# Data-driven Discovery of Closure Models

arXiv: https://arxiv.org/abs/1803.09318 · HTML: https://mlsanity.com/paper/1803.09318/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Mar 2018.

## Built on by (most cited first, 20 of 20)

- [Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders](https://mlsanity.com/paper/2201.05136/) (2022)
- [CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representations](https://mlsanity.com/paper/2206.02607/) (2022)
- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
- [On the Benefits of Memory for Modeling Time-Dependent PDEs](https://mlsanity.com/paper/2409.02313/) (2024)
- [Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data](https://mlsanity.com/paper/2205.13545/) (2022)
- [Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference](https://mlsanity.com/paper/2107.02597/) (2021)
- [Neural Ideal Large Eddy Simulation: Modeling Turbulence with Neural Stochastic Differential Equations](https://mlsanity.com/paper/2306.01174/) (2023)
- [A Multifidelity deep operator network approach to closure for multiscale systems](https://mlsanity.com/paper/2303.08893/) (2023)
- [Deep Learning of Chaotic Systems from Partially-Observed Data](https://mlsanity.com/paper/2205.08384/) (2022)
- [Generalized Neural Closure Models with Interpretability](https://mlsanity.com/paper/2301.06198/) (2023)
- [Operator inference of non-Markovian terms for learning reduced models from partially observed state trajectories](https://mlsanity.com/paper/2103.01362/) (2021)
- [Error Bounds of the Invariant Statistics in Machine Learning of Ergodic Ito Diffusions](https://mlsanity.com/paper/2105.10102/) (2021)
- [Physics Guided Machine Learning for Variational Multiscale Reduced Order Modeling](https://mlsanity.com/paper/2205.12419/) (2022)
- [Systematic construction of continuous-time neural networks for linear dynamical systems](https://mlsanity.com/paper/2403.16215/) (2024)
- [Differentiable Physics-Neural Models enable Learning of Non-Markovian Closures for Accelerated Coarse-Grained Physics Simulations](https://mlsanity.com/paper/2511.21369/) (2025)
- [Variational multiscale reinforcement learning for discovering reduced order closure models of nonlinear spatiotemporal transport systems](https://mlsanity.com/paper/2207.12854/) (2022)
- [Attention-Enhanced Convolutional Autoencoder and Structured Delay Embeddings for Weather Prediction](https://mlsanity.com/paper/2511.12682/) (2025)
- [The impact of observation density on Bayesian inversion of latent dynamics in shock-dominated flows](https://mlsanity.com/paper/2605.19076/) (2026)
- [Flux-form spatiotemporal neural operators for coarse-grained dynamics of multiscale PDEs](https://mlsanity.com/paper/2608.18148/) (2026)
- [Counterfactual Predictions in Scientific Emulators Without Controlled Experiments](https://mlsanity.com/paper/2610.02252/) (2026)

## Abstract (arXiv)

Derivation of reduced order representations of dynamical systems requires the modeling of the truncated dynamics on the retained dynamics. In its most general form, this so-called closure model has to account for memory effects. In this work, we present a framework of operator inference to extract the governing dynamics of closure from data in a compact, non-Markovian form. We employ sparse polynomial regression and artificial neural networks to extract the underlying operator. For a special class of non-linear systems, observability of the closure in terms of the resolved dynamics is analyzed and theoretical results are presented on the compactness of the memory. The proposed framework is evaluated on examples consisting of linear to nonlinear systems with and without chaotic dynamics, with an emphasis on predictive performance on unseen data.