# A Causality-Based Learning Approach for Discovering the Underlying Dynamics of Complex Systems from Partial Observations with Stochastic Parameterization

arXiv: https://arxiv.org/abs/2208.09104 · HTML: https://mlsanity.com/paper/2208.09104/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 127 references that are in the map.
- First posted on arXiv 19 Aug 2022.

## Builds on (most cited first, 10 of 10)

- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [Extracting Sparse High-Dimensional Dynamics from Limited Data](https://mlsanity.com/paper/1707.08528/) (2017)
- [Machine Learning for Stochastic Parameterization: Generative Adversarial Networks in the Lorenz '96 Model](https://mlsanity.com/paper/1909.04711/) (2019)
- [Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation](https://mlsanity.com/paper/2007.07383/) (2020)
- [Data-Driven POD-Galerkin Reduced Order Model for Turbulent Flows](https://mlsanity.com/paper/1907.09909/) (2019)
- [Sparse Identification of Nonlinear Dynamical Systems via Reweighted ℓ1-regularized Least Squares](https://mlsanity.com/paper/2005.13232/) (2020)
- [Using Data Assimilation to Train a Hybrid Forecast System that Combines Machine-Learning and Knowledge-Based Components](https://mlsanity.com/paper/2102.07819/) (2021)
- [Data-driven super-parameterization using deep learning: Experimentation with multi-scale Lorenz 96 systems and transfer-learning](https://mlsanity.com/paper/2002.11167/) (2020)
- [Learning Dynamics from Noisy Measurements using Deep Learning with a Runge-Kutta Constraint](https://mlsanity.com/paper/2109.11446/) (2021)
- [Efficient Construction of Local Parametric Reduced Order Models Using Machine Learning Techniques](https://mlsanity.com/paper/1511.02909/) (2015)

## Built on by (most cited first, 8 of 8)

- [Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methods](https://mlsanity.com/paper/2311.11798/) (2023)
- [CEBoosting: Online Sparse Identification of Dynamical Systems with Regime Switching by Causation Entropy Boosting](https://mlsanity.com/paper/2304.07863/) (2023)
- [CGKN: A Deep Learning Framework for Modeling Complex Dynamical Systems and Efficient Data Assimilation](https://mlsanity.com/paper/2410.20072/) (2024)
- [CGNSDE: Conditional Gaussian Neural Stochastic Differential Equation for Modeling Complex Systems and Data Assimilation](https://mlsanity.com/paper/2404.06749/) (2024)
- [Modeling Partially Observed Nonlinear Dynamical Systems and Efficient Data Assimilation via Discrete-Time Conditional Gaussian Koopman Network](https://mlsanity.com/paper/2507.08749/) (2025)
- [Active Learning of Model Discrepancy with Bayesian Experimental Design](https://mlsanity.com/paper/2502.05372/) (2025)
- [Minimum Reduced-Order Models via Causal Inference](https://mlsanity.com/paper/2407.00271/) (2024)
- [A Continuous-Time Ensemble Kalman-Bucy Smoother for Causal Inference and Model Discovery](https://mlsanity.com/paper/2604.25157/) (2026)

## Abstract (arXiv)

Discovering the underlying dynamics of complex systems from data is an important practical topic. Constrained optimization algorithms are widely utilized and lead to many successes. Yet, such purely data-driven methods may bring about incorrect physics in the presence of random noise and cannot easily handle the situation with incomplete data. In this paper, a new iterative learning algorithm for complex turbulent systems with partial observations is developed that alternates between identifying model structures, recovering unobserved variables, and estimating parameters. First, a causality-based learning approach is utilized for the sparse identification of model structures, which takes into account certain physics knowledge that is pre-learned from data. It has unique advantages in coping with indirect coupling between features and is robust to the stochastic noise. A practical algorithm is designed to facilitate the causal inference for high-dimensional systems. Next, a systematic nonlinear stochastic parameterization is built to characterize the time evolution of the unobserved variables. Closed analytic formula via an efficient nonlinear data assimilation is exploited to sample the trajectories of the unobserved variables, which are then treated as synthetic observations to advance a rapid parameter estimation. Furthermore, the localization of the state variable dependence and the physics constraints are incorporated into the learning procedure, which mitigate the curse of dimensionality and prevent the finite time blow-up issue. Numerical experiments show that the new algorithm succeeds in identifying the model structure and providing suitable stochastic parameterizations for many complex nonlinear systems with chaotic dynamics, spatiotemporal multiscale structures, intermittency, and extreme events.