WP4 – Modelling through Control and Machine Learning
Control theory provides a principled framework for regulating dynamical systems, while Machine Learning enables complex patterns and predictive models to be extracted from data. By integrating these two fields, we develop efficient, robust, and adaptive methodologies that combine data-driven learning with feedback mechanisms. This synergy provides a foundation for systems capable of adapting their behavior in real time while maintaining reliable performance in dynamic and uncertain environments.
Task 4.1 – Learning unknown physics from observations
Physical models often contain parameters or mechanisms that cannot be measured directly. By combining observed data with ODE/PDE models and optimization, these unknown components can be identified while preserving the underlying physical description of the system.
Task 4.2 – Building models when the equations are unknown
Sometimes there is enough data to observe a system, but not enough knowledge to write down the equations governing it. ResNets can then act as surrogate dynamical models, reconstructing continuous trajectories from discrete observations and generating synthetic data describing possible evolutions.
Task 4.3 – Combining physics and Machine Learning
Physics-based models provide structure and interpretability, while Machine Learning can capture information that the equations miss. Hybrid PDE-Machine Learning models combine these two sources of knowledge, allowing data and physical laws to correct and complement each other.
Publications
Liu K., & Zuazua E. (2026). Moments, Time-Inversion and Source Identification for the Heat Equation. Inv. Problems, 42(1), 015009. arXiv:2507.02677
Lyu, K., Biccari, U., Wang JM (2026). Operator learning for prescribed-time stabilization of reaction-diffusion systems. Submitted. arXiv:2602.23157
Liu K., & Zuazua E. (2026). A PDE Perspective on Generative Diffusion Models. Submitted. arXiv:2511.05940
Liverani L., & Zuazua E. (2026). HYCO: A Formalism for Hybrid-Cooperative PDE Modelling. Submitted. arXiv:2602.23859
Chen, J., Biccari, U., & Wang, J. (2026). Learning the Riccati solution operator for time-varying LQR via Deep Operator Networks. Submitted. arXiv:2604.18507
Liu K., & Zuazua E. (2026). Geometric Asymptotics of Score Mixing and Guidance in Diffusion Models. Submitted. arXiv:2605.12231
Li, Z., Liu, K., Song, Y., Yue, H., & Zuazua, E. (2025). Deep Neural ODE Operator Networks for PDEs. Math. Models Methods Appl. Sci., 36(08), 1739-1782. arXiv:2510.15651
Hernández, M. & Zuazua, E. (2025). Random Batch Methods for Discretized PDEs on Graphs. Submitted. arXiv:2506.11809
Lyu, K., Biccari, U., & Wang, J. (2025). Robust stabilization of hyperbolic PDE-ODE systems via Neural Operator-approximated gain kernels. Submitted. arXiv:2508.03242
Jin, S., Ma, C., & Zuazua, E. (2025). Transmutation based Quantum Simulation for Non-unitary Dynamics. Submitted. arXiv:2601.03616
Lü, Q., Ma, B., & Zuazua, E. (2025) Mean-Square Stability of Continuous-Time Stochastic Model Predictive Control . Submitted. arXiv:2512.03516
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