WP2 – Complexity of ResNet Dynamics

Residual Neural Networks (ResNets) leverage shortcut connections to facilitate gradient propagation, enabling faster convergence and enhanced robustness. Their residual structure establishes a natural connection with time-discrete dynamical systems, revealing distinctive properties that can be studied and exploited through control-theoretic tools. Building on simultaneous control theory and universal approximation principles, we create a rigorous framework for the design of ResNet architectures with enhanced explainability, efficiency, and performance.

Discover our research tasks and outcomes in terms of publications and computational code for this working package.

 

Research tasks

• Task 2.1 – From data complexity to network complexity. Complex data generally require complex learning dynamics, but the relation is far from straightforward. We connect properties such as data clustering and class structure with the complexity of the trainable parameters.
• Task 2.2 – Robustness, explainability and feature relevance. Why are some neural networks robust to perturbations while others fail after very small changes in their inputs? Sensitivity and adjoint analysis provide a common framework for connecting robustness with generalization, regularization, explainability and the relevance of individual features.

 

Publications

• Zuazua, E. (2026). Machine Learning and Control: Foundations, Advances, and Perspectives. Proceedings of the International Congress of Mathematicians 2026. SIAM. International Mathematical Union, 7, 267-286. arXiv: 2510.03303
• Biccari, U., Ibanez-de-Opakua, A., Mato, J. M., Millet, O., Roberto, M., & Zuazua, E. (2026). Fair feature attribution for multi-output prediction: a Shapley-based perspective. arXiv preprint. arXiv: 2602.22882
• Li, Z., Liu, K., Liverani, L., & Zuazua, E. (2026). Universal Approximation of Dynamical Systems by Semiautonomous Neural ODEs and Applications. SIAM J. Numer. Anal., 64(1), 193-223. https://doi.org/10.1137/24M1679690 arXiv: 2407.17092
• Alcalde, A., Ji, Z., & Zuazua, E. (2026). Reachability and asymptotics of Gaussian Transformer dynamics. (Submitted), arXiv: 2606.07600
• Ibañez de Opakua, A., Bizkarguenaga, M., de Diego, Á., Conde, R., González-Valle, B., Gil-Redondo, R., Santana, T., Seco, M. L., Cenarro, A., Lope, V., Margarita, S., Pelusi, S., Habisch, H., Haudum, C., Verheyen, N., Obermayer-Pietsch, B., Sorando-Fernandez, M. P., Arana-Arri, E., Meijide, S., Salazar, A. M., Hernandez, G., Baldán-Martín, M., Martínez-González, P. J., Suárez-Trujillo, F., Audije-Gil, J., Arriaga, J. I., Iriberri, M., Moran, M. Á., Camino, X., Gracia, A., Bernardo-Seisdedos, G., Sousa, A., Oliveira, N., Verde, I., Zaro Bastanzuri, M. J., González Fernández, M. R., Fernández-Bergés, D., Navarrete-Arias, L., Llamas-Velasco, M., Biccari, U., Morales, R., Madl, T., Unda-Urzaiz, M., Carracedo, A., Diercks, T., Castelló, A., Schäfer, H., Spraul, M., Cannet, C., Tasic, L., Sanchez-Pernaute, R., Buguianessi, E., Anstee, Q., Embade, N., Lu, S. C., Mera, F., Valenti, L., Luchinat, C., Cibrián, D., de la Fuente, H., Civeira, F., Arenas, D., Gisbert, J. P., Chaparro, M., Giskeodegard, G., Bathen, T., Zuazua, E., Mato, J. M., & Millet, O. (2026). Mapping metabolic aging and disease-associated acceleration using an interpretable NMR-based clock. (Submitted)
• Alvarez-López, A., Orive-Illera, R., & Zuazua, E. (2025). Cluster-based classification with neural odes via control. J. Mach. Learn., 4(2), 128-156. https://doi.org/doi.org/10.4208/jml.241114 arXiv: 2312.13807
• Biccari, U. (2025). Spiking Neural Networks: a theoretical framework for Universal Approximation and training. (Submitted). arXiv: 2509.21920
• Hernández, M. & Zuazua, E. (2025). Constructive Universal Approximation and Finite Sample Memorization by Narrow Deep ReLU Networks. arXiv: 2409.06555
• Liu, K. & Zuazua, E. (2025). Representation and regression problems in neural networks: Relaxation, generalization, and numerics. Math. Models Methods Appl. Sci., 35(6), 1471–1521. https://doi.org/10.1142/S0218202525500228
• Morales, R. (2025). SHAP values through general Fourier representations: theory and applications. (Submitted). arXiv: 2511.00185
• Cheng D., Ji Z. (2025). On universal eigenvalues and eigenvectors of hypermatricesJ. Frankl. Inst. 362(17), 108126, 10.1016/j.jfranklin.2025.108126
• D. Cheng, X. Zhang, Z. Ji & C. Li (2025). Observer-Based Realization of Control Systems IEEE Trans. Automat. Control, 71(2), 1084-1098, doi: 10.1109/TAC.2025.3604656

 

CoDeFeL GitHub – WP2

• U. Biccari & E. Zuazua. Attention rollout for Machine Learning interpretability
• U. Biccari, R. Morales & E. Zuazua. SHAP values for multi-output ML models
• R. Morales. Fourier SHAP values
• U. Biccari. Spiking Neural Networks