# Time Dependence in Non-Autonomous Neural ODEs

arXiv: https://arxiv.org/abs/2005.01906 · HTML: https://mlsanity.com/paper/2005.01906/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 May 2020.

## Built on by (most cited first, 7 of 7)

- [TO-FLOW: Efficient Continuous Normalizing Flows with Temporal Optimization adjoint with Moving Speed](https://mlsanity.com/paper/2203.10335/) (2022)
- [Learning ODEs via Diffeomorphisms for Fast and Robust Integration](https://mlsanity.com/paper/2107.01650/) (2021)
- [STR-GODEs: Spatial-Temporal-Ridership Graph ODEs for Metro Ridership Prediction](https://mlsanity.com/paper/2107.04980/) (2021)
- [Imbedding Deep Neural Networks](https://mlsanity.com/paper/2202.00113/) (2022)
- [Temporal Weights](https://mlsanity.com/paper/2301.04126/) (2022)
- [Weight-Parameterization in Continuous Time Deep Neural Networks for Surrogate Modeling](https://mlsanity.com/paper/2507.22045/) (2025)
- [Constructive interpolation and generalization rates for neural ODEs: a control perspective](https://mlsanity.com/paper/2606.00469/) (2026)

## Abstract (arXiv)

Neural Ordinary Differential Equations (ODEs) are elegant reinterpretations of deep networks where continuous time can replace the discrete notion of depth, ODE solvers perform forward propagation, and the adjoint method enables efficient, constant memory backpropagation. Neural ODEs are universal approximators only when they are non-autonomous, that is, the dynamics depends explicitly on time. We propose a novel family of Neural ODEs with time-varying weights, where time-dependence is non-parametric, and the smoothness of weight trajectories can be explicitly controlled to allow a tradeoff between expressiveness and efficiency. Using this enhanced expressiveness, we outperform previous Neural ODE variants in both speed and representational capacity, ultimately outperforming standard ResNet and CNN models on select image classification and video prediction tasks.