GS-PINN: Greedy Sampling for Parameter Estimation in Partial Differential Equations

TL;DR

This paper introduces a greedy sampling method using the Discrete Empirical Interpolation Method to improve parameter estimation in PDEs with physics-informed neural networks, reducing sample size and error.

Contribution

It proposes a novel greedy sampling approach for PDE parameter estimation that outperforms random sampling, supported by a Python package for implementation.

Findings

Greedy samples lead to lower estimation error than random samples.

The method reduces the number of samples needed for accurate parameter estimation.

The approach is validated across multiple PDEs with consistent improvements.

Abstract

Partial differential equation parameter estimation is a mathematical and computational process used to estimate the unknown parameters in a partial differential equation model from observational data. This paper employs a greedy sampling approach based on the Discrete Empirical Interpolation Method to identify the most informative samples in a dataset associated with a partial differential equation to estimate its parameters. Greedy samples are used to train a physics-informed neural network architecture which maps the nonlinear relation between spatio-temporal data and the measured values. To prove the impact of greedy samples on the training of the physics-informed neural network for parameter estimation of a partial differential equation, their performance is compared with random samples taken from the given dataset. Our simulation results show that for all considered partial differential equations, greedy samples outperform random samples, i.e., we can estimate parameters with a significantly lower number of samples while simultaneously reducing the relative estimation error. A Python package is also prepared to support different phases of the proposed algorithm, including data prepossessing, greedy sampling, neural network training, and comparison.