The Hamilton-Jacobi Theory of Deep Learning 文章

arXiv:2605.28983v1 Announce Type: cross 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 architectures (RNNs, LSTMs, SSMs) each discretize the same class of Hamilton--Jacobi equations, with architecture-dependent Hamiltonian and viscosity. A single deformation parameter $\varepsilon$ unifies all four perspectives (network, tropical algebra, viscous PDE, convex optimization) in a commutative diagram closed under Lipschitz conditions.