Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Active Learning for Neural PDE Solvers
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Introduces AL4PDE, a modular pool-based active learning benchmark for neural PDE surrogates, evaluating uncertainty- and feature-based batch AL methods on six parametric PDEs with U-Net, FNO and SineNet surrogates.
Problem
Neural PDE surrogates need large simulator-generated training sets, and active learning for selecting informative initial conditions and PDE parameters has barely been studied for autoregressive neural PDE solvers.
Approach
A modular AL benchmark for solver-in-the-loop neural PDE surrogates that adapts uncertainty (query-by-committee ensembles) and feature-based batch selection (LCMD, Core-Set, BAIT) to autoregressive rollouts, using spatially averaged features for translation invariance and Gaussian sketching for dimensionality reduction.
Key results
- AL reduces average error by up to 71% compared to random sampling and can reach the same accuracy with up to four times fewer data points.
- SBAL, BAIT, and LCMD perform best overall; on CE and Burgers AL reaches lower error with about a quarter of the data.
- AL especially reduces higher error quantiles (95% and 99%), while the 50% quantile sometimes increases; differences on CNS were not significant.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use active learning instead of random sampling | Neural PDE solvers in AL4PDE · Burgers and CE | RMSE: up to 71% | Helped |
| Use active learning instead of random sampling | Neural PDE solvers in AL4PDE · CNS tasks | RMSE | No clear effect |
| Use SBAL, BAIT, or LCMD instead of random sampling | Neural PDE solvers in AL4PDE · KS | RMSE | Mixed |
| Use Top-K or Core-Set instead of random sampling | Neural PDE solvers in AL4PDE · PDE benchmarks | RMSE | Hurt |
| Use LHS instead of random sampling | Neural PDE solvers in AL4PDE · PDE benchmarks | RMSE | No clear effect |
| Use active learning instead of random sampling | Neural PDE solvers in AL4PDE · PDE benchmarks | higher-quantile RMSE | Helped |
| Use SBAL with absolute-difference uncertainty rather than variance uncertainty (ablation) | Neural PDE solvers · Burgers | MAE | Helped |
| Use variance uncertainty rather than absolute-difference uncertainty (ablation) | Neural PDE solvers · Burgers | MAE | No clear effect |
| Use SBAL with SineNet or FNO rather than U-Net (ablation) | SBAL · 2D CNS | RMSE | Helped |
| Use an ensemble size of two models (ablation) | SBAL · PDE benchmarks | RMSE | Helped |
| Use PINO loss as the SBAL uncertainty estimate (ablation) | SBAL · Neural PDE solvers | RMSE | Hurt |
| Use spatially averaged last-layer features rather than full feature vectors or bottleneck features (ablation) | LCMD · PDE benchmarks | RMSE | Helped |
Surprising findings
- Latin Hypercube sampling showed no notable difference from random sampling.
- Using the PINO physics loss as an uncertainty estimate for SBAL was not an effective selection criterion for autoregressive solvers.
Limitations
- Only rectangular domains with periodic boundary conditions (except 1D CNS) and regular grids were considered.
- Stable training of surrogates can be difficult (e.g., 2D CNS), which hurts AL since the model is retrained repeatedly.
- Hyperparameters for AL are hard to select on growing, unseen datasets.
Experimental setup
Datasets: Burgers (PDEBench) Kuramoto-Sivashinsky (JAX-CFD) Combined equation 1D/2D/3D compressible Navier-Stokes (PDEBench)
Benchmarks: Burgers KS CE 1D CNS 2D CNS 3D CNS
Metrics: RMSE MAE RMSE quantiles (50/95/99%) runtime
Baselines: Random sampling Latin Hypercube sampling Top-K Core-Set BAIT LCMD SBAL
Builds on: SBAL LCMD Core-Set BAIT PDEBench
Closest papers by meaning
- Active Learning with Selective Time-Step Acquisition for PDEs (ICML 2025 · similarity 0.93)
- Breakeven complexity: A new perspective on neural partial differential equation solvers (NeurIPS 2026 · similarity 0.89)
- Learning a Neural Solver for Parametric PDEs to Enhance Physics-Informed Methods (ICLR 2025 · similarity 0.89)
- Compute-optimal data scaling for neural surrogates via multi-fidelity training (NeurIPS 2026 · similarity 0.88)
- MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation (ICML 2025 · similarity 0.88)
- Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations (ICML 2024 · similarity 0.88)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.