The Hamilton-Jacobi Theory of Deep Learning

• URL: http://arxiv.org/abs/2605.28983v1
• Date: Wed, 27 May 2026 18:38:23 GMT
• Title: The Hamilton-Jacobi Theory of Deep Learning
• Abstract summary: Training a neural network is identified, exactly, as a search through Hamilton--Jacobi initial-value problems.<n>The correspondence is exact for log-sum-exp layers and structural for broader architectures.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $π_j$) whose entropy landscape undergoes fold bifurcations as $\varepsilon$ increases, each merging attribution basins.

Related papers

• Hamilton-Zero: A Neural Tensor-Network Foundation Model for Ground States of Arbitrary Quadratic Qubit Hamiltonians [10.606940539062064]
  We train a foundation model with $sim0.5$B variational parameters, trained with contemporary techniques from large language models and deep reinforcement learning.<n>We prove that the resulting variational principle on this manifold preserves the $textspin-1/2$ sector's ground-state upper bound.<n>We fine-tune our model on system sizes of up to 1024 qubits, and evaluate on systems up to 8100 qubits.
  arXiv  Detail & Related papers  (2026-08-12T10:42:38Z)
• Improved constant factors for qubitized Hamiltonian simulation [0.0]
  Quantum signal processing serves as theally optimal technique for Hamiltonian simulation on a quantum computer.<n>There is a gap between state-of-the-art bounds and the optimal constant multiplicative factor, which is approximately equal to 1.<n>Our improvement resides in a careful treatment of the Bessel tail in the Jacobi-Anger series using Kapteyn's and Watson's inequalities.
  arXiv  Detail & Related papers  (2026-08-03T18:00:03Z)
• How Neural Reward Models Learn Features for Policy Optimization: A Single-Index Analysis [53.063298916923976]
  We study the feedback in a Gaussian single-index model with $r*(x) = *(langle *, xrangle)$ and $x sim N(0, I_d)$.<n>We analyze a two-stage neural reward model that first learns the hidden direction $*$ from reward-weighted samples and then fits the readout layer by weighted ridge regression.
  arXiv  Detail & Related papers  (2026-05-23T22:00:38Z)
• Intrinsic Wasserstein Rates for Score-Based Generative Models on Smooth Manifolds [61.14405512940818]
  We show that Score-based generative models are trained in high-dimensional ambient spaces.<n>We use a ReLU implementation of nearest-projection coordinates via finite intrinsic anchors and Gauss Newton.
  arXiv  Detail & Related papers  (2026-05-15T10:20:05Z)
• An Information-Minimal Geometry for Qubit-Efficient Optimization [0.0]
  We recast qubit-efficient optimization as a geometry problem.<n>Local-consistency problem coincides exactly with the Sherali-Adams level-2 polytope $mathrmSA(2)$.
  arXiv  Detail & Related papers  (2025-11-11T15:38:57Z)
• Expressive Power of Deep Networks on Manifolds: Simultaneous Approximation [2.815765641180636]
  We show that a constant-depth $mathrmReLUk-1$ network with bounded weights can approximate any function in the Sobolev space.<n>We also prove that our construction is nearly optimal by showing the required number of parameters matches up to a logarithmic factor.
  arXiv  Detail & Related papers  (2025-09-11T11:28:20Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• A Mean-Field Analysis of Neural Stochastic Gradient Descent-Ascent for Functional Minimax Optimization [90.87444114491116]
  This paper studies minimax optimization problems defined over infinite-dimensional function classes of overparametricized two-layer neural networks. We address (i) the convergence of the gradient descent-ascent algorithm and (ii) the representation learning of the neural networks. Results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $O(alpha-1)$, measured in terms of the Wasserstein distance.
  arXiv  Detail & Related papers  (2024-04-18T16:46:08Z)
• A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization [53.044526424637866]
  In this paper we consider finding an approximate second-order stationary point (SOSP) that minimizes a twice different subject general non conic optimization. In particular, we propose a Newton-CG based-augmentedconjugate method for finding an approximate SOSP.
  arXiv  Detail & Related papers  (2023-01-10T20:43:29Z)
• Spectral Analysis of Product Formulas for Quantum Simulation [0.0]
  We show that the Trotter step size needed to estimate an energy eigenvalue within precision can be improved in scaling from $epsilon$ to $epsilon1/2$ for a large class of systems. Results partially generalize to diabatic processes, which remain in a narrow energy band separated from the rest of the spectrum by a gap.
  arXiv  Detail & Related papers  (2021-02-25T03:17:25Z)
• Tightening the Dependence on Horizon in the Sample Complexity of Q-Learning [59.71676469100807]
  This work sharpens the sample complexity of synchronous Q-learning to an order of $frac|mathcalS|| (1-gamma)4varepsilon2$ for any $0varepsilon 1$. Our finding unveils the effectiveness of vanilla Q-learning, which matches that of speedy Q-learning without requiring extra computation and storage.
  arXiv  Detail & Related papers  (2021-02-12T14:22:05Z)
• Learning elliptic partial differential equations with randomized linear algebra [2.538209532048867]
  We show that one can construct an approximant to $G$ that converges almost surely. The quantity $0Gamma_epsilonleq 1$ characterizes the quality of the training dataset.
  arXiv  Detail & Related papers  (2021-01-31T16:57:59Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.