Sharp numerical approximation of the Hardy constant
L. Ignat, E. Zuazua (2025) Sharp numerical approximation of the Hardy constant Abstract. We study the P1 finite element approximation of the best constant in the classical Hardy inequality over…
Clustering in Pure-Attention Hardmax Transformers and its Role in Sentiment Analysis
A. Alcalde Zafra, G. Fantuzzi, E. Zuazua (2025) Clustering in Pure-Attention Hardmax Transformers and its Role in Sentiment Analysis, SIAM Mathematics of Data Science, SIMODS, arxiv: 2407.01602 Abstract. Transformers are…
A potential game perspective in Federated Learning
K. Liu, Z. Wang, E. Zuazua (2025) A potential game perspective in Federated Learning, arxiv:2411.11793 Abstract. Federated learning (FL) is an emerging paradigm for training machine learning models across distributed…
Optimal convergence rates for the finite element approximation of the Sobolev constant
L. Ignat, E. Zuazua (2025) Optimal convergence rates for the finite element approximation of the Sobolev constant Abstract. We establish optimal convergence rates for the P1 finite element approximation of…
Boundary observation and control for fractional heat and wave equations
U. Biccari, E. Zuazua (2025) Boundary observation and control for fractional heat and wave equations, https://doi.org/10.48550/arXiv.2504.17413 Abstract. We establish boundary observability and control for the fractional heat equation over arbitrary…
Turnpike in optimal control and beyond: A survey
E. Trélat, E. Zuazua (2025)Turnpike in optimal control and beyond: A survey https://hal.science/hal-05002039 Abstract. The turnpike principle is a fundamental concept in optimal control theory, stating that for a wide…
Representation and Regression Problems in Neural Networks: Relaxation, Generalization, and Numerics
K. Liu, E. Zuazua (2025) Representation and Regression Problems in Neural Networks: Relaxation, Generalization, and Numerics, Math. Models Methods Appl. Sci., Vol. 35, No. 6, pp. 1471-1521, arXiv:2412.01619v1, https://doi.org/10.1142/S0218202525500228 Abstract. In…
Exact Sequence Classification with Hardmax Transformers
A. Alcalde Zafra, G. Fantuzzi, E. Zuazua (2025) Exact Sequence Classification with Hardmax Transformers, arxiv: 2502.02270 Abstract. We prove that hardmax attention transformers perfectly classify datasets of $N$ labeled sequences…
FedADMM-InSa: An Inexact and Self-Adaptive ADMM for Federated Learning
Y. Song, Z. Wang, E. Zuazua (2025) FedADMM-InSa: An Inexact and Self-Adaptive ADMM for Federated Learning, Neural Networks, pp. 106-772, ISSN 0893-6080, https://doi.org/10.1016/j.neunet.2024.106772 Abstract. Federated learning (FL) is a promising…
Universal Approximation of Dynamical Systems by Semi-Autonomous Neural ODEs and Applications
Z. Li, K. Liu, L. Liverani, E. Zuazua (2025)Universal Approximation of Dynamical Systems by Semi-Autonomous Neural ODEs and Applications, arXiv: 2407.17092 Abstract. In this paper, we introduce semi-autonomous neural ordinary…
Deep Neural Networks: Multi-Classification and Universal Approximation
M. Hernandez, E. Zuazua (2025)Deep Neural Networks: Multi-Classification and Universal Approximation arXiv:2409.06555 Abstract. We demonstrate that a ReLU deep neural network with a width of 2 and a depth of…
Approximate and Weighted Data Reconstruction Attack in Federated Learning
Z. Wang, Y. Song, E. Zuazua (2025) Approximate and Weighted Data Reconstruction Attack in Federated Learning arXiv:2308.06822 Abstract. Federated Learning (FL) is a distributed learning paradigm that enables multiple clients…
Progress and future directions in machine learning through control theory
E. Zuazua (2024) Progress and future directions in machine learning through control theory Proceedings book FGS 2024, pp. 116, https://hdl.handle.net/10651/74677 Abstract. This paper presents our recent advancements at the intersection…
Stability and Convergence of a Randomized Model Predictive Control Strategy
D.W.M. Veldman, A. Borkowski, E. Zuazua (2024) Stability and Convergence of a Randomized Model Predictive Control Strategy , IEEE Trans. Automat., Vol. 69, No. 9, pp. 6253-6260 Abstract. This paper is…
Control of neural transport for normalising flows
