arXiv subjects

Martín Hernández

Publications and source records attributed to Martín Hernández.

6 recordsLinked to original sources

arXiv2026

Convergence, design and training of continuous-time dropout as a random batch method

We study continuous-time dropout in controlled differential equations. We introduce a random-batch approximation of additive vector fields. On each time interval of length $h$, a random subset of components is activated and rescaled by its inclusion probabilities, yielding an unbiased approximation of the full field. We prove a uniform-in-time mean-square trajectory error of order $\mathcal O(h)$. At the distribution level, we derive Wasserstein and pointwise density estimates, together with global $L^1$ bounds under moment assumptions. For supervised training, we prove uniform $\mathcal O(\sqrt h)$ root-mean-square fluctuations of the randomized objective and an $\mathcal O(h)$ weak error for its expectation, leading to consistency of optimal values and near-minimizers and, under quadratic growth, convergence in control space. We further characterize how the sampling law affects the error constant through inclusion probabilities and field variance, compare fixed-size sampling, fixed partitions, and Bernoulli dropout, and derive a work--accuracy model for selecting the switching scale. Numerical experiments with neural ODEs illustrate trajectory and transport errors, sampling effects, and objective consistency at a fixed control.

cs.LG↗

arXiv2025

Random domain decomposition for parabolic PDEs on graphs

The simulation of complex systems, such as gas transport in large pipeline networks, often involves solving PDEs posed on intricate graph structures. Such problems require considerable computational and memory resources. The Random Batch Method (RBM) has shown promise in addressing these challenges via stochastic decomposition techniques. In this paper, we apply the RBM at the PDE level for parabolic equations on graphs, without assuming any preliminary discretization in space or time. We consider a non-overlapping domain decomposition in which the PDE coefficients and source terms are randomized. We prove that the resulting RBM-based scheme converges, in the mean-square sense and uniformly in time, to the true PDE solution with first-order accuracy in the RBM step size. Numerical experiments confirm this convergence rate and demonstrate substantial reductions in both memory usage and computational time compared to solving on the full graph. Moreover, these advantages persist across different time discretization schemes.

math.NA↗

arXiv2025

Constructive Universal Approximation and Finite Sample Memorization by Narrow Deep ReLU Networks

We present a fully constructive analysis of deep ReLU neural networks for classification and function approximation tasks. First, we prove that any dataset with $N$ distinct points in $\mathbb{R}^d$ and $M$ output classes can be exactly classified using a multilayer perceptron (MLP) of width $2$ and depth at most $2N + 4M - 1$, with all network parameters constructed explicitly. This result is sharp with respect to width and is interpreted through the lens of simultaneous or ensemble controllability in discrete nonlinear dynamics. Second, we show that these explicit constructions yield uniform bounds on the parameter norms and, in particular, provide upper estimates for minimizers of standard regularized training loss functionals in supervised learning. As the regularization parameter vanishes, the trained networks converge to exact classifiers with bounded norm, explaining the effectiveness of overparameterized training in the small-regularization regime. We also prove a universal approximation theorem in $L^p(Ω; \mathbb{R}_+)$ for any bounded domain $Ω\subset \mathbb{R}^d$ and $p \in [1, \infty)$, using MLPs of fixed width $d + 1$. The proof is constructive, geometrically motivated, and provides explicit estimates on the network depth when the target function belongs to the Sobolev space $W^{1,p}$. We also extend the approximation and depth estimation results to $L^p(Ω; \mathbb{R}^m)$ for any $m \geq 1$. Our results offer a unified and interpretable framework connecting controllability, expressivity, and training dynamics in deep neural networks.

stat.ML↗

arXiv2025

Random Batch Methods for Discretized PDEs on Graphs

Gas transport and other complex real-world challenges often require solving and controlling partial differential equations (PDEs) defined on graph structures, which typically demand substantial memory and computational resources. The Random Batch Method (RBM) offers significant relief from these demands by enabling the simulation of large-scale systems with reduced computational cost. In this paper, we analyze the application of RBM for solving PDEs on one-dimensional graphs, specifically concentrating on the heat equation. Our approach involves a two-step process: initially discretizing the PDE to transform it into a finite-dimensional problem, followed by the application of the RBM. We refer to this integrated approach as discretize+RBM. We establish the convergence of this method in expectation, under the appropriate selection and simultaneous reduction of the switching parameter in RBM and the discretization parameters. Moreover, we extend these findings to include the optimal control of the heat equation on graphs, enhancing the practical utility of our methodology. The efficacy and computational efficiency of our proposed solution are corroborated through numerical experiments that not only demonstrate convergence but also show significant reductions in computational costs. Our algorithm can be viewed as a randomized variant of domain decomposition, specifically adapted for PDEs defined on graph structures. It is sufficiently versatile to be applied to a wide range of linear PDEs -- not just the heat equation -- while maintaining comparable analytical guarantees and convergence properties.

math.NA↗

arXiv2025

Mini-batch descent in semiflows

This paper investigates the application of mini-batch gradient descent to semiflows (gradient flows). Given a loss function (potential), we introduce a continuous version of mini-batch gradient descent by randomly selecting sub-loss functions over time, defining a piecewise flow. We prove that, under suitable assumptions on the potential generating the semiflow, the \textit{mini-batch descent flow} trajectory closely approximates the original semiflow trajectory on average. In addition, we study a randomized minimizing movement scheme that also approximates the semiflow of the full loss function. We illustrate the versatility of this approach across various problems, including constrained optimization, sparse inversion, and domain decomposition. Finally, we validate our results with several numerical examples.

math.OC↗

arXiv2024

Averaged observations and turnpike phenomenon for parameter-dependent systems

Our main contribution in this article is the achievement of the turnpike property in its integral and exponential forms for parameter-dependent systems with averaged observations in the cost functional. Namely, under suitable assumptions with respect to the matrices that defined the dynamics and the cost functional, we prove that the optimal control and state for the evolutionary problem converge in average to the optimal pair of an associated stationary problem. Moreover, we characterize the closeness between these two optimal solutions, proving that over a large time interval, they are exponentially close.

math.OC↗