Flexible Active Learning of PDE Trajectories
Keywords: Active learning, Partial Differential Equation (PDE)
TL;DR: We propose an active learning framework that reduces the data acquisition cost for PDE surrogate modeling by selectively querying time steps along a trajectory from numerical solvers.
Abstract: Accurately solving partial differential equations (PDEs) is critical for 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 Heat equation, Korteweg–De Vries equation, Kuramoto–Sivashinsky equation, and the incompressible Navier-Stokes equation. Extensive experiments validate that our approach outperforms existing methods, offering a cost-efficient solution to surrogate modeling for PDEs.
Supplementary Material: zip
Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
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Submission Number: 9048
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