# Data-Driven Theory-guided Learning of Partial Differential Equations using SimultaNeous Basis Function Approximation and Parameter Estimation (SNAPE)

arXiv: https://arxiv.org/abs/2109.07471 · HTML: https://mlsanity.com/paper/2109.07471/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 59 references that are in the map.
- First posted on arXiv 14 Sep 2021.

## Builds on (most cited first, 10 of 10)

- [Explaining Explanations: An Overview of Interpretability of Machine Learning](https://mlsanity.com/paper/1806.00069/) (2018)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Model-Agnostic Interpretability of Machine Learning](https://mlsanity.com/paper/1606.05386/) (2016)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [Explainable Machine Learning for Scientific Insights and Discoveries](https://mlsanity.com/paper/1905.08883/) (2019)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)

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

- [Data-driven Nonlinear Model Reduction to Spectral Submanifolds in Mechanical Systems](https://mlsanity.com/paper/2110.01929/) (2021)
- [Physics-informed AI and ML-based sparse system identification algorithm for discovery of PDE's representing nonlinear dynamic systems](https://mlsanity.com/paper/2410.10023/) (2024)

## Abstract (arXiv)

The measured spatiotemporal response of various physical processes is utilized to infer the governing partial differential equations (PDEs). We propose SimultaNeous Basis Function Approximation and Parameter Estimation (SNAPE), a technique of parameter estimation of PDEs that is robust against high levels of noise nearly 100 %, by simultaneously fitting basis functions to the measured response and estimating the parameters of both ordinary and partial differential equations. The domain knowledge of the general multidimensional process is used as a constraint in the formulation of the optimization framework. SNAPE not only demonstrates its applicability on various complex dynamic systems that encompass wide scientific domains including Schr"odinger equation, chaotic duffing oscillator, and Navier-Stokes equation but also estimates an analytical approximation to the process response. The method systematically combines the knowledge of well-established scientific theories and the concepts of data science to infer the properties of the process from the observed data.