# Multi-Resolution Active Learning of Fourier Neural Operators

arXiv: https://arxiv.org/abs/2309.16971 · HTML: https://mlsanity.com/paper/2309.16971/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 49 references that are in the map.
- First posted on arXiv 29 Sep 2023.

## Builds on (most cited first, 24 of 24)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [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)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Deep Bayesian Active Learning with Image Data](https://mlsanity.com/paper/1703.02910/) (2017)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Deep Batch Active Learning by Diverse, Uncertain Gradient Lower Bounds](https://mlsanity.com/paper/1906.03671/) (2019)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Adversarial Active Learning for Deep Networks: a Margin Based Approach](https://mlsanity.com/paper/1802.09841/) (2018)
- [Discriminative Active Learning](https://mlsanity.com/paper/1907.06347/) (2019)
- [Deep Active Learning over the Long Tail](https://mlsanity.com/paper/1711.00941/) (2017)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Kernel Methods are Competitive for Operator Learning](https://mlsanity.com/paper/2304.13202/) (2023)
- [Multi-Fidelity Bayesian Optimization via Deep Neural Networks](https://mlsanity.com/paper/2007.03117/) (2020)
- [Discovering and forecasting extreme events via active learning in neural operators](https://mlsanity.com/paper/2204.02488/) (2022)
- [Deep Multi-Fidelity Active Learning of High-dimensional Outputs](https://mlsanity.com/paper/2012.00901/) (2020)
- [Batch Multi-Fidelity Active Learning with Budget Constraints](https://mlsanity.com/paper/2210.12704/) (2022)
- [DeepOHeat: Operator Learning-based Ultra-fast Thermal Simulation in 3D-IC Design](https://mlsanity.com/paper/2302.12949/) (2023)
- [Residual Gaussian Process: A Tractable Nonparametric Bayesian Emulator for Multi-fidelity Simulations](https://mlsanity.com/paper/2104.03743/) (2021)
- [Infinite-Fidelity Coregionalization for Physical Simulation](https://mlsanity.com/paper/2207.00678/) (2022)
- [A Kernel Approach for PDE Discovery and Operator Learning](https://mlsanity.com/paper/2210.08140/) (2022)

## Built on by (most cited first, 22 of 22)

- [Active Learning for Neural PDE Solvers](https://mlsanity.com/paper/2408.01536/) (2024)
- [Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems](https://mlsanity.com/paper/2402.11722/) (2024)
- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learning](https://mlsanity.com/paper/2509.12344/) (2025)
- [The False Promise of Zero-Shot Super-Resolution in Machine-Learned Operators](https://mlsanity.com/paper/2510.06646/) (2025)
- [Self-Attention to Operator Learning-based 3D-IC Thermal Simulation](https://mlsanity.com/paper/2510.15968/) (2025)
- [Spectral Convolutional Conditional Neural Processes](https://mlsanity.com/paper/2404.13182/) (2024)
- [Arbitrarily-Conditioned Multi-Functional Diffusion for Multi-Physics Emulation](https://mlsanity.com/paper/2410.13794/) (2024)
- [A Physics-informed Multi-resolution Neural Operator](https://mlsanity.com/paper/2510.23810/) (2025)
- [Active Learning with Selective Time-Step Acquisition for PDEs](https://mlsanity.com/paper/2511.18107/) (2025)
- [Learning where to learn: Training data distribution optimization for scientific machine learning](https://mlsanity.com/paper/2505.21626/) (2025)
- [PICore: Physics-Informed Unsupervised Coreset Selection for Data Efficient Neural Operator Training](https://mlsanity.com/paper/2507.17151/) (2025)
- [Accelerating PDE Surrogates via RL-Guided Mesh Optimization](https://mlsanity.com/paper/2603.02066/) (2026)
- [COMPOL: A Unified Neural Operator Framework for Scalable Multi-Physics Simulations](https://mlsanity.com/paper/2501.17296/) (2025)
- [Thompson Sampling in Function Spaces via Neural Operators](https://mlsanity.com/paper/2506.21894/) (2025)
- [Multi-Fidelity Flow Matching: Cascaded Refinement of PDE Solutions](https://mlsanity.com/paper/2605.16118/) (2026)
- [Deep Gaussian Processes for Functional Maps](https://mlsanity.com/paper/2510.22068/) (2025)
- [SpectraKAN: Conditioning Spectral Operators](https://mlsanity.com/paper/2602.05187/) (2026)
- [Resolution-free neural surrogates for geometric parameterization and mapping with spatially varying fields](https://mlsanity.com/paper/2605.28551/) (2026)
- [Sequential operator learning under dependent data](https://mlsanity.com/paper/2608.24426/) (2026)
- [Learnable composition for neural operators](https://mlsanity.com/paper/2609.03069/) (2026)
- [Helix-FNO: Spectral-Domain Operator Learning Coupled with a High-Fidelity Mechanistic Model for Fast Surrogate Simulation](https://mlsanity.com/paper/2609.22129/) (2026)

## Abstract (arXiv)

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.