PICore: Physics-Informed Unsupervised Coreset Selection for Data Efficient Neural Operator Training

TL;DR

PICore introduces a physics-informed unsupervised coreset selection framework to reduce both data labeling and training time in neural operator learning for PDEs. By evaluating unlabeled inputs with a PDE residual loss, PICore selects a small, highly informative subset and subsequently labels only that subset with solvers, enabling efficient supervised training on a compact dataset. Across four PDE benchmarks and multiple selection strategies, PICore achieves competitive accuracy while delivering up to 78% end-to-end training speedups, outperforming many unsupervised baselines and approaching supervised coreset performance at reduced data budgets. The method is architecture-agnostic and scales to different neural operators (e.g., FNO, UNO), with code available for replication and extension.

Abstract

Neural operators offer a powerful paradigm for solving partial differential equations (PDEs) that cannot be solved analytically by learning mappings between function spaces. However, there are two main bottlenecks in training neural operators: they require a significant amount of training data to learn these mappings, and this data needs to be labeled, which can only be accessed via expensive simulations with numerical solvers. To alleviate both of these issues simultaneously, we propose PICore, an unsupervised coreset selection framework that identifies the most informative training samples without requiring access to ground-truth PDE solutions. PICore leverages a physics-informed loss to select unlabeled inputs by their potential contribution to operator learning. After selecting a compact subset of inputs, only those samples are simulated using numerical solvers to generate labels, reducing annotation costs. We then train the neural operator on the reduced labeled dataset, significantly decreasing training time as well. Across four diverse PDE benchmarks and multiple coreset selection strategies, PICore achieves up to 78% average increase in training efficiency relative to supervised coreset selection methods with minimal changes in accuracy. We provide code at https://github.com/Asatheesh6561/PICore.

Figures (8)

• Figure 1: Overview of the PICore Framework. Given a set of initial conditions and a pre-trained (warm-started) neural operator $\mathcal{G}_\theta$, we compute the physics-informed loss $\mathcal{L}_{PI}(a_i; \theta)$ for each initial condition $a_i$. These losses are passed to a coreset selection algorithm $\mathcal{A}$, which identifies the most informative samples that deviate most from the underlying PDE. Each selected sample is assigned a weight $\gamma_j$ and is then simulated using a numerical solver to obtain the PDE solutions. The resulting labeled subset is used to update $\mathcal{G}_\theta$ using a weighted data loss, enabling efficient training by focusing on the most impactful data points. In the figure, blue arrows represent forward passes and red lines represent backward passes respectively.
• Figure 2: Training convergence of coreset selection v.s. active learning using FNO at a 20% selection ratio.
• Figure 3: Training convergence of coreset selection v.s. active learning using UNO at a 20% selection ratio.
• Figure 4: Test NRMSE on the Advection dataset at resolution 64 across varying coreset percentages (20%–100%) between unsupervised and PICore-based coreset selection methods using both FNO and UNO architectures.
• Figure 5: Test NRMSE on the Burgers dataset at resolution 64 across varying coreset percentages (20%–100%) between unsupervised and PICore-based coreset selection methods using both FNO and UNO architectures.