Variational Free Energy Method Based On Deep Generative Models And Its Applications In Physics

With development of the internet and arrival of the big data era,modern deep learning has made great progress in fields such as image recognition and natural language processing.Its application scenarios have permeated many aspects of our daily lives and even have the potential to bring about a new round of revolutionary changes in human production,sparking extensive discussions across society.All of this is possible based on the powerful approximation capabilities of deep neural networks,as well as the rapid development of computing resources and infrastructure.In recent years,deep learning tools have also begun to be used in scientific research,including physics.The so-called”AI for science” research paradigm has already begun to have a significant impact on fields such as biology,energy,and pharmaceuticals,which are closely related to human development.Unlike traditional machine learning tasks that are mainly based on massive ”fuzzy”data such as images,text,and speech,natural science problems are often based on precise physical principles.In this paper,we introduce a generic computational framework,the variational free energy method,which can be used to study the properties of a large class of strongly correlated physical systems.This method is rooted in the variational free energy principle in statistical physics and has fundamental theoretical significance in studying the macroscopic behavior of large numbers of interacting degrees of freedom.Thanks to the development of deep learning models and computing infrastructures,we are able to truly unleash the power of the variational free energy method and make new and inspiring progress in many fundamentally important physical problems.The arrangement of this thesis is as follows:In Chapter 1,we first introduce the basic components of deep learning,including neural network models,objective functions,gradient computation based on automatic differentiation,and optimization algorithms,using a simple example of image recognition.They also form the backbone of the variational free energy method.We then focus on introducing the basics and latest developments of automatic differentiation and deep generative models.In short,generative models can be used to represent various probability distributions encountered in physical problems,and automatic differentiation is the underlying computational engine for optimizing the model.Both of them are indispensable for the variational free energy framework.In Chapter 2,we introduce the basic theory of neural canonical transformation and emphasize its close connection with modern generative models,especially flow models.We then discuss how to combine it with the variational free energy principle to construct a generic computational framework that can be used to study the finitetemperature properties of interacting fermions in the continuum.Finally,we demonstrate the implementation of this computational framework by studying the example of a two-dimensional quantum dot.In Chapter 3,we use the variational free energy method based on neural canonical transformation to study a fundamentally important problem in physics: the quasiparticle effective mass of interacting uniform electron gas.By directly extracting information on the effective mass from the low-temperature entropy of the system,the variational free energy method helps to clarify many discrepancies that exist in previous theoretical and numerical work.To achieve this goal,the variational free energy calculation in this chapter has been optimized in many technical aspects compared to the quantum dot example in the previous chapter.In Chapter 4,we use the variational free energy method to study the finite-temperature properties of dense hydrogen.Hydrogen is the most abundant element in the universe and the simplest element on the periodic table.However,under high pressure,dense hydrogen composed of a large number of interacting protons and electrons exhibits rich phases of matter,including metallization and high-temperature superconductivity,which are of great significance in physics.On the other hand,quantitative calculations of the phase diagram and equation of state of dense hydrogen are also important for planetary science,nuclear fusion,and other disciplines.We have conducted a preliminary exploration of the atomic and molecular liquid phases of dense hydrogen in the intermediate temperature range,and further research is still ongoing.In Chapter 5,we summarize recent developments of the variational free energy method and provide an outlook on its application prospects.