In recent years, substantial research on the methods for learning Hamiltonian equations has been conducted.

In recent years, substantial research on the methods for learning Hamiltonian equations has been conducted.

Physics-informed neural networks (PINNs) are a widely used tool in today's Machine Learning (ML) landscape. They consist of Neural Networks (NNs) that, during the training phase, learn to solve the differential equations governing the physical laws of a system in a model-independent way. When these differential equations correspond to Hamilton equations of motion, we refer to them as Hamiltonian Neural Networks (HNNs). The HNN paradigm was introduced in [1] and in the present work we closely follow the strategy proposed in [2]. The key advantages of HNNs over standard numerical integrators can be summarized as follows: the predicted solution is analytical in time and not limited to a discrete set of time steps; conservation laws, symmetries, constraints and prior knowledge of the system can be easily incorporated at the level of the architecture and of the loss function to improve the predictability of the HNN; the minimization process of the loss function occurs under the constraint that the solution satisfies the system of equations at all times simultaneously and independently, thus avoiding any iterative mechanism.

Optimal control deals with optimization problems in which variables steer a dynamical system, and its outcome contributes to the objective function. Two classical approaches to solving these problems are Dynamic Programming and the Pontryagin Maximum Principle. In both approaches, Hamiltonian equations offer an interpretation of optimality through auxiliary variables known as costates. However, Hamiltonian equations are rarely used due to their reliance on forward-backward algorithms across the entire temporal domain. This paper introduces a novel neural-based approach to optimal control, with the aim of working forward-in-time. Neural networks are employed not only for implementing state dynamics but also for estimating costate variables. The parameters of the latter network are determined at each time step using a newly introduced local policy referred to as the time-reversed generalized Riccati equation. This policy is inspired by a result discussed in the Linear Quadratic (LQ) problem, which we conjecture stabilizes state dynamics. We support this conjecture by discussing experimental results from a range of optimal control case studies.

We provide an Information-Geometric formulation of Classical Mechanics on the Riemannian manifold of probability distributions, which is an affine manifold endowed with a dually-flat connection. In a non-parametric formalism, we consider the full set of positive probability functions on a finite sample space, and we provide a specific expression for the tangent and cotangent spaces over the statistical manifold, in terms of a Hilbert bundle structure that we call the Statistical Bundle. In this setting, we compute velocities and accelerations of a one-dimensional statistical model using the canonical dual pair of parallel transports and define a coherent formalism for Lagrangian and Hamiltonian mechanics on the bundle. Finally, in a series of examples, we show how our formalism provides a consistent framework for accelerated natural gradient dynamics on the probability simplex, paving the way for direct applications in optimization, game theory and neural networks.