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.

翻译：准确求解偏微分方程对于理解复杂的科学和工程现象至关重要，但传统的数值求解器计算成本高昂。代理模型提供了一种更高效的替代方案，但其发展受到从数值求解器生成足够训练数据成本的阻碍。本文提出了一种用于偏微分方程代理建模的主动学习新框架，以降低这一成本。与现有针对偏微分方程的主动学习方法总是获取整个偏微分方程轨迹不同，我们的方法STAP（面向偏微分方程的选择性时间步获取）策略性地仅利用数值求解器生成最重要的时间步，同时采用代理模型逼近剩余时间步。这降低了每个轨迹的成本，从而允许主动学习算法在相同预算下尝试更多样化的轨迹集。为适应这一新框架，我们开发了一种获取函数，通过近似其产生的方差缩减来估计一组时间步的效用。我们在多个基准偏微分方程上验证了该方法的有效性。