# Machine Learning for Prediction with Missing Dynamics

arXiv: https://arxiv.org/abs/1910.05861 · HTML: https://mlsanity.com/paper/1910.05861/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Oct 2019.

## Built on by (most cited first, 28 of 28)

- [A Framework for Machine Learning of Model Error in Dynamical Systems](https://mlsanity.com/paper/2107.06658/) (2021)
- [Auto-differentiable Ensemble Kalman Filters](https://mlsanity.com/paper/2107.07687/) (2021)
- [Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations](https://mlsanity.com/paper/2108.03561/) (2021)
- [Learning Stochastic Dynamics with Statistics-Informed Neural Network](https://mlsanity.com/paper/2202.12278/) (2022)
- [Discrepancy Modeling Framework: Learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effects](https://mlsanity.com/paper/2203.05164/) (2022)
- [The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimation](https://mlsanity.com/paper/2103.11488/) (2021)
- [Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Series](https://mlsanity.com/paper/2111.13037/) (2021)
- [Online model error correction with neural networks in the incremental 4D-Var framework](https://mlsanity.com/paper/2210.13817/) (2022)
- [Reduced-Order Autodifferentiable Ensemble Kalman Filters](https://mlsanity.com/paper/2301.11961/) (2023)
- [On Fast Simulation of Dynamical System with Neural Vector Enhanced Numerical Solver](https://mlsanity.com/paper/2208.03680/) (2022)
- [Finite Expression Methods for Discovering Physical Laws from Data](https://mlsanity.com/paper/2305.08342/) (2023)
- [A non-intrusive machine learning framework for debiasing long-time coarse resolution climate simulations and quantifying rare events statistics](https://mlsanity.com/paper/2402.18484/) (2024)
- [Deep learning delay coordinate dynamics for chaotic attractors from partial observable data](https://mlsanity.com/paper/2211.11061/) (2022)
- [Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case](https://mlsanity.com/paper/2110.00546/) (2021)
- [Error Bounds of the Invariant Statistics in Machine Learning of Ergodic Ito Diffusions](https://mlsanity.com/paper/2105.10102/) (2021)
- [Learning nonlinear integral operators via Recurrent Neural Networks and its application in solving Integro-Differential Equations](https://mlsanity.com/paper/2310.09434/) (2023)
- [iPINNER: An Iterative Physics-Informed Neural Network with Ensemble Kalman Filter](https://mlsanity.com/paper/2506.00731/) (2025)
- [Learning solution operator of dynamical systems with diffusion maps kernel ridge regression](https://mlsanity.com/paper/2512.17203/) (2025)
- [Regression-based projection for learning Mori-Zwanzig operators](https://mlsanity.com/paper/2205.05135/) (2022)
- [The Mori-Zwanzig formulation of deep learning](https://mlsanity.com/paper/2209.05544/) (2022)
- [Reservoir Computing with Error Correction: Long-term Behaviors of Stochastic Dynamical Systems](https://mlsanity.com/paper/2305.00669/) (2023)
- [A Weak Penalty Neural ODE for Learning Chaotic Dynamics from Noisy Time Series](https://mlsanity.com/paper/2511.06609/) (2025)
- [Deep neural network based adaptive learning for switched systems](https://mlsanity.com/paper/2207.04623/) (2022)
- [Drivetrain simulation using variational autoencoders](https://mlsanity.com/paper/2501.17653/) (2025)
- [Bayesian Symbolic Regression for Missing Physics](https://mlsanity.com/paper/2603.14918/) (2026)
- [Experimental Design for Missing Physics](https://mlsanity.com/paper/2604.01231/) (2026)
- [Memory-Conditioned Diffusion Model for Generalized Langevin Dynamics](https://mlsanity.com/paper/2609.28371/) (2026)
- [Learning Closure of Dynamical Systems with Kernel Ridge Regression](https://mlsanity.com/paper/2610.02564/) (2026)

## Abstract (arXiv)

This article presents a general framework for recovering missing dynamical systems using available data and machine learning techniques. The proposed framework reformulates the prediction problem as a supervised learning problem to approximate a map that takes the memories of the resolved and identifiable unresolved variables to the missing components in the resolved dynamics. We demonstrate the effectiveness of the proposed framework with a theoretical guarantee of a path-wise convergence of the resolved variables up to finite time and numerical tests on prototypical models in various scientific domains. These include the 57-mode barotropic stress models with multiscale interactions that mimic the blocked and unblocked patterns observed in the atmosphere, the nonlinear Schr"odinger equation which found many applications in physics such as optics and Bose-Einstein-Condense, the Kuramoto-Sivashinsky equation which spatiotemporal chaotic pattern formation models trapped ion mode in plasma and phase dynamics in reaction-diffusion systems. While many machine learning techniques can be used to validate the proposed framework, we found that recurrent neural networks outperform kernel regression methods in terms of recovering the trajectory of the resolved components and the equilibrium one-point and two-point statistics. This superb performance suggests that recurrent neural networks are an effective tool for recovering the missing dynamics that involves approximation of high-dimensional functions.