# Model discovery in the sparse sampling regime

arXiv: https://arxiv.org/abs/2105.00400 · HTML: https://mlsanity.com/paper/2105.00400/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 39 references that are in the map.
- First posted on arXiv 2 May 2021.

## Builds on (most cited first, 18 of 18)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Graph networks as learnable physics engines for inference and control](https://mlsanity.com/paper/1806.01242/) (2018)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [Data-Driven Sparse Sensor Placement for Reconstruction](https://mlsanity.com/paper/1701.07569/) (2017)
- [Extracting Sparse High-Dimensional Dynamics from Limited Data](https://mlsanity.com/paper/1707.08528/) (2017)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [Differentiable Physics-informed Graph Networks](https://mlsanity.com/paper/1902.02950/) (2019)
- [Stability selection enables robust learning of partial differential equations from limited noisy data](https://mlsanity.com/paper/1907.07810/) (2019)
- [Sparsely constrained neural networks for model discovery of PDEs](https://mlsanity.com/paper/2011.04336/) (2020)
- [Optimal Sensor Placement and Enhanced Sparsity for Classification](https://mlsanity.com/paper/1310.4217/) (2013)

## Built on by (most cited first, 2 of 2)

- [GN-SINDy: Greedy Sampling Neural Network in Sparse Identification of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2405.08613/) (2024)
- [Sparsistent Model Discovery](https://mlsanity.com/paper/2106.11936/) (2021)

## Abstract (arXiv)

To improve the physical understanding and the predictions of complex dynamic systems, such as ocean dynamics and weather predictions, it is of paramount interest to identify interpretable models from coarsely and off-grid sampled observations. In this work, we investigate how deep learning can improve model discovery of partial differential equations when the spacing between sensors is large and the samples are not placed on a grid. We show how leveraging physics informed neural network interpolation and automatic differentiation, allow to better fit the data and its spatiotemporal derivatives, compared to more classic spline interpolation and numerical differentiation techniques. As a result, deep learning-based model discovery allows to recover the underlying equations, even when sensors are placed further apart than the data's characteristic length scale and in the presence of high noise levels. We illustrate our claims on both synthetic and experimental data sets where combinations of physical processes such as (non)-linear advection, reaction, and diffusion are correctly identified.