We study the problem of approximating a given diffeomorphism and its associated pushforward probability measure using the flow of a Neural ODE with a piecewise-constant velocity field. This setup models conditional sampling tasks where one seeks both an inverse map and a valid generative transform. The core contribution is an explicit constructive method: decomposing the target map into compressible and incompressible components and realizing each via simple flow elements. The compressible part is captured exactly through the gradient of a convex potential, while the incompressible part is realized via shear flows and permutations. Under stronger regularity assumptions on the target map (e.g., for the Knöthe–Rosenblatt rearrangement with regular input and output measures), we provide a probabilistic construction that avoids the exponential scaling ("curse of dimensionality") in the number of parameters in the neural network.

Bio: Domènec Ruiz-Balet is a mathematician and assistant professor at Universitat de Barcelona. He earned his doctorate from Universidad Autónoma de Madrid under the supervision of Enrique Zuazua and later made postdoctoral stages at Imperial College and Université Paris Dauphine. He is working on evolution equations that appear in machine learning with focus on deep neural networks and transformers and certain aspects of game theory.