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Active Learning with Selective Time-Step Acquisition for PDEs

ICML 2025 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural PDE solvers · 0 citations (OpenAlex)

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Proposes STAP, an active learning add-on for PDE trajectory surrogates that acquires only selected time steps from the numerical solver, filling skipped steps with the surrogate, and selects them with a variance-reduction acquisition function.

Problem

Reducing the cost of generating numerical-solver training data for autoregressive neural surrogate models of PDE trajectories, where existing active learning acquires entire trajectories.

Approach

Instead of acquiring full trajectories, select a sampling pattern of time steps per initial condition; skipped steps are filled by the surrogate, and the pattern is chosen by maximizing a committee-based variance-reduction acquisition function, greedily optimized with bit-flip search and used as an add-on to full-trajectory AL methods.

Key results

• SBAL+STAP achieves the best averaged log RMSE on all five PDEs, e.g., Burgers -3.674 vs. -3.388 for SBAL and KS -0.349 vs. -0.275 for SBAL.
• Its variant STAP 10 reduces batch-selection time to 15.4 s versus 62.5 s for QbC/SBAL in INS while still improving over the base method.
• In the multistep FNO setting, SBAL+STAP improves Burgers 8L/8 log RMSE to -2.058 versus -1.893 for SBAL.

Measured effects reported in the paper

Surprising findings

• Random Bernoulli sampling of time steps alone often improves over full-trajectory SBAL, suggesting sparse time-step sampling is inherently advantageous.
• STAP still outperforms baselines when the initial training set has only one trajectory, despite erroneous surrogate-generated inputs.

Limitations

• Skipped time steps are filled with surrogate predictions, which can produce out-of-distribution or unstable synthetic inputs (e.g., KdV simulation crashes).
• Acquisition function is a heuristic approximation to expected error reduction under committee assumptions, not a provably exact criterion.
• Experiments are limited to a fixed set of benchmark PDEs with a single FNO architecture and committee size M=2.

Experimental setup

Datasets: Burgers (simulated, FDM) KdV (simulated, pseudospectral) Kuramoto-Sivashinsky (simulated, ETD-RK4) incompressible Navier-Stokes (simulated, Crank-Nicolson) compressible Navier-Stokes (simulated)

Benchmarks: Burgers KdV Kuramoto-Sivashinsky incompressible Navier-Stokes compressible Navier-Stokes

Metrics: log RMSE log NRMSE log MAE log quantile RMSE (99%, 95%, 50%) wall-clock time

Baselines: Random QbC LCMD SBAL Bernoulli sampling Ber(p)

Builds on: Query-by-Committee Expected error reduction Fourier Neural Operator SBAL LCMD

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