Active Learning with Selective Time-Step Acquisition for PDEs

arXiv stat.ML / 4/17/2026

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Key Points

The paper addresses the high computational cost of generating training data for PDE surrogate models by reframing active learning to reduce solver queries.
It proposes STAP (Selective Time-Step Acquisition), which requests only the most important time steps from a numerical solver and uses the surrogate to estimate the rest.
By cutting per-trajectory cost, the method enables the active learning algorithm to explore a more diverse set of PDE trajectories under a fixed compute budget.
The authors introduce an acquisition function that approximates expected variance reduction for a set of selected time steps.
Experiments on multiple benchmark PDEs show that STAP improves the efficiency of active learning for PDE surrogate modeling.

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.