arXiv subjects
Publications and source records attributed to Enrique Zuazua.
We analyze the control properties of heat equations with memory terms. We recall previous results showing that if the moving support of the control covers the whole domain where heat diffuses, the system is null controllable when the memory kernel is polynomial. We formulate the problem of extending this result to the case of some more general memory kernels, in particular analytic ones.
We investigate the controllability of a generalized diffusive Lotka-Volterra competition model for two species, incorporating boundary controls and an interior multiplicative control. Considering a smooth, bounded N-dimensional domain, we analyze ecologically pertinent scenarios characterized by constraints on both the controls and system states. Our results demonstrate how integrated control strategies can effectively overcome the limitations identified in previous studies. We prove two main results: (1) asymptotic controllability to single-species survival steady states under arbitrary system parameters, ensured by a combination of boundary and interior controls which act jointly to stabilize the system; and (2) finite-time controllability to a specific heterogeneous coexistence steady state via a two-phase strategy - first steering the system near the target with boundary control, then activating an interior multiplicative control in a localized region. The strong synergy between the two control mechanisms is crucial in both cases. We also analyze extinction dynamics and homogeneous coexistence, and support our findings with numerical simulations. The work concludes with perspectives for future research.
Detecting malware, especially ransomware, is essential to securing today's interconnected ecosystems, including cloud storage, enterprise file-sharing, and database services. Training high-performing artificial intelligence (AI) detectors requires diverse datasets, which are often distributed across multiple organizations, making centralization necessary. However, centralized learning is often impractical due to security, privacy regulations, data ownership issues, and legal barriers to cross-organizational sharing. Compounding this challenge, ransomware evolves rapidly, demanding models that are both robust and adaptable. In this paper, we evaluate Federated Learning (FL) using the Sherpa.ai FL platform, which enables multiple organizations to collaboratively train a ransomware detection model while keeping raw data local and secure. This paradigm is particularly relevant for cybersecurity companies (including both software and hardware vendors) that deploy ransomware detection or firewall systems across millions of endpoints. In such environments, data cannot be transferred outside the customer's device due to strict security, privacy, or regulatory constraints. Although FL applies broadly to malware threats, we validate the approach using the Ransomware Storage Access Patterns (RanSAP) dataset. Our experiments demonstrate that FL improves ransomware detection accuracy by a relative 9% over server-local models and achieves performance comparable to centralized training. These results indicate that FL offers a scalable, high-performing, and privacy-preserving framework for proactive ransomware detection across organizational and regulatory boundaries.
Federated Learning (FL) enables collaborative decentralized training across multiple parties (nodes) while keeping raw data private. There are two main paradigms in FL: Horizontal FL (HFL), where all participant nodes share the same feature space but hold different samples, and Vertical FL (VFL), where participants hold complementary features for the same samples. While HFL is widely adopted, VFL is employed in domains where nodes hold complementary features about the same samples. Still, VFL presents a significant limitation: the vast number of communications required during training. This compromises privacy and security, and can lead to high energy consumption, and in some cases, make model training unfeasible due to the high number of communications. In this paper, we introduce Sherpa.ai Blind Vertical Federated Learning (SBVFL), a novel paradigm that leverages a distributed training mechanism enhanced for privacy and security. Decoupling the vast majority of node updates from the server dramatically reduces node-server communication. Experiments show that SBVFL reduces communication by ~99% compared to standard VFL while maintaining accuracy and robustness. Therefore, SBVFL enables practical, privacy-preserving VFL across sensitive domains, including healthcare, finance, manufacturing, aerospace, cybersecurity, and the defense industry.
Operator learning has emerged as a promising paradigm for developing efficient surrogate models to solve partial differential equations (PDEs). However, existing approaches often overlook the domain knowledge inherent in the underlying PDEs and hence suffer from challenges in capturing temporal dynamics and generalization issues beyond training time frames. This paper introduces a deep neural ordinary differential equation (ODE) operator network framework, termed NODE-ONet, to alleviate these limitations. The framework adopts an encoder-decoder architecture comprising three core components: an encoder that spatially discretizes input functions, a neural ODE capturing latent temporal dynamics, and a decoder reconstructing solutions in physical spaces. Theoretically, error analysis for the encoder-decoder architecture is investigated. Computationally, we propose novel physics-encoded neural ODEs to incorporate PDE-specific physical properties. Such well-designed neural ODEs significantly reduce the framework's complexity while enhancing numerical efficiency, robustness, applicability, and generalization capacity. Numerical experiments on nonlinear diffusion-reaction and Navier-Stokes equations demonstrate high accuracy, computational efficiency, and prediction capabilities beyond training time frames. Additionally, the framework's flexibility to accommodate diverse encoders/decoders and its ability to generalize across related PDE families further underscore its potential as a scalable, physics-encoded tool for scientific machine learning.
