# Active Learning with Selective Time-Step Acquisition for PDEs

arXiv: https://arxiv.org/abs/2511.18107 · HTML: https://mlsanity.com/paper/2511.18107/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 59 references that are in the map.
- First posted on arXiv 22 Nov 2025.

## Builds on (most cited first, 30 of 31)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Deep Batch Active Learning by Diverse, Uncertain Gradient Lower Bounds](https://mlsanity.com/paper/1906.03671/) (2019)
- [BatchBALD: Efficient and Diverse Batch Acquisition for Deep Bayesian Active Learning](https://mlsanity.com/paper/1906.08158/) (2019)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Grammar Variational Autoencoder](https://mlsanity.com/paper/1703.01925/) (2017)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Bayesian Optimization with Unknown Constraints](https://mlsanity.com/paper/1403.5607/) (2014)
- [Deep transfer operator learning for partial differential equations under conditional shift](https://mlsanity.com/paper/2204.09810/) (2022)
- [Lie Point Symmetry Data Augmentation for Neural PDE Solvers](https://mlsanity.com/paper/2202.07643/) (2022)
- [Prediction-Oriented Bayesian Active Learning](https://mlsanity.com/paper/2304.08151/) (2023)
- [Predictive Entropy Search for Bayesian Optimization with Unknown Constraints](https://mlsanity.com/paper/1502.05312/) (2015)
- [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)
- [Pool-Based Sequential Active Learning for Regression](https://mlsanity.com/paper/1805.04735/) (2018)
- [Discovering and forecasting extreme events via active learning in neural operators](https://mlsanity.com/paper/2204.02488/) (2022)
- [Active learning of deep surrogates for PDEs: Application to metasurface design](https://mlsanity.com/paper/2008.12649/) (2020)
- [Active Training of Physics-Informed Neural Networks to Aggregate and Interpolate Parametric Solutions to the Navier-Stokes Equations](https://mlsanity.com/paper/2005.05092/) (2020)
- [Deep Multi-Fidelity Active Learning of High-dimensional Outputs](https://mlsanity.com/paper/2012.00901/) (2020)
- [Multi-Resolution Active Learning of Fourier Neural Operators](https://mlsanity.com/paper/2309.16971/) (2023)
- [Active Learning for Neural PDE Solvers](https://mlsanity.com/paper/2408.01536/) (2024)
- [Batch Multi-Fidelity Active Learning with Budget Constraints](https://mlsanity.com/paper/2210.12704/) (2022)
- [Multi-Fidelity Active Learning with GFlowNets](https://mlsanity.com/paper/2306.11715/) (2023)
- [Disentangled Multi-Fidelity Deep Bayesian Active Learning](https://mlsanity.com/paper/2305.04392/) (2023)
- [Feasibility Study on Active Learning of Smart Surrogates for Scientific Simulations](https://mlsanity.com/paper/2407.07674/) (2024)

## Built on by (most cited first, 4 of 4)

- [Learning Where to Simulate: Generative Active Sampling for Online PDE Surrogate Training](https://mlsanity.com/paper/2606.09949/) (2026)
- [Debiased neural operators for estimating functionals](https://mlsanity.com/paper/2604.19296/) (2026)
- [Gradient-Informed Temporal Sampling Improves Rollout Accuracy in PDE Surrogate Training](https://mlsanity.com/paper/2603.18237/) (2026)
- [Data-Efficient Neural Operator Training via Physics-Based Active Learning](https://mlsanity.com/paper/2605.21348/) (2026)

## Abstract (arXiv)

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 in PDE surrogate modeling that reduces this cost. Unlike the existing AL methods for PDEs that always acquire entire PDE trajectories, our approach, STAP (**S**elective **T**ime-Step **A**cquisition for **P**DEs), strategically generates only the most important time steps with the numerical solver, while employing the surrogate model to approximate the remaining steps. This 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.