Exact Fixed-Point Constraints in Neural-ODEs with Provable Universality
For researchers using Neural-ODEs in scientific applications requiring known fixed points, this provides a principled way to incorporate such constraints without losing expressivity.
The paper introduces a technique to enforce exact fixed-point constraints in Neural-ODEs, proving that this does not compromise their universal approximation property. The method is validated on two physical models.
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.