# Stochastic Physics-Informed Neural Ordinary Differential Equations

arXiv: https://arxiv.org/abs/2109.01621 · HTML: https://mlsanity.com/paper/2109.01621/

## Key facts

- Cited by 19 papers in the map (data through 7 Oct 2026).
- Builds on 14 of its 87 references that are in the map.
- First posted on arXiv 3 Sep 2021.

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

- [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)
- [Deep learning for time series classification: a review](https://mlsanity.com/paper/1809.04356/) (2018)
- [Tutorial on Variational Autoencoders](https://mlsanity.com/paper/1606.05908/) (2016)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Adversarial Variational Bayes: Unifying Variational Autoencoders and Generative Adversarial Networks](https://mlsanity.com/paper/1701.04722/) (2017)
- [A Tutorial on Kernel Density Estimation and Recent Advances](https://mlsanity.com/paper/1704.03924/) (2017)
- [Nonlinear System Identification: A User-Oriented Roadmap](https://mlsanity.com/paper/1902.00683/) (2019)
- [Deep Directed Generative Models with Energy-Based Probability Estimation](https://mlsanity.com/paper/1606.03439/) (2016)
- [Review of Deep Learning](https://mlsanity.com/paper/1804.01653/) (2018)
- [A Generalized Unscented Transformation for Probability Distributions](https://mlsanity.com/paper/2104.01958/) (2021)

## Built on by (most cited first, 19 of 19)

- [Physics-Informed Neural Networks with Hard Linear Equality Constraints](https://mlsanity.com/paper/2402.07251/) (2024)
- [Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KAN](https://mlsanity.com/paper/2407.18373/) (2024)
- [Neural Schr"odinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assembly](https://mlsanity.com/paper/2307.14442/) (2023)
- [Learning Neural Differential Algebraic Equations via Operator Splitting](https://mlsanity.com/paper/2403.12938/) (2024)
- [Complex Physics-Informed Neural Network](https://mlsanity.com/paper/2502.04917/) (2025)
- [Learning Effective SDEs from Brownian Dynamics Simulations of Colloidal Particles](https://mlsanity.com/paper/2205.00286/) (2022)
- [ControlSynth Neural ODEs: Modeling Dynamical Systems with Guaranteed Convergence](https://mlsanity.com/paper/2411.02292/) (2024)
- [Physics-informed time series analysis with Kolmogorov-Arnold Networks under Ehrenfest constraints](https://mlsanity.com/paper/2509.18483/) (2025)
- [Neural Ordinary Differential Equations for Learning and Extrapolating System Dynamics Across Bifurcations](https://mlsanity.com/paper/2507.19036/) (2025)
- [Position: A Dynamical Systems Perspective is Needed to Advance Time Series Modeling](https://mlsanity.com/paper/2602.16864/) (2026)
- [HyperSBINN: A Hypernetwork-Enhanced Systems Biology-Informed Neural Network for Efficient Drug Cardiosafety Assessment](https://mlsanity.com/paper/2408.14266/) (2024)
- [ICODE: Modeling Dynamical Systems with Extrinsic Input Information](https://mlsanity.com/paper/2411.13914/) (2024)
- [Unified Spatiotemporal Physics-Informed Learning (USPIL): A Framework for Modeling Complex Predator-Prey Dynamics](https://mlsanity.com/paper/2509.13425/) (2025)
- [Stochastic Port-Hamiltonian Neural Networks: Universal Approximation with Passivity Guarantees](https://mlsanity.com/paper/2603.10078/) (2026)
- [An analysis of Universal Differential Equations for data-driven discovery of Ordinary Differential Equations](https://mlsanity.com/paper/2306.10335/) (2023)
- [Forecasting with an N-dimensional Langevin Equation and a Neural-Ordinary Differential Equation](https://mlsanity.com/paper/2405.07359/) (2024)
- [Perception-Informed Neural Networks: Beyond Physics-Informed Neural Networks](https://mlsanity.com/paper/2505.03806/) (2025)
- [Learning Biomolecular Motion: The Physics-Informed Machine Learning Paradigm](https://mlsanity.com/paper/2511.06585/) (2025)
- [Geometric Kolmogorov--Arnold Network (GeoKAN)](https://mlsanity.com/paper/2605.06740/) (2026)

## Abstract (arXiv)

Stochastic differential equations (SDEs) are used to describe a wide variety of complex stochastic dynamical systems. Learning the hidden physics within SDEs is crucial for unraveling fundamental understanding of these systems' stochastic and nonlinear behavior. We propose a flexible and scalable framework for training artificial neural networks to learn constitutive equations that represent hidden physics within SDEs. The proposed stochastic physics-informed neural ordinary differential equation framework (SPINODE) propagates stochasticity through the known structure of the SDE (i.e., the known physics) to yield a set of deterministic ODEs that describe the time evolution of statistical moments of the stochastic states. SPINODE then uses ODE solvers to predict moment trajectories. SPINODE learns neural network representations of the hidden physics by matching the predicted moments to those estimated from data. Recent advances in automatic differentiation and mini-batch gradient descent with adjoint sensitivity are leveraged to establish the unknown parameters of the neural networks. We demonstrate SPINODE on three benchmark in-silico case studies and analyze the framework's numerical robustness and stability. SPINODE provides a promising new direction for systematically unraveling the hidden physics of multivariate stochastic dynamical systems with multiplicative noise.