Authors: Ziqian Li, Kang Liu, Yongcun Song, Hangrui Yue, and Enrique Zuazua Learning a family of PDE solutions by making time evolution part of the model. A simulation rarely ends…
Authors: Albert Alcalde, Giovanni Fantuzzi, Enrique Zuazua Modern machine learning systems often process a large amount of input data to produce a short output. A document may contain thousands of…
Authors: Kang Liu, Ziqi Wang, Enrique Zuazua A shared AI model needs more than a way to exchange updates: it needs participants willing to do the work. Our CoDeFeL research…
Authors: Xinyu Cheng, Yunzhang Li, Weiran Xiong, Enrique Zuazua Partial differential equations (PDEs) describe how quantities evolve in space and time, with applications in fluid dynamics, finance, control, physics, and…
Author: Antonio Álvarez-López A model can remember every example in its training set and still fail on the next one. This familiar tension between interpolation and generalization becomes particularly interesting…
Author: Bo You Every forecast begins from an estimated state rather than nature’s exact one. Weather stations leave spatial gaps, ocean sensors sample only a small fraction of the water…
Authors: Umberto Biccari, Roberto Morales, Enrique Zuazua Many artificial-intelligence systems make several predictions at once. A medical model might use the same blood tests to estimate several health indicators; a…
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Modern machine learning often begins with an implicit assumption: somewhere, somehow, the data can be brought together. But in many of the settings where artificial intelligence is most needed (hospitals,…
Authors: Umberto Biccari, Jun Chen, Roberto Morales, Enrique Zuazua Imagine trying to infer the properties of a material or a fluid when no single model can see all the measurements.…
Author: Enrique Zuazua Read the complete contribution on SIAM News.
Authors: Umberto Biccari An accurate prediction does not reveal the path that produced it. Consider a Transformer that predicts biological age from clinical measurments. A blood panel goes in, an…
Authors: Roberto Morales Modern machine learning often begins with an implicit assumption: somewhere, somehow, the data can be brought together. But in many of the settings where artificial intelligence is…
Authors: Jun Chen, Umberto Biccari, Junmin Wang Imagine a system that must adapt as conditions change. A robot may encounter a different load, an aircraft may move into a new…
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…
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…
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…
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…
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…
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…
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…
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…
T. Crin-Barat, N. De Nitti, E. Zuazua (2025) On the decay of one-dimensional locally and partially dissipated and hyperbolic systems, ESAIM:COCV, https://arxiv.org/abs/2206.00555 Abstract. We study the time-asymptotic behavior of linear…
J.A. Bárcena-Petisco, M. Cavalcante, G.M. Coclite, N. de Nitti, E. Zuazua (2025) Control of Hyperbolic and Parabolic Equations on Networks and Singular limits, MCRF, Vol. 15, No. 1, March 2025,…
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…
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…
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…
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…
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…
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…
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…
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…
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…
DyCMaMod. Dinámica y control para el aprendizaje automático y la Modelización Project reference: PID2023-146872OB-I00 Funded by: MINECO. Ministerio de Economía, Comercio y Empresa Principal Investigator (PI): Prof. Enrique Zuazua (University…
WP1 | WP2 | WP3 | WP4 • CoDeFeL GitHub • CoDeFeL Blog • Technology Transfer • Data Management Plan Machine Learning (ML) is transforming Applied Mathematics, creating powerful…
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…
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…
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…
C. Esteve, E. Zuazua. Reachable set for Hamilton-Jacobi equations with non-smooth Hamiltonian and scalar conservation laws Nonlinear Analysis, Vol. 227, https://doi.org/10.1016/j.na.2022.113167 Abstract. We give a full characterization of the range…
I. Mazari, D. Ruiz-Balet, E. Zuazua (2023)Constrained control of gene-flow models, Ann. Inst. Henri Poincare (C) Anal. Non Lineaire ⟨hal-02373668⟩ Abstract. In ecology and population dynamics, gene-flow refers to the transfer…
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…
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…
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…
DyCon: Dynamic Control is an European project funded by the European Research Council – ERC (2016 - 2022), focused at making a breakthrough contribution in the broad area of Control of Partial Differential Equations (PDE) and their numerical approximation methods by addressing key unsolved issues appearing systematically in real-life applications. The field of PDEs, together with numerical approximation, simulation methods and control theory, has evolved significantly to address the industrial demands.
