2 citations · 3 across the 16 of their papers we have counts for
16 papers
Constructive interpolation and generalization rates for neural ODEs: a control perspective
Antonio Álvarez-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…
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…
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 Structure-Preserving Numerical Scheme for Optimal Control and Design of Mixing in Incompressible Flows
Weiwei Hu, Ziqian Li, Yubiao Zhang +1
We develop a structure-preserving computational framework for optimal mixing control in incompressible flows. Our approach exactly conserves the continuous system's key invariants…
A PDE Perspective on Generative Diffusion Models
Kang Liu, Enrique Zuazua
Score-based diffusion models have emerged as a powerful class of generative methods, with successful applications across diverse domains. Despite their empirical success, their mat…
Deep Neural ODE Operator Networks for PDEs
Ziqian Li, Kang Liu, Yongcun Song +2
Operator learning has emerged as a promising paradigm for developing efficient surrogate models to solve partial differential equations (PDEs). However, existing approaches often o…
HYCO: Hybrid-Cooperative Learning for Data-Driven PDE Modeling
Lorenzo Liverani, Enrique Zuazua
We introduce Hybrid-Cooperative Learning (HYCO), a framework for data-driven PDE modeling in which a physics-based solver and a flexible synthetic model are trained as two independ…
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…
Random Batch Methods for Discretized PDEs on Graphs
Martin Hernandez, Martín Hernández, Enrique Zuazua
Diffusion and transport processes on networks arise in applications such as pipeline systems, traffic and information flow, and energy and thermal networks. In these settings the d…
Large-Time Asymptotics for Hyperbolic Systems with Non-Symmetric Relaxation: An Algorithmic Approach
Timothée Crin-Barat, Lorenzo Liverani, Ling-Yun Shou +1
We study the stability of one-dimensional linear hyperbolic systems with non-symmetric relaxation. Introducing a new frequency-dependent Kalman stability condition, we prove an abs…
Representation and Regression Problems in Neural Networks: Relaxation, Generalization, and Numerics
Kang Liu, Enrique Zuazua
In this work, we address three non-convex optimization problems associated with the training of shallow neural networks (NNs) for exact and approximate representation, as well as f…
Constructive Universal Approximation and Finite Sample Memorization by Narrow Deep ReLU Networks
Martín Hernández, Enrique Zuazua
We present a fully constructive analysis of deep ReLU neural networks for classification and function approximation tasks. First, we prove that any dataset with distinct points…
Universal Approximation of Dynamical Systems by Semi-Autonomous Neural ODEs and Applications
Ziqian Li, Kang Liu, Lorenzo Liverani +1
In this paper, we introduce semi-autonomous neural ordinary differential equations (SA-NODEs), a variation of the vanilla NODEs, employing fewer parameters. We investigate the univ…
Interplay between depth and width for interpolation in neural ODEs
Antonio Álvarez-López, Arselane Hadj Slimane, Enrique Zuazua
Neural ordinary differential equations (neural ODEs) have emerged as a natural tool for supervised learning from a control perspective, yet a complete understanding of their optima…
Cluster-based classification with neural ODEs via control
Antonio Álvarez-López, Rafael Orive-Illera, Enrique Zuazua
We address binary classification using neural ordinary differential equations from the perspective of simultaneous control of data points. We consider a single-neuron architect…