Approximating velocity fields with planted attractors via Neural-ODEs for classification purposes

Feliciano Giuseppe Pacifico; Duccio Fanelli; Lorenzo Buffoni; Lorenzo Chicchi; Diego Febbe; Raffaele Marino

cond-mat.dis-nnJun 22, 2026

Approximating velocity fields with planted attractors via Neural-ODEs for classification purposes

Authors: Feliciano Giuseppe Pacifico, Duccio Fanelli, Lorenzo Buffoni, Lorenzo Chicchi, Diego Febbe, Raffaele Marino

Organizations: Department of Informatics and Computer Science, University of Pisa, Italy · Department of Physics and Astronomy, University of Florence, Sesto Fiorentino, Italy · INFN, Italy

Abstract

In this work, Neural ODEs equipped with a curated collection of equilibrium points have been successfully employed for classification tasks. The planted attractors serve as indicators for the target classes, while the velocity field leveraging the universal approximation capabilities of the architecture shapes the dynamical landscape. This process defines the basins of attraction of the trained model, effectively directing each input (provided as an initial condition) toward its corresponding destination target.

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May 11, 2026 · cond-mat.dis-nnJ/K move · Enter open · S save

Exact Fixed-Point Constraints in Neural-ODEs with Provable Universality

Feliciano Giuseppe Pacifico, Duccio Fanelli, Lorenzo Buffoni, Lorenzo Chicchi+2

Department of Informatics and Computer Science, University of Pisa, Italy · Department of Physics and Astronomy and INFN, University of Florence, Sesto Fiorentino, Italy

We introduce a technique that enables Neural-ODEs to approximate arbitrary velocity fields with a priori planted fixed-points. Specifically, a recipe is given to explicitly accommodate for a finite collection of points in the reference multi-dimensional space of the Neural-ODE where the velocity field is exactly equal to zero. In this way, the gradient-based training is rigorously constrained inside the prescribed hypothesis class while leaving the expressive power of the Neural-ODE unaltered. We rigorously prove the universality of the Neural-ODE under any local constraints in the velocity field and give a computationally convenient way of imposing the fixed points. Our method is then tested on two paradigmatic physical models.

Neural Ordinary Differential Equations Fixed-Point Iteration Neural Control Velocity Field Gradient-Based Optimization Neural-Network Approximation Structural Constraints Exact

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Abstract

Explore similar work

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