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…
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,…
Project name: DASEL, Ciencia de Datos para Redes Eléctricas Project reference: TED2021-131390B-I00 AEI: TED2021-131390B-I00/ AEI/10.13039/501100011033 DASEL Funding source: MINECO. Ministry of Science, Innovation and Universities. Proyectos de Transición Ecológica y…
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…
Spain 21.11.2022 Spectral inequalities for pseudo-differential operators and control theory on compact manifolds Author: Duvan Cardona In this post, we explain some results related to the null-controllability of diffusion models…
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…
Domènec Ruiz-Balet, , Enrique Zuazua. Control of certain parabolic models from biology and social sciences(2022) Math Control and Related Fields, Vol. 12, No. 4, pp 955-1038. doi: https://doi.org/10.3934/mcrf.2022032 Abstract. These…
R. Bianchini, Crin-Barat T., M. Paicu. Relaxation approximation and asymptotic stability of stratified solutions to the IPM equation (2022) Abstract. We prove the nonlinear asymptotic stability of stably stratified solutions…
Crin-Barat T., L. Shou. Diffusive Relaxation Limit of the Multi-Dimensional Hyperbolic Jin-Xin System (2022) Abstract. In this paper we study the diffusive relaxation limit of the Jin-Xin system toward viscous…
Spain 10.08.2022 Constructing the Scattering Transform Author: Javier Minguillón Sánchez 1 Introduction As Mallat presents it in [8], the scattering transform appears from the pursuit of a representation of…
Spain 05.08.2022 Null Controllability of a Nonlinear Age, Space and Two-sex Structured Population Dynamics Model Author: Yacouba Simpore Motivation and description of two sex structured population dynamics model Motivation Malaria…
Spain 26.07.2022 Long-time behavior of locally and partially dissipated hyperbolic systems Author: Timothée Crin-Barat In this post we are interested in the long-time behavior of solutions to locally and partially…
Project name: KiLearn - Kinetic equations and Learning control Project reference: PID2020-112617GB-C22 AEI: PID2020-112617GB-C22/AEI/10.13039/501100011033 KiLearn Funding source: MINECO, Ministry of Science, Innovation and Universities Duration: September 2021 - May 2025…
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…
D. Pighin. Nonuniqueness of minimizers for semilinear optimal control problems (2022) J. Eur. Math. Soc, , published online first. https://doi.org/10.4171/jems/1232 Abstract. A counterexample to uniqueness of global minimizers of semilinear…
Model Predictive Control with Random Batch Method for Linear-Quadratic Optimal Control: Introduction and Matlab Implementation Spain. 02.05.2022 Author: Alexandra Borkowski Optimal Control Problems in Practice Imagine that you are given…
C. Esteve, H. Kouhkouh, D. Pighin, E. Zuazua. The turnpike property and the long-time behavior of the Hamilton-Jacobi-Bellman equation for finite-dimensional LQ control problems. Math. Control Signals Syst. (2022). https://doi.org/10.1007/s00498-022-00325-2 Abstract:…
Spain. 29.03.2022 Author: Elisa Affili Introduction In this post, we present some basic facts about the study of geometry on spheres. In fact, geometry on the sphere is surprisingly different…
Esteve C., Geshkovski G., Pighin D., Zuazua E. . Turnpike in Lipschitz-nonlinear optimal control Nonlinearity. Vol 5. No. 34, pp. 1652-1701 (2022) https://doi.org/10.1088/1361-6544/ac4e61 Abstract. We present a new proof of…
Shi Jin, Yuhua Zhu, Enrique Zuazua. The Vlasov-Fokker-Planck Equation with High Dimensional Parametric Forcing Term Numer. Math. 150, 479–519 (2022). https://doi.org/10.1007/s00211-021-01257-w Abstract. We consider the Vlasov-Fokker-Planck equation with random electric…
