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arXiv:2207.04941 · Physical Review X

Multiscale Data-Driven Energy Estimation and Generation

Tanguy Marchand, Misaki Ozawa, Giulio Biroli, Stéphane Mallat

Published Nov 30, 2023• 20 citations

Abstract

We develop a multiscale approach to estimate high-dimensional probability distributions.Our approach applies to cases in which the energy function (or Hamiltonian) is not known from the start.Using data acquired from experiments or simulations we can estimate the energy function.We obtain a representation of the approximate probability distribution based on a multiscale cascade of conditional probabilities.This representation allows for fast sampling of many-body systems in various domains, from statistical physics to cosmology.Our method-the wavelet-conditional renormalization group (WCRG)-proceeds scale by scale, estimating models for the conditional probabilities of "fast degrees of freedom" conditioned by coarse-grained fields.These probability distributions are modeled by energy functions associated with scale interactions, and are represented in an orthogonal wavelet basis.The WCRG decomposes the microscopic energy function as a sum of interaction energies at all scales and can efficiently generate new samples by going from coarse to fine scales.Near phase transitions, this representation of the approximate probability distribution completely avoids the "critical slowing-down" of direct estimation and sampling algorithms.This is explained theoretically by combining results from RG and wavelet theories, and verified numerically for the Gaussian and φ 4 -field theories.We show that multiscale WCRG energy-based models are more general than local potential models and can capture the physics of complex many-body interacting systems at all length scales.This is demonstrated for weak-gravitational-lensing fields reflecting darkmatter distributions in cosmology, which include long-range interactions with long-tail probability distributions.The WCRG has a large number of potential applications in nonequilibrium systems, where the underlying distribution is not known a priori.Finally, we discuss the connection between the WCRG and deep network architectures.

DOI: 10.1103/physrevx.13.041038

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sweeps at each scale, yielding an overall sampling complexity of

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, thus making WCRG much more efficient than standard local MCMC methods. These results show that learned multiscale sampling can overcome critical slowing down in frustrated systems for which conventional cluster algorithms fail. By assessing the sampling accuracy of different observables, we also clarify the main tradeoff of the WCRG method: the accuracy of the fast sampling scheme depends on the expressiveness of the energy-based model used to estimate the wavelet conditional distributions.

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We present a variational renormalization group (RG) approach based on a reversible generative model with hierarchical architecture. The model performs hierarchical change-of-variables transformations from the physical space to a latent space with reduced mutual information. Conversely, the neural network directly maps independent Gaussian noises to physical configurations following the inverse RG flow. The model has an exact and tractable likelihood, which allows unbiased training and direct access to the renormalized energy function of the latent variables. To train the model, we employ probability density distillation for the bare energy function of the physical problem, in which the training loss provides a variational upper bound of the physical free energy. We demonstrate practical usage of the approach by identifying mutually independent collective variables of the Ising model and performing accelerated hybrid Monte Carlo sampling in the latent space. Lastly, we comment on the connection of the present approach to the wavelet formulation of RG and the modern pursuit of information preserving RG.