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
Principal Investigators (PI): Miguel Escobedo (UPV/EHU), Enrique Zuazua (Deusto Foundation)
About the Project
KiLearn The project arises from the necessity to deepen the understanding of several fundamental questions in Machine Learning, quantum gases, and protein’s dynamics through a detailed study of the mathematical properties of some models that are currently used in those fields. These questions include the time evolution of a Bose-Einstein condensate, the fragmentation dynamics of proteins from experiments, or the impact of the architecture on the performances of a Neural Network.
The increasing interconnection between Machine Learning and kinetic theory motivates a research path that develops on the boundary between these two worlds and has the potential of leading to deep and breakthrough results in both areas. Starting from this hypothesis, the general objective of the research project KiLearn is to develop new Machine Learning-inspired mathematical tools for the analysis of Kinetic Equations and, at the same time, to extend the present knowledge in Machine Learning and Neural Networks taking advantage of the vast theoretical baggage that kinetic theory has to offer.
Project Members
• Miguel Escobedo – UPV/EHU, University of the Basque Country
• Enrique Zuazua – University of Deusto
• Umberto Biccari – University of Deusto
• Iker Pastor – University of Deusto
• Martin Lazar, Dubrovnik University
• Jon Asier Bárcena-Petisco, UPV/EHU
• Nicola de Nitti, FAU
• Carlos Esteve, UAM
• Borjan Geshkovski, UAM
• Domènec Ruiz-Balet, UAM
• Yongcun Song, FAU
KiLearn Toolbox
• Learning with Neural ODEs Toolbox
Author: Borjan Geshkovski
Language: Python
A toolbox for learning with neural ODEs.
• The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approach
Author: Yongcun Song
Language: Python / MATLAB
A source code to study the combination of the alternating direction method of multipliers (ADMM) with physics-informed neural networks (PINNs) for a general class of nonsmooth partial differential equation (PDE)-constrained optimization problems, where additional regularization can be employed for constraints on the control or design variables.
• PINNS Wave Equation
Author: Daniël Veldman
Language: Python & MATLAB
An implementation of Physics-Informed Neural Networks (PINNs) to solve various forward and inverse problems for the 1 dimensional wave equation.
• pyControls Framework
Author: Daniël Veldman
Language: C++ & Python
The pyControls package provides a framework for simulations of gas networks with arbirary geometry and edge dynamics. This package provides a high-order solver for the isoeuler equations and a reader for the GasLib scenarios. Furthermore, the network is easily plotted either in 2D or 3D manner.
• Model Predictive Control
Author: Daniël Veldman
Language: MATLAB
Model Predictive Control for course discretizations of the heat and wave equation.
• A sheep herding game
Author: Daniël Veldman
Language: MATLAB
A sheep herding game in MATLAB for the Long Night of Sciences (Lange Nacht der Wissenschaften) Erlangen-Furth-Nürnberg 2022.
• RBM-MPC
Author: Daniël Veldman
Language: MATLAB
This is the code we used for the numerical experiments with RBM-MPC, a combination of Model Predictive Control (MPC) and Random Batch Methods (RBMs).
• HJ – Inverse design
Author: Daniël Veldman
Language: MATLAB
Our goal here is to study the inverse design problem associated to Hamilton Jacobi Equations (HJ)
• The interplay of control and deep learning
Author: Borjan Geshkovski
Deep supervised learning by merging the latter with well-known subfields of mathematical control theory and numerical analysis.
• Hamilton-Jacobi Equations: Inverse Design
Author: Carlos Esteve
Source code to study the inverse design problem associated to Hamilton Jacobi Equations (HJ).
• Randomized time-splitting in linear-quadratic optimal control
Author: Daniël Veldman
Solving an optimal control problem for a large-scale dynamical system can be computationally demanding.
• Control of Advection-Diffusion Equations on Networks and Singular Limits
Author: Jon Asier Bárcena-Petisco, Márcio Cavalcante, Giuseppe Maria Coclite, Nicola de Nitti and Enrique Zuazua
We define an upper bound and a lower bound on the time in which the information propagates across the network (i.e., the maximal and minimal travel time of the characteristics across the network).
• Averaged dynamics and control for heat equations with random diffusion
Author: Jon Asier Bárcena Petisco, Enrique Zuazua
to illustrate the effect of averaging in the dynamics, let us study the dynamics of (1) when =^ and =0. As averaging and the Fourier transform commute, we work on the Fourier transform of the fundamental solution of the heat equation.
