# Bulk-boundary decomposition of neural networks

arXiv: https://arxiv.org/abs/2511.02003 · HTML: https://mlsanity.com/paper/2511.02003/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 58 references that are in the map.
- First posted on arXiv 3 Nov 2025.

## Builds on (most cited first, 26 of 26)

- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [A survey of loss functions for semantic segmentation](https://mlsanity.com/paper/2006.14822/) (2020)
- [Statistical physics of inference: Thresholds and algorithms](https://mlsanity.com/paper/1511.02476/) (2015)
- [Interpretable Deep Learning: Interpretation, Interpretability, Trustworthiness, and Beyond](https://mlsanity.com/paper/2103.10689/) (2021)
- [Non-Gaussian processes and neural networks at finite widths](https://mlsanity.com/paper/1910.00019/) (2019)
- [Asymptotics of Wide Networks from Feynman Diagrams](https://mlsanity.com/paper/1909.11304/) (2019)
- [Neural Networks and Quantum Field Theory](https://mlsanity.com/paper/2008.08601/) (2020)
- [An exact mapping between the Variational Renormalization Group and Deep Learning](https://mlsanity.com/paper/1410.3831/) (2014)
- [Nonlocality and Nonlinearity Implies Universality in Operator Learning](https://mlsanity.com/paper/2304.13221/) (2023)
- [Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators](https://mlsanity.com/paper/2106.00694/) (2021)
- [Building Quantum Field Theories Out of Neurons](https://mlsanity.com/paper/2112.04527/) (2021)
- [Sampling with flows, diffusion and autoregressive neural networks: A spin-glass perspective](https://mlsanity.com/paper/2308.14085/) (2023)
- [Nonperturbative renormalization for the neural network-QFT correspondence](https://mlsanity.com/paper/2108.01403/) (2021)
- [Neural Network Field Theories: Non-Gaussianity, Actions, and Locality](https://mlsanity.com/paper/2307.03223/) (2023)
- [Unified field theoretical approach to deep and recurrent neuronal networks](https://mlsanity.com/paper/2112.05589/) (2021)
- [Statistical mechanics of complex neural systems and high dimensional data](https://mlsanity.com/paper/1301.7115/) (2013)
- [A survey and taxonomy of loss functions in machine learning](https://mlsanity.com/paper/2301.05579/) (2023)
- [A Correspondence Between Random Neural Networks and Statistical Field Theory](https://mlsanity.com/paper/1710.06570/) (2017)
- [Conformal Fields from Neural Networks](https://mlsanity.com/paper/2409.12222/) (2024)
- [A duality connecting neural network and cosmological dynamics](https://mlsanity.com/paper/2202.11104/) (2022)
- [The Hintons in your Neural Network: a Quantum Field Theory View of Deep Learning](https://mlsanity.com/paper/2103.04913/) (2021)
- [Bayesian RG Flow in Neural Network Field Theories](https://mlsanity.com/paper/2405.17538/) (2024)
- [Applications of Statistical Field Theory in Deep Learning](https://mlsanity.com/paper/2502.18553/) (2025)
- [Emergent field theories from neural networks](https://mlsanity.com/paper/2411.08138/) (2024)
- [Revisiting Deep Information Propagation: Fractal Frontier and Finite-size Effects](https://mlsanity.com/paper/2508.03222/) (2025)
- [Dynamic neuron approach to deep neural networks: Decoupling neurons for renormalization group analysis](https://mlsanity.com/paper/2410.00396/) (2024)

## Abstract (arXiv)

We present the bulk--boundary decomposition as a new framework for understanding the training dynamics of deep neural networks. Starting from the stochastic gradient descent formulation, we show that the Lagrangian can be reorganized into a data-independent bulk term and a data-dependent boundary term. The bulk captures the intrinsic dynamics set by network architecture and activation functions, while the boundary reflects stochastic interactions from training samples at the input and output layers. This decomposition exposes the local and homogeneous structure underlying deep networks. As a physical consequence of locality and homogeneity, we derive the energy continuity equation within a deep neural network.