# Deep-Learning Discovers Macroscopic Governing Equations for Viscous Gravity Currents from Microscopic Simulation Data

arXiv: https://arxiv.org/abs/2106.00009 · HTML: https://mlsanity.com/paper/2106.00009/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 41 references that are in the map.
- First posted on arXiv 31 May 2021.

## Builds on (most cited first, 8 of 8)

- [Explainable Machine Learning for Scientific Insights and Discoveries](https://mlsanity.com/paper/1905.08883/) (2019)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm](https://mlsanity.com/paper/2001.07305/) (2020)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data](https://mlsanity.com/paper/1908.04463/) (2019)
- [Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data](https://mlsanity.com/paper/2011.11981/) (2020)
- [Data-driven PDE discovery with evolutionary approach](https://mlsanity.com/paper/1903.08011/) (2019)

## Built on by (most cited first, 3 of 3)

- [Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion](https://mlsanity.com/paper/2208.03322/) (2022)
- [AutoKE: An automatic knowledge embedding framework for scientific machine learning](https://mlsanity.com/paper/2205.05390/) (2022)
- [Discovering Governing Equations by Machine Learning implemented with Invariance](https://mlsanity.com/paper/2203.15586/) (2022)

## Abstract (arXiv)

Although deep-learning has been successfully applied in a variety of science and engineering problems owing to its strong high-dimensional nonlinear mapping capability, it is of limited use in scientific knowledge discovery. In this work, we propose a deep-learning based framework to discover the macroscopic governing equation of viscous gravity current based on high-resolution microscopic simulation data without the need for prior knowledge of underlying terms. For two typical scenarios with different viscosity ratios, the deep-learning based equations exactly capture the same dominated terms as the theoretically derived equations for describing long-term asymptotic behaviors, which validates the proposed framework. Unknown macroscopic equations are then obtained for describing short-term behaviors, and additional deep-learned compensation terms are eventually discovered. Comparison of posterior tests shows that the deep-learning based PDEs actually perform better than the theoretically derived PDEs in predicting evolving viscous gravity currents for both long-term and short-term regimes. Moreover, the proposed framework is proven to be very robust against non-biased data noise for training, which is up to 20%. Consequently, the presented deep-learning framework shows considerable potential for discovering unrevealed intrinsic laws in scientific semantic space from raw experimental or simulation results in data space.