Publications
• L. Ignat, E. Zuazua (2025) Optimal convergence rates for the finite element approximation of the Sobolev constant, preprint, arxiv:2504.17413
• L. Ignat, E. Zuazua (2025) Sharp numerical approximation of the Hardy constant, preprint, arXiv:2506.19422
• U. Biccari, E. Zuazua (2025) Boundary observation and control for fractional heat and wave equations, preprint, arXiv:2504.17413
• N. Nitti, E. Zuazua (2023) On the Controllability of Entropy Solutions of Scalar Conservation Laws at a Junction via Lyapunov Methods Vietnam J. Math., Vol. 51, No. 1, pp. 71-88, https://doi.org/10.1007/s10013-022-00598-9
• 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
• T. Liard, E. Zuazua (2023) Analysis and numerical solvability of backward-forward conservation laws SIAM Journal on Mathematical Analysis, Vol. 55, No. 3, pp. 1949-1968
• 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
• I. Mazari, D. Ruiz-Balet, E. Zuazua (2023) Constrained control of gene-flow models, Ann. Inst. Henri Poincare (C) Anal. Non Lineaire, Vol. 40, No. 3, pp. 717-766 ⟨hal-02373668⟩
• C. Zhang, E. Zuazua (2023) A quantitative analysis of Koopman operator methods for system identification and predictions Comptes Rendus, Mécanique, Vol. 351, No. S1, pp. 1-31. In memory of R. Glowinski
• Jin, Y. Zhu, E. Zuazua (2022) The Vlasov-Fokker-Planck Equation with High Dimensional Parametric Forcing Term, Numer. Math. 150, 479–519 https://doi.org/10.1007/s00211-021-01257-w
• U. Biccari, M. Warma, E. Zuazua (2022) Control and Numerical approximation of Fractional Diffusion Equations, Handb. Numer. Anal. Elsevier. ISSN:1570-8659, DOI: https://doi.org/10.1016/bs.hna.2021.12.001
• 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: https://doi.org/10.1007/s00211-022-01290-3
• C. Esteve, H. Kouhkouh, D. Pighin, E. Zuazua (2022) The turnpike property and the long-time behavior of the Hamilton-Jacobi-Bellman equation for finite-dimensional LQ control problems, Math. Control Signals Syst., https://doi.org/10.1007/s00498-022-00325-2
• Z. Han, E. Zuazua (2022) Slow decay and Turnpike for Infinite-horizon Hyperbolic Linear Quadratic problems, SIAM J. Control Optim. Vol. 60, No. 4, pp. 2440-2468. https://dx.doi.org/10.1137/21M1441985
• D. Ruiz-Balet, E. Zuazua (2022) Control of certain parabolic models from biology and social sciences, Math Control and Related Fields, Vol. 12, No. 4, pp. 955-1038. doi: https://doi.org/10.3934/mcrf.2022032
• C. Esteve, E. Zuazua (2022) Differentiability with respect to the initial condition for Hamilton-Jacobi equations SIAM Journal on Mathematical Analysis, Vol. 54, No. 5, pp. 5388-5423 https://doi.org/10.1137/22M1469353
• U. Biccari, E. Zuazua (2022) Multilevel selective harmonic modulation by duality, IFAC-PapersOnLine, Vol. 55, No. 16, pp. 56-61, https://doi.org/10.1016/j.ifacol.2022.08.081
• U. Biccari, V. Hernández-Santamaría, J. Vancostenoble (2022) Existence and cost of boundary controls for a degenerate/singular parabolic equation, Math. Control Relat. F., Vol. 12, No. 2, pp. 495-530
• 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
• D. Ruiz-Balet, E. Affili, E. Zuazua (2022) Interpolation and approximation via Momentum ResNets and Neural ODEs, IEEE Control Syst. Lett., Vol. 162, ISSN 0167-6911, https://doi.org/10.1016/j.sysconle.2022.105182
• B. Geshkovski, E. Zuazua (2022) Turnpike in Optimal Control of PDES, ResNets, and beyond, Acta Numer., Vol. 31, pp. 135-263. doi:10.1017/S0962492922000046
• U. Biccari (2022) Internal control for a non-local Schrödinger equation involving the fractional Laplace operator, Evol. Eq. Control. Theo., Vol. 11, No. 1, pp. 301-324
• B. Geshkovski, E. Zuazua (2022) Optimal actuator design via Brunovsky’s normal form, IEEE Automatic Control, Vol. 67, No. 12, DOI: 10.1109/TAC.2022.3181222
• C. Esteve, B. Geshkovski, D. Pighin, E. Zuazua (2022) Turnpike in Lipschitz-nonlinear optimal control, Nonlinearity. Vol 5, No. 34, pp. 1652-1701, https://doi.org/10.1088/1361-6544/ac4e61