Universal Approximation Property of Neural Ordinary Differential Equations
This work provides stronger approximation guarantees for NODEs, which is incremental but important for applications in invertible neural networks and related domains.
The paper addresses the limited representation power of Neural Ordinary Differential Equations (NODEs) by proving their sup-universality for approximating a large class of diffeomorphisms, complementing prior L^p-universality results.
Neural ordinary differential equations (NODEs) is an invertible neural network architecture promising for its free-form Jacobian and the availability of a tractable Jacobian determinant estimator. Recently, the representation power of NODEs has been partly uncovered: they form an $L^p$-universal approximator for continuous maps under certain conditions. However, the $L^p$-universality may fail to guarantee an approximation for the entire input domain as it may still hold even if the approximator largely differs from the target function on a small region of the input space. To further uncover the potential of NODEs, we show their stronger approximation property, namely the $\sup$-universality for approximating a large class of diffeomorphisms. It is shown by leveraging a structure theorem of the diffeomorphism group, and the result complements the existing literature by establishing a fairly large set of mappings that NODEs can approximate with a stronger guarantee.