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

2026-05-11

AI Summary

This work addresses the challenge of precisely embedding prescribed fixed points into Neural Ordinary Differential Equations (Neural ODEs) without compromising their universal approximation capability for arbitrary vector fields. To this end, we propose an analytically constructed constraint mechanism on the velocity field that rigorously enforces exactness at a finite set of fixed points during training while preserving the model’s expressive power. Theoretically, we provide the first rigorous proof that Neural ODEs subject to local constraints—such as exact fixed points—retain universal approximation properties. Practically, we introduce a computationally efficient implementation strategy. Experiments on two canonical physical systems demonstrate that our approach enables high-fidelity, structurally consistent dynamical modeling.

Abstract

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.

Problem

• Neural-ODEs
• fixed-point constraints
• velocity field
• universality
• expressive power

Innovation

• Neural-ODEs
• fixed-point constraints
• velocity field
• universality
• expressive power