Active Learning with Selective Time-Step Acquisition for PDEs

TL;DR

The paper tackles the data inefficiency of constructing PDE surrogate models by introducing Selective Time-Step Acquisition for PDEs (STAP), which adaptively queries only a subset of time steps per trajectory and uses surrogates for the remaining steps under a fixed budget. Central to STAP is a variance-reduction based acquisition function that selects both the initial condition from a pool and the time-step subset in a partially observable trajectory, ensuring informative and diverse data collection. Empirical results across Burgers, KdV, KS, INS, and CNS show that STAP with SBAL outperforms full-trajectory AL baselines, achieving lower average errors and better tail quantiles, while reducing solver calls. The approach advances data-efficient PDE surrogate modeling and enables broader, cost-effective exploration of trajectory data, with potential extensions to multi-fidelity settings and other time-series PDE tasks.

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 (AL) in PDE surrogate modeling that reduces this cost. Unlike the existing AL methods for PDEs that always acquire entire PDE trajectories, our approach strategically generates only the most important time steps with the numerical solver, while employing the surrogate model to approximate the remaining steps. This dramatically 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, including the Burgers' equation, Korteweg-De Vries equation, Kuramoto-Sivashinsky equation, the incompressible Navier-Stokes equation, and the compressible Navier-Stokes equation. Experiments show that our approach improves performance by large margins over the best existing method. Our method not only reduces average error but also the 99\%, 95\%, and 50\% quantiles of error, which is rare for an AL algorithm. All in all, our approach offers a data-efficient solution to surrogate modeling for PDEs.

Figures (15)

• Figure 1: PCA of surrogate model hidden layer's activation patterns on states of the incompressible Navier-Stokes equation. The left figure highlights states within 10 trajectories, and the right figure highlights the same number of states chosen randomly.
• Figure 2: Illustrated overview of STAP. This illustration describes one round of AL.
• Figure 3: Log RMSE of AL strategies, measured across 10 rounds of acquisition. Each round incurs constant cost of data acquisition, namely the budget $B$.
• Figure 4: Quantiles of log RMSE on Burgers measured across 10 rounds of acquisition.
• Figure 5: Timesteps chosen by SBAL+. Each row corresponds to an acquired trajectory, where the black cells indicate the selected time steps. We show twenty trajectories acquired in the first rounds of active learning.