Active Learning with Selective Time-Step Acquisition for PDEs

2025-11-22

AI Summary

To address the high computational cost of data generation in PDE surrogate modeling, this paper proposes a time-dimensional active learning framework. The method selectively samples critical time steps at fine granularity: a novel multi-step variance-reduction acquisition function quantifies each time point’s contribution to global uncertainty reduction, enabling targeted high-value data generation; numerical solvers are applied only at selected time steps, while the surrogate model interpolates intermediate states. Evaluated on multiple canonical PDE benchmarks, the approach significantly outperforms state-of-the-art methods—reducing average error and the 99th, 95th, and 50th percentile errors substantially—while achieving superior data efficiency and modeling accuracy.

Technology Category

• Machine Learning: Active Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Web Mining and Content Analysis: Web data generation and simulation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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.

Problem

• Reducing computational cost of generating PDE training data
• Selectively acquiring critical time steps instead of full trajectories
• Developing active learning framework for efficient PDE surrogate modeling

Innovation

• Selectively acquires key time steps for PDE trajectories
• Uses surrogate model to approximate remaining time steps
• Develops acquisition function estimating variance reduction utility