We study the $P_1$ finite element approximation of the best constant in the classical Hardy inequality over bounded domains containing the origin in $\mathbb{R}^N$, for $N \geq 3$. Despite the fact that this constant is not attained in the associated Sobolev space $H^1$, our main result establishes an explicit, sharp, and dimension-independent rate of convergence proportional to $1/|\log h|^2$. The analysis carefully combines an improved Hardy inequality involving a reminder term with logarithmic weights, approximation estimates for Hardy-type singular radial functions constituting minimizing sequences, properties of piecewise linear and continuous finite elements, and weighted Sobolev space techniques. We also consider other closely related spectral problems involving the Laplacian with singular quadratic potentials obtaining sharp convergence rates.
We investigate the controllability of the competition-diffusion Lotka-Volterra system. Our primary focus is on the one-dimensional setting with Dirichlet boundary controls, interpreted as ecological management policies regulating the density of species at the habitat boundaries and satisfying bilateral constraints. We show that the system can be steered from any initial state to a constant steady state representing the extinction of the less competitive species. In contrast, we prove that controllability toward a steady state where the more competitive species vanishes is generally not achievable when the inter-species competition rates are too unbalanced. This obstruction is due to the existence of barrier solutions, which we explicitly construct based on the spectral properties of the associated reaction-diffusion operators. Our theoretical results are illustrated through numerical simulations and are accompanied by a discussion of open problems and potential directions for future research.
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.
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.
We address binary classification using neural ordinary differential equations from the perspective of simultaneous control of $N$ data points. We consider a single-neuron architecture with parameters fixed as piecewise constant functions of time. In this setting, the model complexity can be quantified by the number of control switches. Previous work has shown that classification can be achieved using a point-by-point strategy that requires $O(N)$ switches. We propose a new control method that classifies any arbitrary dataset by sequentially steering clusters of $d$ points, thereby reducing the complexity to $O(N/d)$ switches. The optimality of this result, particularly in high dimensions, is supported by some numerical experiments. Our complexity bound is sufficient but often conservative because same-class points tend to appear in larger clusters, simplifying classification. This motivates studying the probability distribution of the number of switches required. We introduce a simple control method that imposes a collinearity constraint on the parameters, and analyze a worst-case scenario where both classes have the same size and all points are i.i.d. Our results highlight the benefits of high-dimensional spaces, showing that classification using constant controls becomes more probable as $d$ increases.
In this work, we address three non-convex optimization problems associated with the training of shallow neural networks (NNs) for exact and approximate representation, as well as for regression tasks. Through a mean-field approach, we convexify these problems and, applying a representer theorem, prove the absence of relaxation gaps. We establish generalization bounds for the resulting NN solutions, assessing their predictive performance on test datasets and, analyzing the impact of key hyperparameters on these bounds, propose optimal choices. On the computational side, we examine the discretization of the convexified problems and derive convergence rates. For low-dimensional datasets, these discretized problems are efficiently solvable using the simplex method. For high-dimensional datasets, we propose a sparsification algorithm that, combined with gradient descent for over-parameterized shallow NNs, yields effective solutions to the primal problems.
We study the stability of one-dimensional linear hyperbolic systems with non-symmetric relaxation. Introducing a new frequency-dependent Kalman stability condition, we prove an abstract decay result underpinning a form of inhomogeneous hypocoercivity. In contrast with the homogeneous setting, the decay rates depend on how the Kalman condition is fulfilled and, in most cases, a loss of derivative occurs: one must assume an additional regularity assumption on the initial data to ensure the decay. Under structural assumptions, we refine our abstract result by providing an algorithm, of wide applicability, for the construction of Lyapunov functionals. This allows us to systematically establish decay estimates for a given system and uncover algebraic cancellations (beyond the reach of the Kalman-based approach) reducing the loss of derivatives in high frequencies. To demonstrate the applicability of our method, we derive new stability results for the Sugimoto model, which describes the propagation of nonlinear acoustic waves, and for a beam model of Timoshenko type with memory.
Given a convex set $Ω$ of $\mathbb{R}^n$, we consider the shape optimization problem of finding a convex subset $ω\subset Ω$, of a given measure, minimizing the $p$-distance functional $$\mathcal{J}_p(ω) := \left(\int_{\mathbb{S}^{n-1}} |h_Ω-h_ω|^p d\mathcal{H}^{n-1}\right)^{\frac{1}{p}},$$ where $1 \le p <\infty$ and $h_ω$ and $h_Ω$ are the support functions of $ω$ and the fixed container $Ω$, respectively. We prove the existence of solutions and show that this minimization problem $Γ$-converges, when $p$ tends to $+\infty$, towards the problem of finding a convex subset $ω\subset Ω$, of a given measure, minimizing the Hausdorff distance to the convex $Ω$. In the planar case, we show that the free parts of the boundary of the optimal shapes, i.e., those that are in the interior of $Ω$, are given by polygonal lines. Still in the $2-d$ setting, from a computational perspective, the classical method based on optimizing Fourier coefficients of support functions is not efficient, as it is unable to efficiently capture the presence of segments on the boundary of optimal shapes. We subsequently propose a method combining Fourier analysis and a recent numerical scheme, allowing to obtain accurate results, as demonstrated through numerical experiments.