Spain. 03.02.2022 Author: Javier Minguillón Sánchez Introduction We can define an operator between function spaces by modifying the frequencies in which a function decomposes via the Fourier transform. If the…
Spain. 24.01.2022 Author: Arnaud Rippol Introduction Pendulums are systems often used in Control Theory or Reinforcement Learning as benchmarks examples to prove the efficiency of some methods or algorithms. The…
Spain. 13.11.2021 Author: Pierre Mordant Introduction Unsupervised learning is the field of machine learning which tries to get as much information as possible on the probability distribution of the training…
Ko Dongnam, Zuazua Enrique. Model predictive control with random batch methods for a guiding problem (2021). Math. Models Methods Appl. Sci., Vol. 31, No. 8, pp. 1569-1592. (2021) DOI: https://doi.org/10.1142/S0218202521500329…
Heiland J., Zuazua E. Classical system theory revisited for Turnpike in standard state space systems and impulse controllable descriptor systems (2021) SIAM J. Control Optim., Vol. 59.5 (2021), pp. 3600-3624 Abstract.…
Jon Asier Bárcena-Petisco. Optimal control for neural ODE in a long time horizon and applications to the classification (2021) Abstract. We study the optimal control in a long time horizon…
J.A. Bárcena-Petisco, Kevin Le Balc'H. Local null controllability of the penalized Boussinesq system with a reduced number of controls (2021) Abstract. In this paper we consider the Boussinesq system with homogeneous…
B. Geshkovski, E. Zuazua. Controllability of one-dimensional viscous free boundary flows. Siam. J. Control. Optim (2021), Vol. 59, No. 3, pp. 1830–1850. https://doi.org/10.1137/19M1285354 Abstract. In this work, we address the…
D. Pighin The turnpike property in semilinear control ESAIM Control Optim. Calc. Var., Vol. 26 (2021), pp. 1-48 Abstract.An exponential turnpike property for a semilinear control problem is proved. The…
Spain. 12.04.2021. Our team members Deyviss Jesús Oroya-Villalta, Carlos Esteve-Yagüe and Umberto Biccari, made a contribution to the DyCon Blog about "Multilevel Selective Harmonic Modulation via Optimal Control": Selective harmonic Modulation (SHM)…
M. Gugat, M. Schuster, E. Zuazua. The Finite-Time Turnpike Phenomenon for Optimal Control Problems: Stabilization by Non-Smooth Tracking Terms, in "Stabilization of Distributed Parameter Systems: Design Methods and Applications". Grigory…
Jon Asier Bárcena-Petisco, Sergio Guerrero and Ademir F. Pazoto. Local null controllability of a model system for strong interaction between internal solitary waves. Commun. Contemp. Math. (2021) https://doi.org/10.1142/S0219199721500036 Abstract. In…
Spain. 10.03.2021. Aicha Karite recent member from our team, made a contribution to the DyCon Blog about "Points of Interest Extraction and Prediction of Next Locations": Generally, a prediction problem…
B. Geshkovski, E. Zuazua. Control and Deep Learning: Some connections (2021) This note is an extended abstract for a talk given by the second author during the workshop ”Challenges in…
Spain. 09.03.2021 Author: Sergi Andreu Perceptrons, Neural Networks and Dynamical Systems // This post is last part of the "Deep Learning and Paradigms" post Binary classification with Neural Networks When dealing with data classification,…
Spain. 01.03.2021 Author: Sergi Andreu Deep Learning & Paradigms By
Spain. 24.02.2021 Author: Sergi Andreu Opening the black box of Deep Learning By
U. Biccari Internal control for a non-local Schrödinger equation involving the fractional Laplace operator (2022), Vol. 11, No. 1: 301-324. doi: 10.3934/eect.2021014 Abstract: We analyze the interior controllability problem for…
