# Uncertainty and Structure in Neural Ordinary Differential Equations

**Authors:** Katharina Ott, Michael Tiemann, Philipp Hennig  
**arXiv:** [2305.13290](https://arxiv.org/abs/2305.13290)  
**Date:** 2023-05-22  
**Summary page:** https://www.summarizepaper.com/en/paper/2305.13290v1/

## Key points

- Neural ordinary differential equations (ODEs) are powerful deep learning models for dynamical systems.
- They learn ODE vector fields from observed trajectories, particularly in inverse problems.
- Basic Bayesian deep learning techniques like the Laplace approximation applied to neural ODEs provide effective uncertainty quantification in scientific and engineering domains.
- Scientific datasets often contain mechanistic knowledge, such as conservation laws, which can significantly impact modeling and prediction quality.
- The study explores the interaction between mechanistic knowledge and uncertainty quantification in symplectic neural ODEs and physical models augmented with neural ODEs.
- Uncertainty quantification reflects mechanistic information more directly than just predictive power alone, enhancing model extrapolation capabilities.
- Incorporating structure derived from mechanistic knowledge can improve the performance of neural ODEs.
- Leveraging both uncertainty quantification and structure is crucial for model development and evaluation in complex scientific domains.

## Summary

Neural ordinary differential equations (ODEs) have emerged as a powerful class of deep learning models for dynamical systems. They offer a unique approach to learning ODE vector fields from observed trajectories, particularly in inverse problems. In the realm of science and engineering, where structured uncertainty estimates are crucial for making accurate predictions, the application of basic Bayesian deep learning techniques such as the Laplace approximation to neural ODEs has proven to be effective in providing meaningful uncertainty quantification. However, scientific datasets often contain more than just raw trajectories; they may also incorporate mechanistic knowledge, such as conservation laws. This additional information can significantly impact the modeling process and the quality of predictions. In this study by Katharina Ott, Michael Tiemann, and Philipp Hennig, the interaction between mechanistic knowledge and uncertainty quantification is explored within two novel neural ODE frameworks: symplectic neural ODEs and physical models augmented with neural ODEs. The findings reveal that uncertainty quantification reflects the influence of mechanistic information more directly than the predictive power of the trained model alone. Conversely, incorporating structure derived from mechanistic knowledge can enhance the extrapolation capabilities of neural ODEs. This symbiotic relationship between uncertainty and structure underscores the importance of leveraging both aspects in model development and evaluation. Through experimental analysis on both low-dimensional ODE problems and a high-dimensional partial differential equation, the authors demonstrate the efficacy of employing the Laplace approach for structured uncertainty quantification in neural ODEs. By integrating mechanistic knowledge with uncertainty estimates, researchers can improve model performance and make more reliable predictions in complex scientific domains.