# Data-Driven Discovery of Coarse-Grained Equations

arXiv: https://arxiv.org/abs/2002.00790 · HTML: https://mlsanity.com/paper/2002.00790/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jan 2020.

## Built on by (most cited first, 14 of 14)

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- [A novel corrective-source term approach to modeling unknown physics in aluminum extraction process](https://mlsanity.com/paper/2209.10861/) (2022)
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- [Learning Nonautonomous Systems via Dynamic Mode Decomposition](https://mlsanity.com/paper/2306.15618/) (2023)
- [Large language models for partial differential equation workflows](https://mlsanity.com/paper/2608.03600/) (2026)
- [Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fields](https://mlsanity.com/paper/2608.27475/) (2026)

## Abstract (arXiv)

Statistical (machine learning) tools for equation discovery require large amounts of data that are typically computer generated rather than experimentally observed. Multiscale modeling and stochastic simulations are two areas where learning on simulated data can lead to such discovery. In both, the data are generated with a reliable but impractical model, e.g., molecular dynamics simulations, while a model on the scale of interest is uncertain, requiring phenomenological constitutive relations and ad-hoc approximations. We replace the human discovery of such models, which typically involves spatial/stochastic averaging or coarse-graining, with a machine-learning strategy based on sparse regression that can be executed in two modes. The first, direct equation-learning, discovers a differential operator from the whole dictionary. The second, constrained equation-learning, discovers only those terms in the differential operator that need to be discovered, i.e., learns closure approximations. We illustrate our approach by learning a deterministic equation that governs the spatiotemporal evolution of the probability density function of a system state whose dynamics are described by a nonlinear partial differential equation with random inputs. A series of examples demonstrates the accuracy, robustness, and limitations of our approach to equation discovery.