mlsanityData to 14 Oct 2026

Universal Approximation Property of Neural Ordinary Differential Equations

arXiv 2012.02414 ↗First posted 4 Dec 2020cs.LG math.DG stat.MLcited by 32

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Abstract (arXiv)

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 Lp-universal approximator for continuous maps under certain conditions. However, the Lp-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.

Key facts. Cited by 32 papers in the map (data through 14 Oct 2026). First posted on arXiv 4 Dec 2020.

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Citing papers per year: 2021: 4; 2022: 8; 2023: 8; 2024: 6; 2025: 3; 2026: 3.

32 papers.

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  1. 2021Statistically Meaningful Approximation: a Case Study on Approximating Turing Machines with Transformerscited by 102
  2. 2021Neural Flows: Efficient Alternative to Neural ODEscited by 101
  3. 2021Momentum Residual Neural Networkscited by 37
  4. 2022Universal approximation property of invertible neural networkscited by 30
  5. 2022Do Residual Neural Networks discretize Neural Ordinary Differential Equations?cited by 27
  6. 2023Distribution learning via neural differential equations: a nonparametric statistical perspectivecited by 22
  7. 2023Implicit regularization of deep residual networks towards neural ODEscited by 20
  8. 2024On the Universality of Volume-Preserving and Coupling-Based Normalizing Flowscited by 18
  9. 2023Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraintscited by 15
  10. 2023Minimum Width of Leaky-ReLU Neural Networks for Uniform Universal Approximationcited by 13
  11. 2022FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantitiescited by 12
  12. 2021Symplectic Adjoint Method for Exact Gradient of Neural ODE with Minimal Memorycited by 11
20 more
  1. 2022Dynamical systems' based neural networkscited by 10
  2. 2024Feedback Favors the Generalization of Neural ODEscited by 8
  3. 2023Controllable Video Generation by Learning the Underlying Dynamical System with Neural ODEcited by 7
  4. 2022Whitening Convergence Rate of Coupling-based Normalizing Flowscited by 6
  5. 2023On the Convergence Rate of Gaussianization with Random Rotationscited by 5
  6. 2023Finding emergence in data by maximizing effective informationcited by 4
  7. 2022Neural Information Squeezer for Causal Emergencecited by 3
  8. 2022Learning via nonlinear conjugate gradients and depth-varying neural ODEscited by 3
  9. 2022Deep Curvilinear Editing: Commutative and Nonlinear Image Manipulation for Pretrained Deep Generative Modelcited by 3
  10. 2023Variational formulations of ODE-Net as a mean-field optimal control problem and existence resultscited by 3
  11. 2024Modeling chaotic Lorenz ODE System using Scientific Machine Learningcited by 1
  12. 2024Universal approximation property of ODENet and ResNet with a single activation functioncited by 1
  13. 2026NeuralFLoC: Neural Flow-Based Joint Registration and Clustering of Functional Datacited by 1
  14. 2026Exact Fixed-Point Constraints in Neural-ODEs with Provable Universalitycited by 1
  15. 2024Deep Koopman-layered Model with Universal Property Based on Toeplitz Matricescited by 0
  16. 2024A comparative study of NeuralODE and Universal ODE approaches to solving Chandrasekhar White Dwarf equationcited by 0
  17. 2025Learning Geometrically-Informed Lyapunov Functions with Deep Diffeomorphic RBF Networkscited by 0
  18. 2025ODE-ViT: Plug & Play Attention Layer from the Generalization of the ViT as an Ordinary Differential Equationcited by 0
  19. 2025Forecasting N-Body Dynamics: A Comparative Study of Neural Ordinary Differential Equations and Universal Differential Equationscited by 0
  20. 2026Approximating velocity fields with planted attractors via Neural-ODEs for classification purposescited by 0