Multi-Resolution Active Learning of Fourier Neural Operators

TL;DR

This work tackles the data-cost bottleneck in training Fourier Neural Operators by introducing MRA-FNO, a probabilistic, multi-resolution framework that jointly learns from low- and high-resolution inputs. It leverages an ensemble-based posterior for uncertainty quantification and uses mutual-information–based acquisition with a cost-annealing strategy to dynamically balance resolution choices. The method yields superior data efficiency and uncertainty calibration across multiple PDE benchmarks, outperforming standard FNO and several probabilistic baselines while enabling scalable multi-fidelity learning. The approach offers practical benefits for physics-informed modeling and simulation pipelines where generating high-fidelity data is expensive, with public code available for replication.

Abstract

Fourier Neural Operator (FNO) is a popular operator learning framework. It not only achieves the state-of-the-art performance in many tasks, but also is efficient in training and prediction. However, collecting training data for the FNO can be a costly bottleneck in practice, because it often demands expensive physical simulations. To overcome this problem, we propose Multi-Resolution Active learning of FNO (MRA-FNO), which can dynamically select the input functions and resolutions to lower the data cost as much as possible while optimizing the learning efficiency. Specifically, we propose a probabilistic multi-resolution FNO and use ensemble Monte-Carlo to develop an effective posterior inference algorithm. To conduct active learning, we maximize a utility-cost ratio as the acquisition function to acquire new examples and resolutions at each step. We use moment matching and the matrix determinant lemma to enable tractable, efficient utility computation. Furthermore, we develop a cost annealing framework to avoid over-penalizing high-resolution queries at the early stage. The over-penalization is severe when the cost difference is significant between the resolutions, which renders active learning often stuck at low-resolution queries and inferior performance. Our method overcomes this problem and applies to general multi-fidelity active learning and optimization problems. We have shown the advantage of our method in several benchmark operator learning tasks. The code is available at https://github.com/shib0li/MRA-FNO.

Figures (9)

• Figure 1: Relative $L_2$ error vs. accumulated data cost. Each method ran 500 active learning steps. Note that different methods can end up with different total data cost (after running the same number of steps).
• Figure 2: Number of resolutions queried by MRA-FNO at different stages during active learning.
• Figure 3: Relative $L_2$ error vs. accumulated data cost.
• Figure 4: The influence of the cost schedule on active learning. We report the result with the exponential decay; see \ref{['eq:decay-func']}. The larger $\alpha$, the faster the schedule converges to the true cost.
• Figure 5: Point-wise error on NS.