Coauthors: Marimuthu Kalimuthu, David Holzmüller (@dholzmueller.bsky.social), Makoto Takamoto, and Mathias Niepert (@mniepert.bsky.social)
Paper: arxiv.org/abs/2408.01536
Code: github.com/dmusekamp/al...
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Limitations:
- Future work is needed to look at the missing advantage of AL on CNS.
- Benchmark does not include irregular grids or complex geometries, which might be
an interesting setting for AL due to the more complex input space. 8/

The generated data is also beneficial for surrogate models which have not been used to select the data. Here, we compare the accuracy of a U-Net with data selected randomly or using an FNO or the U-Net itself as the base model. 7/

A look at the distribution of the selected parameters shows that the standard deviation between random repetitions is small, indicating that the AL procedure reliably produces very similar datasets. 6/

The experiments show that AL reduces the average errors by up to 71% compared to random sampling for the same amount of selected data. Especially, Stochastic Batch Active Learning and LCMD perform well. 5/

To facilitate the research of AL on autoregressive neural PDE solvers, we introduce AL4PDE, an extensible, modular benchmark framework. It provides:
- Parametric PDEs such as incompressible Navier-Stokes
- Surrogate models (U-Net, FNO, SineNet).
- AL algorithms such as SBAL, CoreSet, or LCMD. 4/

AL presents a promising solution by only selecting the most informative training samples, reducing the number of simulations required to train neural PDE solvers. As high-dimensional, spatio-temporal time series, PDEs are a challenging domain for AL algorithms. 3/

Solving partial differential equations (PDEs) is fundamental in science & engineering. Neural PDE solvers can offer advantages such as speed and differentiability but require large datasets from costly numerical simulations. 2/

AL presents a promising solution by only selecting the most informative training samples, reducing the number of simulations required to train neural PDE solvers. As high-dimensional, spatio-temporal time series, PDEs are a challenging domain for AL algorithms. 3/

Solving partial differential equations (PDEs) is fundamental in science & engineering. Neural PDE solvers can offer advantages such as speed and differentiability but require large datasets from costly numerical simulations. 2/