Deep Learning the Functional Renormalization Group

Abstract

We perform a data-driven dimensionality reduction of the scale-dependent four-point vertex function characterizing the functional renormalization group (FRG) flow for the widely studied two-dimensional t-t 0 Hubbard model on the square lattice. We demonstrate that a deep learning architecture based on a neural ordinary differential equation solver in a low-dimensional latent space efficiently learns the FRG dynamics that delineates the various magnetic and d-wave superconducting regimes of the Hubbard model. We further present a dynamic mode decomposition analysis that confirms that a small number of modes are indeed sufficient to capture the FRG dynamics. Our Letter demonstrates the possibility of using artificial intelligence to extract compact representations of the four-point vertex functions for correlated electrons, a goal of utmost importance for the success of cutting-edge quantum field theoretical methods for tackling the many-electron problem.

Machine learning renormalization group for statistical physics

“…However, to better match the teacher model p tch (x), the ⟨x 0 x 2 ⟩ and ⟨x 1 x 3 ⟩ correlations should be further weakened relative to others because there is no direct coupling between these diagonal spins in the original teacher model equation (11). This is where the E representation plays a role, as the z 2 0 and z 3 0 fluctuations will mediate some negative correlation in ⟨x 0 x 2 ⟩ and ⟨x 1 x 3 ⟩ due to the minus signs in equation (12), which helps to bring the student model's visible distribution p std (x) closer to that of the teacher model p tch (x). This argument explains why including more point group representations to the hidden spin in the student model is generally helpful to improve its expressive power.…”

“…Prior research has demonstrated that neural networks can learn to perform hierarchical feature extraction at the configuration level [1][2][3][4][5][6][7][8][9][10][11]. However, a more fascinating aspect of RG is its ability to quantitatively analyze the flow of physics theory in the parameter space at the model level [12,13]. Therefore, this research aims to develop a novel machine-learning RG (MLRG) method that can automatically formulate RG flow equations, discover RG monotones, propose effective theories, identify critical points, and estimate critical exponents, all starting from the symmetry and dimension of the physical system.…”

Exploring explicit coarse-grained structure in artificial neural networks

“…(1) i (10, 6) on each of which the distillation process is performed and 10 distilled images are obtained, and then we obtain the distilled dataset D (2) (10, 100) which contains 100 images for each class at the second level. Repeat this divideand-conquer strategy, we can obtain dataset D (3) (10, 20), D (4) (10,4), and finally reach the highest distilled dataset D (5) (10, 1), which contains only a single distilled image for each class and can be regarded as the typical representatives hosting the essential features of that class. The whole process is illustrated in Fig.…”

“…In the past decade, machine learning has drawn great attention from almost all natural science and engineering communities, such as mathematics [1][2][3], physics [4][5][6][7][8][9][10], biology [11][12][13], and materials sciences [14][15][16], and has been widely used in various aspects of modern society, e.g., automatic driving systems, face recognition, fraud detection, expert recommendation system, speech enhancement, and natural language processing, etc. Especially, the deep learning techniques based on the artificial neural networks [17,18] have become the most popular and dominant machine learning approaches progressively, and their interactions with many-body physics have been intensively explored in recent years.…”

Activation Function Dependence of Data‐Driven Spectra Prediction of Nanostructures

“…[ 3 ] As a data‐driven method, Deep Learning can learn from the data, and solve problems with former mastered knowledge. Recently, more and more Deep Learning methods were introduced to physics research, [ 4 ] which provides a new perspective for researchers. In the photonic devices design area, as on‐demand design of chiral metamaterials, [ 5 ] plasmonic nanostructure design, [ 6 ] and inverse design of Fabry‐Perot‐cavity‐based color filters have been proposed.…”