The Propagation Field: A Geometric Substrate Theory of Deep Learning

2026-05-08

AI Summary

This work addresses the limitation of conventional deep learning, which optimizes only the endpoint mapping from input to output while neglecting the geometric structure of internal signal propagation, thereby constraining model generalization, robustness, and calibration. The authors propose a "propagation field" perspective, modeling neural networks as dynamic fields composed of hidden-state trajectories and local Jacobian operators, and introduce observable quantities such as path sensitivity and solver consistency. Building on this framework, they design field-aware loss functions and field-preserving regularization methods to explicitly model and control the internal geometric structure of networks. Experiments demonstrate significant improvements in unseen-path generalization, out-of-distribution robustness, and calibration across multi-path tasks; when integrated with DER++ on Split CIFAR-100, the approach enhances average accuracy, backward transfer performance, and field preservation.

Abstract

Modern deep learning treats neural networks primarily as endpoint functions from inputs to outputs. Inspired by the shift from force to geometry in physics, we ask whether a network should instead be understood through the geometry of its internal propagation. We define a neural propagation field as the collection of hidden-state trajectories and local Jacobian operators across depth. Endpoint losses constrain only the boundary behavior of this field, leaving its interior geometry underdetermined. We show that endpoint-equivalent models can differ by orders of magnitude in trajectory and Jacobian structure, and introduce observable field metrics such as path sensitivity, solver consistency, and trajectory/Jacobian retention. In controlled teacher-flow and PDE systems, endpoint fitting fails to recover the underlying propagation law. In real multi-path tasks, field-aware objectives improve unseen-path generalization, OOD robustness, and calibration when aligned with the observation structure, but can collapse when over-constrained. In continual learning, field-preservation regularization complements replay and distillation: on Split CIFAR-100, DER++ with field preservation improves average accuracy, backward transfer, and field-retention metrics. These results identify propagation-field quality as a measurable and trainable property of neural networks beyond endpoint performance.

Problem

• propagation field
• neural networks
• geometry
• generalization
• continual learning

Innovation

• propagation field
• neural geometry
• Jacobian structure
• field-aware learning
• trajectory retention