The Hamilton-Jacobi Theory of Deep Learning
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, with ReLU, sigmoid, SiLU, and GELU each an exact limit, gradient, or moment of the same object, and exact in composition across depth and width, with a quantified error at finite depth that vanishes in the joint limit. It is structural fo...
| Authors | Jose Marie Antonio Miñoza, Erika Fille T. Legara, Christopher P. Monterola |
|---|---|
| Primary category | cs.LG |
| Published | 27 May 2026 |
| Last updated at source | 4 August 2026 |
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