D. Ruiz-Balet, E. Zuazua (2024)Control of neural transport for normalising flows J. Math. Pures Appl., Vol. 181, pp. 58-90, ISSN 0021-7824. https://doi.org/10.1016/j.matpur.2023.10.005 Abstract. Inspired by normalizing flows, we analyze the…
Optimal Placement and Shape Design of Sensors via Geometric Criteria
I. Ftouhi, E. Zuazua (2024) Optimal Placement and Shape Design of Sensors via Geometric Criteria, 2024 European Control Conference (ECC), Stockholm, Sweden, 2024, pp. 834-837, doi: 10.23919/ECC64448.2024.10591236 Abstract. The optimal…
Controllability properties from the exterior under positivity constraints for a 1-D fractional heat equation
H. Antil, U. Biccari, R. Ponce, M. Warma, S. Zamorano (2024) Controllability properties from the exterior under positivity constraints for a 1-D fractional heat equation, Vol. 13, pp. 893-924, https://doi.org/10.3934/eect.2024010…
Interplay between depth and width for interpolation in neural ODEs
A. Alvarez-Lopez, A. Hadj Slimane, E. Zuazua (2024) Interplay between depth and width for interpolation in neural ODEs, Neural Networks, Vol. 180, pp. 106-640, https://doi.org/10.1016/j.neunet.2024.106640, ISSN 0893-6080 Abstract. Neural ordinary…
Uniform Turnpike Property and Singular Limits
M Hernandez-Salinas, E. Zuazua (2024) Uniform Turnpike Property and Singular Limits Acta Appl. Math., Vol. 190, No. 3, https://doi.org/10.1007/s10440-024-00640-7 Abstract. Motivados por límites singulares para problemas de control óptimo a…
Large-time asymptotics in deep learning
C. Esteve, B. Geshkovski, D. Pighin, E. Zuazua (2025) Large-time asymptotics in deep learning, https://hal.archives-ouvertes.fr/hal-02912516 Abstract. We consider the neural ODE perspective of supervised learning and study the impact of the…
Multilevel control by duality
U. Biccari, E. Zuazua (2023) Multilevel control by duality Systems & Control Letters, Vol. 175, 105-502, https://doi.org/10.1016/j.sysconle.2023.105502 Abstract. We discuss the multilevel control problem for linear dynamical systems, consisting in designing…
Neural ODE Control for Classification, Approximation and Transport
D. Ruiz-Balet, E. Zuazua (2023)Neural ODE Control for Classification, Approximation and Transport, SIAM Review, Vol. 65, No. 3, pp. 735-773, https://doi.org/10.1137/21M1411433 Abstract. We analyze Neural Ordinary Differential Equations (NODEs) from…
Optimal design of sensors via geometric criteria
I. Ftouhi, E. Zuazua (2023) Optimal design of sensors via geometric criteria, J. Geom. Anal., Vol. 33, No. 253, https://doi.org/10.1007/s12220-023-01301-1 Abstract. We consider a convex set Ω and look for…
Multiplicity of solutions for fractional q(.)-Laplacian equations
A. Rahmoune, U. Biccari (2023) Multiplicity of solutions for fractional -Laplacian equations., J Elliptic Parabol Equ, Vol. 9, pp. 1101–1129, https://doi.org/10.1007/s41808-023-00239-3 Abstract. In this paper, we deal with the following elliptic…
Multilevel Selective Harmonic Modulation via Optimal Control
U. Biccari, C. Esteve-Yagüe, D.J. Oroya-Villalta (2022) Multilevel Selective Harmonic Modulation via Optimal Control., Appl Math Optim, Vol. 86, No. 43, https://doi.org/10.1007/s00245-022-09917-5 Abstract. We consider the Selective Harmonic Modulation (SHM) problem,…
Multilevel selective harmonic modulation by duality
U. Biccari, E. Zuazua Multilevel selective harmonic modulation by duality (2022) IFAC-PapersOnLine, Vol. 55, No. 16, pp. 56-61, https://doi.org/10.1016/j.ifacol.2022.08.081. Abstract. We address the Selective Harmonic Modulation (SHM) problem in power…
Differentiability with respect to the initial condition for Hamilton-Jacobi equations
C. Esteve, E. Zuazua. Differentiability with respect to the initial condition for Hamilton-Jacobi equations (2022) SIAM Journal on Mathematical Analysis, Vol. 54, No. 5, pp. 5388-5423 https://doi.org/10.1137/22M1469353 Abstract. We prove…
A framework for randomized time-splitting in linear-quadratic optimal control
D.W.M. Veldman, E. Zuazua (2022) A framework for randomized time-splitting in linear-quadratic optimal control Numerische Mathematik, Vol. 151, No. 2, DOI: 10.1007/s00211-022-01290-3 Abstract. Inspired by the successes of stochastic algorithms in…