# How to Learn and Generalize From Three Minutes of Data: Physics-Constrained and Uncertainty-Aware Neural Stochastic Differential Equations

arXiv: https://arxiv.org/abs/2306.06335 · HTML: https://mlsanity.com/paper/2306.06335/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 72 references that are in the map.
- First posted on arXiv 10 Jun 2023.

## Builds on (most cited first, 30 of 48)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Offline Reinforcement Learning: Tutorial, Review, and Perspectives on Open Problems](https://mlsanity.com/paper/2005.01643/) (2020)
- [Conservative Q-Learning for Offline Reinforcement Learning](https://mlsanity.com/paper/2006.04779/) (2020)
- [Deep Reinforcement Learning in a Handful of Trials using Probabilistic Dynamics Models](https://mlsanity.com/paper/1805.12114/) (2018)
- [When to Trust Your Model: Model-Based Policy Optimization](https://mlsanity.com/paper/1906.08253/) (2019)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [MOPO: Model-based Offline Policy Optimization](https://mlsanity.com/paper/2005.13239/) (2020)
- [MOReL : Model-Based Offline Reinforcement Learning](https://mlsanity.com/paper/2005.05951/) (2020)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [COMBO: Conservative Offline Model-Based Policy Optimization](https://mlsanity.com/paper/2102.08363/) (2021)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Uncertainty-Based Offline Reinforcement Learning with Diversified Q-Ensemble](https://mlsanity.com/paper/2110.01548/) (2021)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Model-Ensemble Trust-Region Policy Optimization](https://mlsanity.com/paper/1802.10592/) (2018)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [Neural Lander: Stable Drone Landing Control using Learned Dynamics](https://mlsanity.com/paper/1811.08027/) (2018)
- [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)
- [Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systems](https://mlsanity.com/paper/2107.08024/) (2021)
- [Learning and Policy Search in Stochastic Dynamical Systems with Bayesian Neural Networks](https://mlsanity.com/paper/1605.07127/) (2016)
- [skrl: Modular and Flexible Library for Reinforcement Learning](https://mlsanity.com/paper/2202.03825/) (2022)
- [MBRL-Lib: A Modular Library for Model-based Reinforcement Learning](https://mlsanity.com/paper/2104.10159/) (2021)

## Built on by (most cited first, 10 of 10)

- [Reference-Free Formula Drift with Reinforcement Learning: From Driving Data to Tire Energy-Inspired, Real-World Policies](https://mlsanity.com/paper/2410.20990/) (2024)
- [Physically Interpretable World Models via Weakly Supervised Representation Learning](https://mlsanity.com/paper/2412.12870/) (2024)
- [Neural Port-Hamiltonian Differential Algebraic Equations for Compositional Learning of Electrical Networks](https://mlsanity.com/paper/2412.11215/) (2024)
- [Multi-Robot Collaboration through Reinforcement Learning and Abstract Simulation](https://mlsanity.com/paper/2503.05092/) (2025)
- [Artifacts of Numerical Integration in Learning Dynamical Systems](https://mlsanity.com/paper/2507.14491/) (2025)
- [Learning Physically Consistent Lagrangian Control Models Without Acceleration Measurements](https://mlsanity.com/paper/2512.03035/) (2025)
- [A Method to Improve the Performance of Reinforcement Learning Based on the Y Operator for a Class of Stochastic Differential Equation-Based Child-Mother Systems](https://mlsanity.com/paper/2311.04014/) (2023)
- [Improving the Noise Estimation of Latent Neural Stochastic Differential Equations](https://mlsanity.com/paper/2412.17499/) (2024)
- [Policy Gradient for Continuous-Time Robust Markov Decision Processes](https://mlsanity.com/paper/2606.04335/) (2026)
- [Continuous-Time Trajectory Generation from Discrete Observations with Stochasticity](https://mlsanity.com/paper/2609.32026/) (2026)

## Abstract (arXiv)

We present a framework and algorithms to learn controlled dynamics models using neural stochastic differential equations (SDEs) -- SDEs whose drift and diffusion terms are both parametrized by neural networks. We construct the drift term to leverage a priori physics knowledge as inductive bias, and we design the diffusion term to represent a distance-aware estimate of the uncertainty in the learned model's predictions -- it matches the system's underlying stochasticity when evaluated on states near those from the training dataset, and it predicts highly stochastic dynamics when evaluated on states beyond the training regime. The proposed neural SDEs can be evaluated quickly enough for use in model predictive control algorithms, or they can be used as simulators for model-based reinforcement learning. Furthermore, they make accurate predictions over long time horizons, even when trained on small datasets that cover limited regions of the state space. We demonstrate these capabilities through experiments on simulated robotic systems, as well as by using them to model and control a hexacopter's flight dynamics: A neural SDE trained using only three minutes of manually collected flight data results in a model-based control policy that accurately tracks aggressive trajectories that push the hexacopter's velocity and Euler angles to nearly double the maximum values observed in the training dataset.