AL-PINN: Active Learning-Driven Physics-Informed Neural Networks for Efficient Sample Selection in Solving Partial Differential Equations

TL;DR

AL-PINN introduces an active learning framework for physics-informed neural networks, significantly reducing training samples needed for solving PDEs by focusing on high-uncertainty regions, thus improving efficiency and accuracy.

Contribution

This paper presents a novel integration of active learning and uncertainty quantification into PINNs, enabling adaptive sample selection for efficient PDE solving.

Findings

Achieves comparable or better accuracy with fewer samples.

Reduces computational costs in solving PDEs.

Effective on benchmark and real-world data.

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving Partial Differential Equations (PDEs) by incorporating physical constraints into deep learning models. However, standard PINNs often require a large number of training samples to achieve high accuracy, leading to increased computational costs. To address this issue, we propose Active Learning-Driven PINNs (AL-PINN), which integrates Uncertainty Quantification (UQ) and Active Learning (AL) strategies to optimize sample selection dynamically. AL-PINN utilizes Monte Carlo Dropout to estimate epistemic uncertainty in the model predictions, enabling the adaptive selection of high-uncertainty regions for additional training. This approach significantly enhances learning efficiency by focusing computational resources on the most informative data points. We evaluate AL-PINN on benchmark PDE problems with known analytical solutions and real-world WeatherBench climate data. Our results demonstrate that AL-PINN achieves comparable or superior accuracy compared to traditional PINNs while reducing the number of required training samples. The proposed framework is particularly beneficial for scientific and engineering applications where data collection is expensive or limited, such as climate modeling, medical simulations, and material science. Our findings highlight the potential of active learning in accelerating PINN-based PDE solvers while maintaining high accuracy and computational efficiency.