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.
Task 2.1 – From data complexity to network complexity
More complex data generally require more 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. Submitted. arXiv:2510.03303 , FAU CRIS
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. Submitted. arXiv:2602.22882
Li, Z., Liu, K., Liverani, L., & Zuazua, E. (2026). Universal Approximation of Dynamical Systems by Semi-Autonomous Neural ODEs and Applications. SIAM J. Numer. Anal., 64(1), 193-223. arXiv:2407.17092
Alcalde A., Ji Z., & Zuazua E. (2026). Reachability and Asymptotics of Gaussian Transformer Dynamics. Submitted. arXiv:2606.07600
Biccari U., Morales R., & Zuazua E. (2026). Mapping Metabolic Aging and Disease-Associated Acceleration Using an Interpretable NMR-Based Clock. Submitted.
Alcalde, A., Fantuzzi, G., & Zuazua, E. (2025). Clustering in Pure-Attention Hardmax Transformers and its Role in Sentiment Analysis. SIAM J. Math. Data Sci., 7(3), 1367-1393. https://doi.org/10.1137/24M167086X arXiv:2407.01602
Alcalde, A., Fantuzzi, G., & Zuazua, E. (2025). Exact sequence interpolation with transformers. Submitted. arXiv:2502.02270 , FAU CRIS
Álvarez-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/10.4208/jml.241114
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. Submitted. 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. arXiv:2412.01619, FAU CRIS
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 hypermatrices. J. Franklin Inst., 362(17), 108126.
Cheng, D., Zhang, X., Ji, Z., & Li, C. (2025). Observer-Based Realization of Control Systems. IEEE Trans. Automat. Control, 71(2), 1084-1098.
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