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Fast renormalizing the structures and dynamics of ultra-large systems via random renormalization group

Yizhou Xu, Yang Tian, Pei Sun

Published Mar 26, 2024• 0 citations
Publisher

Abstract

Abstract Criticality and symmetry, studied by the renormalization groups, lie at the heart of modern physics theories of matters and complex systems. However, surveying these properties with massive experimental data is bottlenecked by the intolerable costs of computing renormalization groups on real systems. Here, we develop a time- and memory-efficient framework, termed as the random renormalization group, for renormalizing ultra-large systems (e.g., with millions of units) within minutes. This framework is based on random projections, hashing techniques, and kernel representations, which support the renormalization governed by linear and non-linear correlations. For system structures, it exploits the correlations among local topology in kernel spaces to unfold the connectivity of units, identify intrinsic system scales, and verify the existences of symmetries under scale transformation. For system dynamics, it renormalizes units into correlated clusters to analyze scaling behaviours, validate scaling relations, and investigate potential criticality. Benefiting from hashing-function-based designs, our framework significantly reduces computational complexity compared with classic renormalization groups, realizing a single-step acceleration of two orders of magnitude. Meanwhile, the efficient representation of different kinds of correlations in kernel spaces realized by random projections ensures the capacity of our framework to capture diverse unit relations. As shown by our experiments, the random renormalization group helps identify non-equilibrium phase transitions, criticality, and symmetry in diverse large-scale genetic, neural, material, social, and cosmological systems.