We study the tracking or sidewise controllability of the heat equation. More precisely, we seek for controls that, acting on part of the boundary of the domain where the heat process evolves, aim to assure that the normal trace or flux on the complementary set tracks a given trajectory. The dual equivalent observability problem is identified. It consists on estimating the boundary sources, localized on a given subset of the boundary, out of boundary measurements on the complementary subset. Classical unique continuation and smoothing properties of the heat equation allow us proving approximate tracking controllability properties and the smoothness of the class of trackable trajectories. We also develop a new transmutation method which allows to transfer known results on the sidewise controllability of the wave equation to the tracking controllability of the heat one. Using the flatness approach we also give explicit estimates on the cost of approximate tracking control. The analysis is complemented with a discussion of some possible variants of these results and a list of open problems.
The Lotka-Volterra model reflects real ecological interactions where species compete for limited resources, potentially leading to coexistence, dominance of one species, or extinction of another. Comprehending the mechanisms governing these systems can yield critical insights for developing strategies in ecological management and biodiversity conservation. In this work, we investigate the controllability of a Lotka-Volterra system modeling weak competition between two species. Through constrained controls acting on the boundary of the domain, we establish conditions under which the system can be steered towards various target states. More precisely, we show that the system is controllable in finite time towards a state of coexistence whenever it exists, and asymptotically controllable towards single-species states or total extinction, depending on domain size, diffusion, and competition rates. Additionally, we determine scenarios where controllability is not possible and, in these cases, we construct barrier solutions that prevent the system from reaching specific targets. Our results offer critical insights into how competition, diffusion, and spatial domain influence species dynamics under constrained controls. Several numerical experiments complement our analysis, confirming the theoretical findings.
Federated learning (FL) is a promising framework for learning from distributed data while maintaining privacy. The development of efficient FL algorithms encounters various challenges, including heterogeneous data and systems, limited communication capacities, and constrained local computational resources. Recently developed FedADMM methods show great resilience to both data and system heterogeneity. However, they still suffer from performance deterioration if the hyperparameters are not carefully tuned. To address this issue, we propose an inexact and self-adaptive FedADMM algorithm, termed FedADMM-InSa. First, we design an inexactness criterion for the clients' local updates to eliminate the need for empirically setting the local training accuracy. This inexactness criterion can be assessed by each client independently based on its unique condition, thereby reducing the local computational cost and mitigating the undesirable straggle effect. The convergence of the resulting inexact ADMM is proved under the assumption of strongly convex loss functions. Additionally, we present a self-adaptive scheme that dynamically adjusts each client's penalty parameter, enhancing algorithm robustness by mitigating the need for empirical penalty parameter choices for each client. Extensive numerical experiments on both synthetic and real-world datasets are conducted. As validated by some numerical tests, our proposed algorithm can reduce the clients' local computational load significantly and also accelerate the learning process compared to the vanilla FedADMM.
We analyse a dynamic control problem for scalar reaction-diffusion equations, focusing on the emulation of pattern formation through the selection of appropriate active controls. While boundary controls alone prove inadequate for replicating the complex patterns seen in biological systems, particularly under natural point-wise constraints of the system state, their combination with the regulation of the diffusion coefficient enables the successful generation of such patterns. Our study demonstrates that the set of steady-states is path-connected, facilitating the use of the staircase method. This approach allows any admissible initial configuration to evolve into any stationary pattern over a sufficiently long time while maintaining the system's natural bilateral constraints. We provide also examples of complex patterns that steady-state configurations can adopt.
A new framework to obtain time-decay estimates for partially dissipative hyperbolic systems set on the real line is developed. Under the classical Shizuta-Kawashima (SK) stability condition, equivalent to the Kalman rank condition in control theory, the solutions of these systems decay exponentially in time for high frequencies and polynomially for low ones. This allows to derive a sharp description of the space-time decay of solutions for large time. However, such analysis relies heavily on the use of the Fourier transform that we avoid here, developing the "physical space version" of the hyperbolic hypocoercivity approach introduced by Beauchard and Zuazua, to prove new asymptotic results in the linear and nonlinear settings. The new physical space version of the hyperbolic hypocoercivity approach allows to recover the natural heat-like time-decay of solutions under sharp rank conditions, without employing Fourier analysis or $L^1$ assumptions on the initial data. Taking advantage of this Fourier-free framework, we establish new enhanced time-decay estimates for initial data belonging to weighted Sobolev spaces. These results are then applied to the nonlinear compressible Euler equations with linear damping. We also prove the logarithmic stability of the nonlinearly damped $p$-system.