Researchers Propose Bulk-Boundary Decomposition Framework for Understanding Neural Network Training
A new theoretical framework called bulk-boundary decomposition reorganizes the stochastic gradient descent Lagrangian into a data-independent 'bulk' term and a data-dependent 'boundary' term to analyze deep neural network training dynamics. The approach draws on concepts from physics, including locality and homogeneity, to derive an energy continuity equation within deep networks. The framework may offer new analytical tools for understanding how network architecture and training data separately influence learning dynamics.
Researchers have introduced the bulk-boundary decomposition as a theoretical framework for analyzing the training dynamics of deep neural networks. Starting from a stochastic gradient descent formulation, the Lagrangian is reorganized into two components: a bulk term that is data-independent and reflects the intrinsic dynamics set by network architecture and activation functions, and a boundary term that is data-dependent and captures stochastic interactions from training samples at the input and output layers. The decomposition is said to expose local and homogeneous structure underlying deep networks. Leveraging these physical properties of locality and homogeneity, the authors derive an energy continuity equation within a deep neural network, analogizing the system to concepts from condensed matter physics and high-energy physics phenomenology. The paper spans 13 pages with 3 figures and was submitted to arXiv in November 2025, with a substantially revised version posted in June 2026.
What's missing
The authors do not appear to provide empirical benchmarks demonstrating practical improvements in training efficiency or model performance resulting from the framework, leaving open the question of whether the decomposition yields actionable insights beyond theoretical understanding. It is also unclear how the framework scales to modern large-scale architectures such as transformers.
What different sources said
- arXiv cs.LGCenter
Bulk-boundary decomposition of neural networks
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