arXiv:2605.28983v1 Announce Type: new Abstract: In this paper, training a neural network is identified, exactly, as a search through Hamilton--Jacobi initial-value problems: each gradient step selects the initial data of a viscous Hamilton--Jacobi equation whose Hopf--Cole propagator best fits the observations; at inference, the input is the spatial point at which that solution is evaluated and the initial condition is already encoded in the weights. The correspondence is exact for log-sum-exp layers and structural for broader architectures: residual networks, transformers, and recurrent archi
The paper provides a novel theoretical framework to understand and potentially optimize deep learning, suggesting a fundamental mathematical correspondence between neural network training and Hamilton-Jacobi theory.
This theoretical breakthrough could lead to more robust, efficient, and interpretable AI models by reframing deep learning optimization within a well-established physics-based mathematical structure.
Our understanding of neural network training shifts from purely empirical optimization to one grounded in continuous mathematics, potentially opening new avenues for algorithm design and performance guarantees.
- · AI researchers
- · Deep learning practitioners
- · Mathematical physicists
- · Software developers building AI tools
- · Empirical AI design methodologies
- · Ad-hoc network architectures
This theoretical mapping could enable the development of new, more efficient training algorithms for deep neural networks.
Improved training efficiency and interpretability could accelerate the development and deployment of advanced AI across various sectors.
A deeper mathematical understanding might reveal fundamental limits or new paradigms for AI capabilities, impacting long-term research trajectories.
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Read at arXiv cs.LG