# Uncertainty and Structure in Neural Ordinary Differential Equations

arXiv: https://arxiv.org/abs/2305.13290 · HTML: https://mlsanity.com/paper/2305.13290/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 39 references that are in the map.
- First posted on arXiv 22 May 2023.

## Builds on (most cited first, 22 of 22)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Laplace Redux -- Effortless Bayesian Deep Learning](https://mlsanity.com/paper/2106.14806/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [Being Bayesian, Even Just a Bit, Fixes Overconfidence in ReLU Networks](https://mlsanity.com/paper/2002.10118/) (2020)
- [How Neural Networks Extrapolate: From Feedforward to Graph Neural Networks](https://mlsanity.com/paper/2009.11848/) (2020)
- [Composable Effects for Flexible and Accelerated Probabilistic Programming in NumPyro](https://mlsanity.com/paper/1912.11554/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting](https://mlsanity.com/paper/2010.04456/) (2020)
- [Approximate Inference Turns Deep Networks into Gaussian Processes](https://mlsanity.com/paper/1906.01930/) (2019)
- [Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning](https://mlsanity.com/paper/2002.08860/) (2020)
- [SDE-Net: Equipping Deep Neural Networks with Uncertainty Estimates](https://mlsanity.com/paper/2008.10546/) (2020)
- [Neural ODE Processes](https://mlsanity.com/paper/2103.12413/) (2021)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [Bayesian Neural Ordinary Differential Equations](https://mlsanity.com/paper/2012.07244/) (2020)
- [An Ode to an ODE](https://mlsanity.com/paper/2006.11421/) (2020)
- [Learning ODE Models with Qualitative Structure Using Gaussian Processes](https://mlsanity.com/paper/2011.05364/) (2020)
- [Improving Robustness and Uncertainty Modelling in Neural Ordinary Differential Equations](https://mlsanity.com/paper/2112.12707/) (2021)

## Built on by (most cited first, 6 of 6)

- [Bayesian Neural Controlled Differential Equations for Treatment Effect Estimation](https://mlsanity.com/paper/2310.17463/) (2023)
- [Generative Uncertainty in Diffusion Models](https://mlsanity.com/paper/2502.20946/) (2025)
- [Bayesian polynomial neural networks and polynomial neural ordinary differential equations](https://mlsanity.com/paper/2308.10892/) (2023)
- [System-Aware Neural ODE Processes for Few-Shot Bayesian Optimization](https://mlsanity.com/paper/2406.02352/) (2024)
- [Safe Active Learning for Gaussian Differential Equations](https://mlsanity.com/paper/2412.09053/) (2024)
- [Learning Hybrid Biophysical Neuron Models with Neural ODEs](https://mlsanity.com/paper/2606.16693/) (2026)

## Abstract (arXiv)

Neural ordinary differential equations (ODEs) are an emerging class of deep learning models for dynamical systems. They are particularly useful for learning an ODE vector field from observed trajectories (i.e., inverse problems). We here consider aspects of these models relevant for their application in science and engineering. Scientific predictions generally require structured uncertainty estimates. As a first contribution, we show that basic and lightweight Bayesian deep learning techniques like the Laplace approximation can be applied to neural ODEs to yield structured and meaningful uncertainty quantification. But, in the scientific domain, available information often goes beyond raw trajectories, and also includes mechanistic knowledge, e.g., in the form of conservation laws. We explore how mechanistic knowledge and uncertainty quantification interact on two recently proposed neural ODE frameworks - symplectic neural ODEs and physical models augmented with neural ODEs. In particular, uncertainty reflects the effect of mechanistic information more directly than the predictive power of the trained model could. And vice versa, structure can improve the extrapolation abilities of neural ODEs, a fact that can be best assessed in practice through uncertainty estimates. Our experimental analysis demonstrates the effectiveness of the Laplace approach on both low dimensional ODE problems and a high dimensional partial differential equation.