Effective Field Theory of Neural Networks: Mathematical Foundations of Deep Learning from a Theoretical Physics Perspective

This article introduces a summer research project by Logan-Arm under the guidance of Professors Kenway and Del Debbio (Source: GitHub, published on June 6, 2026). Its core is to apply the effective field theory (EFT) framework from theoretical physics to neural networks, aiming to establish a rigorous theoretical framework to understand their generalization ability, training dynamics, and limiting behavior, as well as to explore the value of interdisciplinary research and the guiding significance of theory for practice.

Deep learning has achieved remarkable practical results, but the theoretical understanding of "why it works" is insufficient. Theoretical physics (especially EFT) excels at handling multi-scale complex systems and provides tools to fill this gap. This project is a practice of such interdisciplinary exploration, attempting to describe neural networks using physical language.

The core of EFT is to describe phenomena with an effective theory at a specific scale (e.g., the four-fermion model in particle physics). Mapping to neural networks: the infinite width limit corresponds to mean field theory (analytically tractable); the Neural Tangent Kernel (NTK) corresponds to linearized dynamics near the mean field; generalization ability is related to the Renormalization Group (RG) (hierarchical structure analogous to coarse-graining for feature extraction).

Based on typical applications of EFT, it is speculated that the project may involve: finite width corrections (analogous to field theory loop corrections), scaling behavior of deep networks (universality), field theory description of activation functions (explaining the effectiveness of ReLU, etc.), and coarse-grained effective equations for training dynamics.

Theoretical understanding can guide architecture design (e.g., theoretical intuition for ResNet/Transformer), rational selection of hyperparameters (reducing trial and error), improve interpretability (necessary for high-risk fields), and discover new paradigms (historical theoretical breakthroughs precede practice).

The intersection of physics and ML has spawned new fields (e.g., physics-informed neural networks). Related research includes NTK theory (since 2018), the connection between RG and DL (teams like Mehta's), and the application of mathematical physics tools such as random matrix theory. The institutions where Professors Kenway and Del Debbio are affiliated have deep accumulations in this area.

This project represents an important direction in DL theoretical research, and the value of interdisciplinary exploration lies in the understanding of deep principles. Insights for young researchers: solid foundations (ML + physics), starting with simple models, combining computation and analysis, and participating in academic communities.