# CODE: A global approach to ODE dynamics learning

arXiv: https://arxiv.org/abs/2511.15619 · HTML: https://mlsanity.com/paper/2511.15619/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 55 references that are in the map.
- First posted on arXiv 19 Nov 2025.

## Builds on (most cited first, 14 of 14)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [Don't Unroll Adjoint: Differentiating SSA-Form Programs](https://mlsanity.com/paper/1810.07951/) (2018)
- [Discretize-Optimize vs. Optimize-Discretize for Time-Series Regression and Continuous Normalizing Flows](https://mlsanity.com/paper/2005.13420/) (2020)
- [Interpretable Polynomial Neural Ordinary Differential Equations](https://mlsanity.com/paper/2208.05072/) (2022)
- [Differentiable Programming for Differential Equations: A Review](https://mlsanity.com/paper/2406.09699/) (2024)
- [Multiple shooting for training neural differential equations on time series](https://mlsanity.com/paper/2109.06786/) (2021)
- [Physics-constrained polynomial chaos expansion for scientific machine learning and uncertainty quantification](https://mlsanity.com/paper/2402.15115/) (2024)
- [Variational multiple shooting for Bayesian ODEs with Gaussian processes](https://mlsanity.com/paper/2106.10905/) (2021)
- [Generalization in Neural Networks: A Broad Survey](https://mlsanity.com/paper/2209.01610/) (2022)
- [Anamnesic Neural Differential Equations with Orthogonal Polynomial Projections](https://mlsanity.com/paper/2303.01841/) (2023)

## Abstract (arXiv)

Ordinary differential equations (ODEs) are a conventional way to describe the observed dynamics of physical systems. Scientists typically hypothesize about dynamical behavior, propose a mathematical model, and compare its predictions to data. However, modern computing and algorithmic advances now enable purely data-driven learning of governing dynamics directly from observations. In data-driven settings, one learns the ODE's right-hand side (RHS). Dense measurements are often assumed, yet high temporal resolution is typically both cumbersome and expensive. Consequently, one usually has only sparsely sampled data. In this work we introduce ChaosODE (CODE), a Polynomial Chaos ODE Expansion in which we use an arbitrary Polynomial Chaos Expansion (aPCE) for the ODE's right-hand side, resulting in a global orthonormal polynomial representation of dynamics. We evaluate the performance of CODE in several experiments on the Lotka-Volterra system, across varying noise levels, initial conditions, and predictions far into the future, even on previously unseen initial conditions. CODE exhibits remarkable extrapolation capabilities even when evaluated under novel initial conditions and shows advantages compared to well-examined methods using neural networks (NeuralODE) or kernel approximators (KernelODE) as the RHS representer. We observe that the high flexibility of NeuralODE and KernelODE degrades extrapolation capabilities under scarce data and measurement noise. Finally, we provide practical guidelines for robust optimization of dynamics-learning problems and illustrate them in the accompanying code.