Active Learning with Selective Time-Step Acquisition for PDEs

TL;DR

This paper introduces STAP, an active learning framework for PDE surrogate modeling that selectively acquires key time steps to reduce computational costs and improve efficiency.

Contribution

The paper proposes a novel selective time-step acquisition method for PDEs that reduces data generation costs and enhances active learning efficiency.

Findings

STAP reduces the number of full trajectory computations needed.

The method achieves comparable accuracy with fewer data points.

Demonstrated effectiveness on multiple benchmark PDEs.

Abstract

Accurately solving partial differential equations (PDEs) is critical to understanding complex scientific and engineering phenomena, yet traditional numerical solvers are computationally expensive. Surrogate models offer a more efficient alternative, but their development is hindered by the cost of generating sufficient training data from numerical solvers. In this paper, we present a novel framework for active learning in PDE surrogate modeling that reduces this cost. Unlike the existing AL methods for PDEs that always acquire entire PDE trajectories, our approach, STAP (**S**elective **T**ime-Step **A**cquisition for **P**DEs), strategically generates only the most important time steps with the numerical solver, while employing the surrogate model to approximate the remaining steps. This reduces the cost incurred by each trajectory and thus allows the active learning algorithm to try out a more diverse set of trajectories given the same budget. To accommodate this novel framework, we develop an acquisition function that estimates the utility of a set of time steps by approximating its resulting variance reduction. We demonstrate the effectiveness of our method on several benchmark PDEs.