The Propagation Field: A Geometric Substrate Theory of Deep Learning

Xingrui Gu

Keywords: Deep Learning, Information Geometry

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.

Paper Type: Short (4 pages)

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Submission Number: 126