# Methods to Recover Unknown Processes in Partial Differential Equations Using Data

arXiv: https://arxiv.org/abs/2003.02387 · HTML: https://mlsanity.com/paper/2003.02387/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Mar 2020.

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

- [WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming](https://mlsanity.com/paper/2211.03134/) (2022)
- [Asymptotic consistency of the WSINDy algorithm in the limit of continuum data](https://mlsanity.com/paper/2211.16000/) (2022)
- [Adaptive Group Lasso Neural Network Models for Functions of Few Variables and Time-Dependent Data](https://mlsanity.com/paper/2108.10825/) (2021)

## Abstract (arXiv)

We study the problem of identifying unknown processes embedded in time-dependent partial differential equation (PDE) using observational data, with an application to advection-diffusion type PDE. We first conduct theoretical analysis and derive conditions to ensure the solvability of the problem. We then present a set of numerical approaches, including Galerkin type algorithm and collocation type algorithm. Analysis of the algorithms are presented, along with their implementation detail. The Galerkin algorithm is more suitable for practical situations, particularly those with noisy data, as it avoids using derivative/gradient data. Various numerical examples are then presented to demonstrate the performance and properties of the numerical methods.