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Synaptic Field Theory for Neural Networks (2503.08827v2)

Published 11 Mar 2025 in hep-th, cond-mat.dis-nn, and hep-ph

Abstract: Theoretical understanding of deep learning remains elusive despite its empirical success. In this study, we propose a novel "synaptic field theory" that describes the training dynamics of synaptic weights and biases in the continuum limit. Unlike previous approaches, our framework treats synaptic weights and biases as fields and interprets their indices as spatial coordinates, with the training data acting as external sources. This perspective offers new insights into the fundamental mechanisms of deep learning and suggests a pathway for leveraging well-established field-theoretic techniques to study neural network training.

Summary

  • The paper proposes a novel theoretical framework that maps neural network training dynamics to a continuum field theory in de Sitter space.
  • It models neural networks as fields, showing that gradient descent dynamics can be described by an action similar to that found in de Sitter field theory.
  • This correspondence allows applying techniques from field theory to neural networks and using insights from NNs to potentially understand field theory dynamics.

Neural Network/de Sitter Space Correspondence: A Theoretical Exploration

The paper authored by Donghee Lee, Hye-Sung Lee, and Jaeok Yi proposes a novel conceptual framework connecting neural network (NN) training dynamics with field theory in de Sitter (dS) space. With the remarkable advancements in machine learning, especially in deep learning, there is an imperative need to develop an in-depth understanding of the underlying principles that govern neural network training processes. This paper represents a significant theoretical investigation that seeks to illuminate these principles through a field-theoretic perspective.

Core Insights and Methodology

The authors initiate their discussion by noting the extensive interest in deep learning's mechanisms, despite the absence of consensus on its foundational principles. They propose that the dynamics of neural network training via gradient descent can be mapped to a continuum limit resembling a field theory in de Sitter space, known for its role in cosmology as a model of an exponentially expanding universe. By establishing a correspondence between DNN dynamics and dS field theory, the paper opens possibilities for new theoretical approaches in both machine learning and high-energy physics.

The methodology involves modeling neural networks as fields by considering layers of neurons as analogous to fields in a spacetime. Synaptic weights and biases act as dynamic variables subject to gradient descent, analogous to the variation of fields under a field theory action. The primary result demonstrates that the learning dynamics of neural networks can be described by an action similar to that in curved spacetime, implicating the metric tensor of de Sitter space, thereby establishing a formal relationship termed the NN/dS correspondence.

Theoretical Implications and Empirical Structure

An explicit example provided in the paper involves constructing a simple neural network model where the continuum limit results in a de Sitter field theory. This example clarifies the corresponding elements, offering a foundational structure for further research. The work introduces a unique dictionary mapping aspects of neural networks to elements of field theory, for instance, equating the neural network weights with fields in the de Sitter setup.

The theoretical implications are broad, suggesting that established techniques in field theory might be applicable to neural networks, potentially unveiling novel learning algorithms or optimization techniques. Conversely, insights from neural networks could enrich the understanding of field theory models, especially those considering dynamics in de Sitter space.

Evaluation and Future Directions

The paper contributes a sophisticated theoretical tool to the discourse on deep learning, positioning it alongside other significant explorations like the AdS/CFT correspondence in theoretical physics. However, challenges remain, particularly the need to generalize these models beyond simplified network architectures to more complex, real-world applicable structures.

Future research, prompted by this correspondence, may involve:

  • Investigating how network architectures can induce specific field-theoretical properties, such as nonlocal interactions or symmetry behaviors.
  • Using the framework to explore optimization processes inspired by physical systems governed by similar dynamics.
  • Generalizing the current model to encompass realistic, non-linear activation functions and validation on empirical datasets.

This conceptual framework might serve as a bridge between machine learning and fundamental physics through mathematical rigour, offering fertile ground for cross-disciplinary innovation and further research in artificial intelligence and theoretical physics.