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arXiv · 2506.21797

Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning

Peihao Wang·Zhangyang Wang
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Abstract

We develop a theoretical framework that explains how discrete symbolic structures can emerge naturally from continuous neural network training dynamics. By lifting neural parameters to a measure space and modeling training as Wasserstein gradient flow, we show that under geometric constraints, such as group invariance, the parameter measure $μ_t$ undergoes two concurrent phenomena: (1) a decoupling of the gradient flow into independent optimization trajectories over some potential functions, and (2) a progressive contraction on the degree of freedom. These potentials encode algebraic constraints relevant to the task and act as ring homomorphisms under a commutative semi-ring structure on the measure space. As training progresses, the network transitions from a high-dimensional exploration to compositional representations that comply with algebraic operations and exhibit a lower degree of freedom. We further establish data scaling laws for realizing symbolic tasks, linking representational capacity to the group invariance that facilitates symbolic solutions. This framework charts a principled foundation for understanding and designing neurosymbolic systems that integrate continuous learning with discrete algebraic reasoning.

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Peihao Wang, Zhangyang Wang. 2025-07-01. Why Neural Network Can Discover Symbolic Structures with Gradient-based Training: An Algebraic and Geometric Foundation for Neurosymbolic Reasoning. https://arxiv.org/abs/2506.21797

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