Lohéac J., Trélat E., Zuazua E. Nonnegative control of finite-dimensional linear systems. Ann. I. H. Poincare-An., Vol. 38, No. 2, pp. 301-346. (2021) DOI: https://doi.org/10.1016/j.anihpc.2020.07.004 Abstract. We consider the controllability…
M. Gnuffi, D. Pighin, N. Sakamoto Rotor imbalance suppression by optimal control Optimal Control Applications and Methods, Vol. 43, No. 1 (2022), pp. 213-242. Abstract. An imbalanced rotor is considered.…
Spain. 04.12.2020. Our team member Jon Asier Bárcena Petisco and our Head Enrique Zuazua made a contribution to the DyCon Blog about "Averaged dynamics and control for heat equations with…
Esteve C., Zuazua E.. The Inverse Problem for Hamilton-Jacobi equations and Semiconcave Envelopes SIAM J. Math. Anal., Vol. 52, No. 6, pp. 5627–5657 (2020). https://doi.org/10.1137/20M1330130 Abstract. We study the inverse…
B. Geshkovski,Null-controllability of perturbed porous medium gas flow. ESAIM:COCV, vol. 26, No. 85 (2020). DOI: 10.1051/cocv/2020009 Abstract: In this work, we investigate the null-controllability of a nonlinear degenerate parabolic equation,…
Spain. 21.07.2019. Interview to Noboru Sakamoto about the DyCon Working Package 2: "Long time horizon control & Turnpike" Read more about DyCon ERC Project
Spain. 29.10.2020. Our team member Carlos Esteve made a contribution to the DyCon Blog about "Q-learning for finite-dimensional problems": Reinforcement Learning (RL) is, together with Supervised Learning and Unsupervised Learning,…
Ruiz-Balet D., Zuazua E.. Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations. J. Math. Pures Appl, vol. 143, pp. 345-375 (2020) https://doi.org/10.1016/j.matpur.2020.08.006 Abstract. Dynamic phenomena in social and…
Spain. 20.10.2020. Jesus Oroya and Umberto Biccari, researchers at our team, made a contribution to the DyCon Blog about "Moving control strategy for memory-type equations" Take a look the detailed…
Monge A., Zuazua E. Sparse source identification of linear diffusion–advection equations by adjoint methods. Syst. Control. Lett, vol. 145 (2020). DOI: https://doi.org/10.1016/j.sysconle.2020.104801 Abstract: We present an algorithm for the time-inversion…
Pighin D., Sakamoto N. The turnpike with lack of observability (2020) Abstract: We obtain turnpike results for optimal control problems with lack of stabilizability in the state equation and/or detectability in…
Maity D., Tucsnak M., Zuazua E. Controllability of a Class of Infinite Dimensional Systems with Age Structure. Control Cybern. Vol. 48 (2020), No. 2, pp. 231-260 hal-01964612v2 Abstract: Given a…
Lance G., Trélat E., Zuazua E. Shape turnpike for linear parabolic PDE models Syst. Control. Lett. Vol. 142 (2020). DOI: 10.1016/j.sysconle.2020.104733 Abstract: We introduce and study the turnpike property for…
Umberto Biccari, Enrique Zuazua. A stochastic approach to the synchronization of coupled oscillators. Frontiers in Energy Research, section Smart Grids. Front. Energy Res. Vol. 8 (2020). DOI: 10.3389/fenrg.2020.00115 Abstract. This paper…
J.-C. Cortes, A. Navarro-Quiles, J.-V. Romero, M.-D. Rosello, E. Zuazua. Full probabilistic solution of a infinite dimensional linear control system with random initial and final conditions. J. Franklin I. (2020)…
Spain. 03.06.2020. Umberto Biccari, and Enrique Zuazua made a contribution to the DyCon Blog about " Synchronized Oscillators" Take a look the detailed explanation at DyCon Blog All posts at…
Spain. 03.06.2020. Ana Navarro, Umberto Biccari, and Enrique Zuazua made a contribution to the DyCon Blog about "Stochastic optimization for simultaneous control" Take a look the detailed explanation at DyCon…
G. Lance, E. Trélat, E. Zuazua Turnpike in optimal shape design IFAC-PapersOnline, Vol. 52, No. 16 (2019), pp. 496-501, ISSN: 24058963, DOI: 10.1137/17M1119160 Abstract: We investigate the turnpike problem in…
