In this paper, we discuss how pure mathematics and theoretical physics can be applied to the study of language models. Using set theory and analysis, we formulate mathematically rigorous definitions of language models, and introduce the concept of the moduli space of distributions for a language model. We formulate a generalized distributional hypothesis using functional analysis and topology. We define the entropy function associated with a language model and show how it allows us to understand many interesting phenomena in languages. We argue that the zero points of the entropy function and the points where the entropy is close to 0 are the key obstacles for an LLM to approximate an intelligent language model, which explains why good LLMs need billions of parameters. Using the entropy function, we formulate a conjecture about AGI. Then, we show how thermodynamics gives us an immediate interpretation to language models. In particular we will define the concepts of partition function, internal energy and free energy for a language model, which offer insights into how language models work. Based on these results, we introduce a general concept of the geometrization of language models and define what is called the Boltzmann manifold. While the current LLMs are the special cases of the Boltzmann manifold.

By bridging deep networks and physics, the programme of geometrization of deep networks was proposed as a framework for the interpretability of deep learning systems. Following this programme we can apply two key ideas of physics, the geometrization of physics and the least action principle, on deep networks and deliver a new picture of deep networks: deep networks as memory space of information, where the capacity, robustness and efficiency of the memory are closely related with the complexity, generalization and disentanglement of deep networks. The key components of this understanding include:(1) a Fisher metric based formulation of the network complexity; (2)the least action (complexity=action) principle on deep networks and (3)the geometry built on deep network configurations. We will show how this picture will bring us a new understanding of the interpretability of deep learning systems.

Achieving disentangled representations of information is one of the key goals of deep network based machine learning system. Recently there are more discussions on this issue. In this paper, by comparing the geometric structure of disentangled representation and the geometry of the evolution of mixed states in quantum mechanics, we give a fibre bundle based geometric picture of disentangled representation which can be regarded as a kind of gauge theory. From this perspective we can build a connection between the disentangled representations and the twins paradox in relativity. This can help to clarify some problems about disentangled representation.

How to understand deep learning systems remains an open problem. In this paper we propose that the answer may lie in the geometrization of deep networks. Geometrization is a bridge to connect physics, geometry, deep network and quantum computation and this may result in a new scheme to reveal the rule of the physical world. By comparing the geometry of image matching and deep networks, we show that geometrization of deep networks can be used to understand existing deep learning systems and it may also help to solve the interpretability problem of deep learning systems.