Universal Approximation Property of Neural Ordinary Differential Equations

• URL: http://arxiv.org/abs/2012.02414v1
• Date: Fri, 4 Dec 2020 05:53:21 GMT
• Title: Universal Approximation Property of Neural Ordinary Differential Equations
• Authors: Takeshi Teshima, Koichi Tojo, Masahiro Ikeda, Isao Ishikawa, Kenta Oono
• Abstract summary: We show that NODEs can form an $Lp$-universal approximator for continuous maps under certain conditions. We also show their stronger approximation property, namely the $sup$-universality for approximating a large class of diffeomorphisms.
• Score: 19.861764482790544
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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