# Learning Where to Simulate: Generative Active Sampling for Online PDE Surrogate Training

arXiv: https://arxiv.org/abs/2606.09949 · HTML: https://mlsanity.com/paper/2606.09949/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 49 references that are in the map.
- First posted on arXiv 8 Jun 2026.

## Builds on (most cited first, 30 of 35)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [A Survey of Deep Active Learning](https://mlsanity.com/paper/2009.00236/) (2020)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Sampling](https://mlsanity.com/paper/2207.02338/) (2022)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [APEBench: A Benchmark for Autoregressive Neural Emulators of PDEs](https://mlsanity.com/paper/2411.00180/) (2024)
- [Semi-Supervised Active Learning with Temporal Output Discrepancy](https://mlsanity.com/paper/2107.14153/) (2021)
- [A Framework and Benchmark for Deep Batch Active Learning for Regression](https://mlsanity.com/paper/2203.09410/) (2022)
- [Stochastic Batch Acquisition: A Simple Baseline for Deep Active Learning](https://mlsanity.com/paper/2106.12059/) (2021)
- [PINNACLE: PINN Adaptive ColLocation and Experimental points selection](https://mlsanity.com/paper/2404.07662/) (2024)
- [PDE-Transformer: Efficient and Versatile Transformers for Physics Simulations](https://mlsanity.com/paper/2505.24717/) (2025)
- [Active learning for data streams: a survey](https://mlsanity.com/paper/2302.08893/) (2023)
- [Active learning of deep surrogates for PDEs: Application to metasurface design](https://mlsanity.com/paper/2008.12649/) (2020)
- [RIGNO: A Graph-based framework for robust and accurate operator learning for PDEs on arbitrary domains](https://mlsanity.com/paper/2501.19205/) (2025)
- [Active Learning for Neural PDE Solvers](https://mlsanity.com/paper/2408.01536/) (2024)
- [Physics-enhanced deep surrogates for partial differential equations](https://mlsanity.com/paper/2111.05841/) (2021)
- [Data efficient surrogate modeling for engineering design: Ensemble-free batch mode deep active learning for regression](https://mlsanity.com/paper/2211.10360/) (2022)
- [Training Deep Surrogate Models with Large Scale Online Learning](https://mlsanity.com/paper/2306.16133/) (2023)
- [On the Benefits of Active Data Collection in Operator Learning](https://mlsanity.com/paper/2410.19725/) (2024)
- [Feasibility Study on Active Learning of Smart Surrogates for Scientific Simulations](https://mlsanity.com/paper/2407.07674/) (2024)
- [Active Learning with Selective Time-Step Acquisition for PDEs](https://mlsanity.com/paper/2511.18107/) (2025)

## Built on by (most cited first, 2 of 2)

- [Staying on the Attractor: Supervising Neural Surrogates of 3D Turbulence Where They Leave It](https://mlsanity.com/paper/2609.32864/) (2026)
- [Scalable Diffusion SBI for Compositional Inference under Simulator Misspecification](https://mlsanity.com/paper/2609.36950/) (2026)

## Abstract (arXiv)

Data-driven PDE surrogates are trained with data produced by numerical PDE solvers. However, when the surrogate's goal is to generalize across a wide range of PDE configurations (e.g., initial conditions and physical coefficients), generating a representative training set is non-trivial. Uniform sampling of configuration parameters often under-represents trajectories exhibiting challenging dynamics, leading to high prediction errors and large error variance in the trained surrogate. Online training, where data generation and surrogate training are coupled, offers a natural advantage by allowing solver parameters to be steered on-the-fly. To efficiently exploit this capability, we introduce Online Generative Active Sampling (OGAS), an active learning method that reactively learns the relationship between configuration parameters and surrogate performance to control the sampling distribution. OGAS trains a fast diffusion model in parallel to the surrogate to act as a conditional sampler, mapping a surrogate-derived difficulty signal (e.g., loss or uncertainty) to configuration parameters. By actively drawing target signals from a prior biased toward high difficulty, OGAS continuously steers data generation toward challenging regimes without delaying the training workflow. We evaluate OGAS across 2D PDEs with distinct challenging dynamics (Kuramoto-Sivashinsky, Navier-Stokes, Gray-Scott) and up to 308 parameters, using multiple surrogate architectures. Across all settings, OGAS consistently improves tail statistics, yielding substantial reductions in errors above the 99th percentile and overall error dispersion compared to uniform sampling. While prioritizing challenging trajectories introduces a trade-off with average error, OGAS effectively ensures worst-case reliability of trained surrogates with negligible wall-time overhead.