15 papers
Constructive interpolation and generalization rates for neural ODEs: a control perspective
Antonio Ãlvarez-López, Lorenzo Liverani, Enrique Zuazua
We study supervised regression with neural ODEs (NODEs) from a control-theoretic perspective to derive explicit population-risk bounds. We focus on a widely used class of non-auton…
Reachability and asymptotics of Gaussian Transformer dynamics
Albert Alcalde, Zhengping Ji, Enrique Zuazua
We formulate data propagation through the Transformer, the machine learning architecture powering large language models, as a nonlinear control system on the space of probability m…
HYCO: A Formalism for Hybrid-Cooperative PDE Modelling
Lorenzo Liverani, Enrique Zuazua
We present Hybrid-Cooperative Learning (HYCO), a hybrid modeling framework that integrates physics-based and data-driven models through mutual regularization. Unlike traditional ap…
A PDE Perspective on Generative Diffusion Models
Kang Liu, Enrique Zuazua
Score-based diffusion models have emerged as a powerful class of generative methods, achieving state-of-the-art performance across diverse domains. Despite their empirical success,…
Hamiltonian Interface Dynamics for Reduced-Order Optimization of Incompressible Mixing
Ziqian Li, Enrique Zuazua
We develop a reduced-order framework for optimizing mixing in two-dimensional incompressible flows. Instead of optimizing the full transport PDE, the method maximizes the length of…
Moments, Time-Inversion and Source Identification for the Heat Equation
Kang Liu, Enrique Zuazua
We address the initial source identification problem for the heat equation, a notably ill-posed inverse problem characterized by exponential instability. Departing from classical T…