Active Learning with Selective Time-Step Acquisition for PDEs

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.

Many scientific and engineering problems rely on partial differential equations (PDEs) — mathematical rules that describe how things like air, water or traffic flow change over time. Unfortunately, computer programs that solve these equations with high accuracy can take hours or even days, so researchers often build faster “surrogate” models that learn from a limited set of high-quality simulations. The catch is that every extra simulation to learn from still costs valuable computer time and energy.Our work introduces a smarter data-collection strategy called Selective Time-Step Acquisition for PDEs (STAP). Instead of asking the expensive simulator to generate every moment in a scenario, STAP first lets a lightweight surrogate guess the easy moments and then pinpoints only the most informative time steps for precise simulation. In this way, each new training example gives more “bang for the buck.”Across five classic PDE benchmarks — from shock-forming Burgers waves to turbulent Navier–Stokes flows — STAP cut simulation costs while also lowering prediction errors compared with existing active-learning methods. By squeezing more knowledge out of fewer simulations, our approach can accelerate research in climate science, engineering design and any field that depends on large-scale PDE modeling.