DOI
10.21203/rs.3.rs-3888068/v1

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Abstract Criticality and symmetry, studied by the renormalization groups, lie at the heart of modern physics theories of matters and complex systems. However, surveying these properties with massive experimental data is bottlenecked by the intolerable costs of computing renormalization groups on real systems. Here, we develop a time- and memory-efficient framework, termed as the random renormalization group, for renormalizing ultra-large systems (e.g., with millions of units) within minutes. This framework is based on random projections, hashing techniques, and kernel representations, which support the renormalization governed by linear and non-linear correlations. For system structures, it exploits the correlations among local topology in kernel spaces to unfold the connectivity of units, identify intrinsic system scales, and verify the existences of symmetries under scale transformation. For system dynamics, it renormalizes units into correlated clusters to analyze scaling behaviours, validate scaling relations, and investigate potential criticality. Benefiting from hashing-function-based designs, our framework significantly reduces computational complexity compared with classic renormalization groups, realizing a single-step acceleration of two orders of magnitude. Meanwhile, the efficient representation of different kinds of correlations in kernel spaces realized by random projections ensures the capacity of our framework to capture diverse unit relations. As shown by our experiments, the random renormalization group helps identify non-equilibrium phase transitions, criticality, and symmetry in diverse large-scale genetic, neural, material, social, and cosmological systems.
Criticality and symmetry, studied by the renormalization groups, lie at the heart of modern physics theories of matters and complex systems. However, surveying these properties with massive experimental data is bottlenecked by the intolerable costs of computing renormalization groups on real systems. Here, we develop a time- and memory-efficient framework, termed as the random renormalization group, for renormalizing ultra-large systems (e.g., with millions of units) within minutes. This framework is based on random projections, hashing techniques, and kernel representations, which support the renormalization governed by linear and non-linear correlations. For system structures, it exploits the correlations among local topology in kernel spaces to unfold the connectivity of units, identify intrinsic system scales, and verify the existences of symmetries under scale transformation. For system dynamics, it renormalizes units into correlated clusters to analyze scaling behaviours, validate scaling relations, and investigate potential criticality. Benefiting from hashing-function-based designs, our framework significantly reduces computational complexity compared with classic renormalization groups, realizing a single-step acceleration of two orders of magnitude. Meanwhile, the efficient representation of different kinds of correlations in kernel spaces realized by random projections ensures the capacity of our framework to capture diverse unit relations. As shown by our experiments, the random renormalization group helps identify non-equilibrium phase transitions, criticality, and symmetry in diverse large-scale genetic, neural, material, social, and cosmological systems.
Exploring and understanding topological phases in systems with strong distributed disorder requires developing fundamentally new approaches to replace traditional tools such as topological band theory. Here, we present a general real-space renormalization group (RG) approach for scattering models, which is capable of dealing with strong distributed disorder without relying on the renormalization of Hamiltonians or wave functions. Such scheme, based on a block-scattering transformation combined with a replica strategy, is applied for a comprehensive study of strongly disordered unitary scattering networks with localized bulk states, uncovering a connection between topological physics and critical behavior. Our RG scheme leads to topological flow diagrams that unveil how the microscopic competition between reflection and non-reciprocity leads to the large-scale emergence of macroscopic scattering attractors, corresponding to trivial and topological insulators. Our findings are confirmed by a scaling analysis of the localization length (LL) and critical exponents, and experimentally validated. The results not only shed light on the fundamental understanding of topological phase transitions and scaling properties in strongly disordered regimes, but also pave the way for practical applications in modern topological condensed-matter and photonics, where disorder may be seen as a useful design degree of freedom, and no longer as a hindrance.
Far from equilibrium, neural systems self-organize across multiple scales. Exploiting multiscale self-organization in neuroscience and artificial intelligence requires a computational framework for modeling the effective non-equilibrium dynamics of stochastic neural trajectories. Non-equilibrium thermodynamics and representational geometry offer theoretical foundations, but we need scalable data-driven techniques for modeling collective properties of high-dimensional neural networks from partial subsampled observations. Renormalization is a coarse-graining technique central to studying emergent scaling properties of many-body and nonlinear dynamical systems. While widely applied in physics and machine learning, coarse-graining complex dynamical networks remains unsolved, affecting many computational sciences. Recent diffusion-based renormalization, inspired by quantum statistical mechanics, coarse-grains networks near entropy transitions marked by maximal changes in specific heat or information transmission. Here I explore diffusion-based renormalization of neural systems by generating symmetry-breaking representations across scales and offering scalable algorithms using tensor networks. Diffusion-guided renormalization bridges microscale and mesoscale dynamics of dissipative neural systems. For microscales, I developed a scalable graph inference algorithm for discovering community structure from subsampled neural activity. Using community-based node orderings, diffusion-guided renormalization generates renormalization group flow through metagraphs and joint probability functions. Towards mesoscales, diffusion-guided renormalization targets learning the effective non-equilibrium dynamics of dissipative neural trajectories occupying lower-dimensional subspaces, enabling coarse-to-fine control in systems neuroscience and artificial intelligence.
Numerical simulations of critical lattice systems are fundamentally limited by critical slowing down, as long-range correlations are typically established through slow temporal equilibration. A physically constrained generative framework that replaces temporal relaxation with a spatial projection mechanism for critical systems is proposed. Using the two-dimensional Ising model at criticality as a benchmark, we introduce an energy-constrained kernel that synthesizes large-scale configurations from compact equilibrated seeds by enforcing Hamiltonian-level observables. The generated configurations are projected onto the nearest-neighbor energy manifold, ensuring thermodynamic consistency while retaining universal critical properties. We show that the resulting configurations reproduce scale-invariant spin correlations, Binder cumulants, and isotropic structure factors for lattice sizes exceeding 10,000, without iterative Monte Carlo equilibration. While not a strict renormalization group transformation, and motivated by renormalization ideas, the method provides a practical inverse mapping that retains universal features of criticality and enables efficient GPU-parallel generation of ultra-large critical ensembles.
Abstract Modern theories of phase transitions and scale invariance are rooted in path integral formulation and renormalization groups (RGs). Despite the applicability of these approaches in simple systems with only pairwise interactions, they are less effective in complex systems with undecomposable high-order interactions (i.e. interactions among arbitrary sets of units). To precisely characterize the universality of high-order interacting systems, we propose a simplex path integral and a simplex RG (SRG) as the generalizations of classic approaches to arbitrary high-order and heterogeneous interactions. We first formalize the trajectories of units governed by high-order interactions to define path integrals on corresponding simplices based on a high-order propagator. Then, we develop a method to integrate out short-range high-order interactions in the momentum space, accompanied by a coarse graining procedure functioning on the simplex structure generated by high-order interactions. The proposed SRG, equipped with a divide-and-conquer framework, can deal with the absence of ergodicity arising from the sparse distribution of high-order interactions and can renormalize a system with intertwined high-order interactions at the p-order according to its properties at the q-order ( p ⩽ q ). The associated scaling relation and its corollaries provide support to differentiate among scale-invariant, weakly scale-invariant, and scale-dependent systems across different orders. We validate our theory in multi-order scale-invariance verification, topological invariance discovery, organizational structure identification, and information bottleneck analysis. These experiments demonstrate the capability of our theory to identify intrinsic statistical and topological properties of high-order interacting systems during system reduction.
Abstract We develop a machine-learning renormalization group (MLRG) algorithm to explore and analyze many-body lattice models in statistical physics. Using the representation learning capability of generative modeling, MLRG automatically learns the optimal renormalization group (RG) transformations from self-generated spin configurations and formulates RG equations without human supervision. The algorithm does not focus on simulating any particular lattice model but broadly explores all possible models compatible with the internal and lattice symmetries given the on-site symmetry representation. It can uncover the RG monotone that governs the RG flow, assuming a strong form of the c-theorem. This enables several downstream tasks, including unsupervised classification of phases, automatic location of phase transitions or critical points, controlled estimation of critical exponents, and operator scaling dimensions. We demonstrate the MLRG method in two-dimensional lattice models with Ising symmetry and show that the algorithm correctly identifies and characterizes the Ising criticality.