-zero EP. We also demonstrate the thermodynamic advantages of implementing a DFA with a physical process that is agnostic about the inputs that it processes.   

Thermodynamic processes of Quantum heat transport in atomic systems

Several computational and analytical methods investigate time-dependent heat transport in nanostructures. Computational calculations consist of a considerable challenge, rendering the perturbative approach an attractive alternative. We study heat transport and quantum thermodynamics by phonons, using the equation of motion with Green's functions in phase space. We analyze heat flowing between out-of-equilibrium reservoirs in a driven system by time-dependent contacts in transient and permanent regimes. We obtain the heat current between reservoirs through Dyson series time-dependent perturbation. Additionally, we analyze the thermal efficiencies of different configurations to model atomic-scale quantum heat engines or refrigerators.   

Towards quantization Conway Game of LifeKrzysztof Pomorski1,3, Dariusz Kotula2,3 1: Institute of PhysicsLodz University of Technology, Lodz, Poland2: Faculty of Computer Science and TelecommunicationCracow University of Technology, Cracow, Poland3: Quantum Hardware Systems, Lodz, Poland

Classical stochastic Conway Game of Life is expressed by the dissipative Schrodinger equation and dissipative tight-binding model. This is conducted at the prize of usage of time dependent anomalous non-Hermitian Schroedinger Hamiltonians as with occurrence of complex value potential that do not preserve the normalization of wave-function and thus allows for mimicking creationism or annihilationism of cellular automaton. At the same time various aspects of thermodynamics were observed in cellular automata system that can be later reformulated by quantum mechanical pictures. The usage of Shannon entropy and mass equivalence to energy points definition of cellular automata temperature. Contrary to intuitive statement the system dynamical equilibrium is always reflected by negative Conway temperatures. Conway-Schrrodinger equation is formulated.   

Entropy production of a qubit out of the perturbative regime

The calculation of Rényi and von Neumann entropy production rates in open quantum systems has been explored by our group for simple systems in the weak coupling limit. However, there have been questions regarding the validity of certain results in higher perturbative order. We therefore investigate entropy flows in an environment that disturbs the quantum system so that perturbation theory is no longer applicable. In this case, our formalism of evaluating powers of density matrices has to be generalized for analogous quasi-density matrices and is thus different from the previous approach. We present the obtained results and discuss differences to the weak coupling case.