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 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. Quantitative consequences include: the minimax optimal generalization rate $O(n^{-1/(d+2)})$ for fixed $t$; adversarial robustness controlled by $varepsilon$; backpropagation as the co-state equation of the Hamiltonian system for residual networks (Pontryagin Maximum Principle); scaling exponents consistent with data intrinsic dimension via PDE quadrature; and a closed-form $O(N)$ influence function (softmax attribution weights $pi_j$) whose entropy landscape undergoes fold bifurcations as $varepsilon$ increases, each merging attribution basins.