Multi-Resolution Active Learning for Efficient Training of Fourier Neural Operators

Core Concepts

The core message of this work is to propose a novel multi-resolution active learning method, called MRA-FNO, that can dynamically select the most valuable input functions and resolutions to train Fourier neural operators (FNOs) efficiently while significantly reducing the data collection cost.

Abstract

The authors present MRA-FNO, a multi-resolution active learning method for training Fourier neural operators (FNOs). FNOs are a popular operator learning framework that can achieve state-of-the-art performance in many tasks. However, collecting training data for FNOs can be a severe bottleneck in practice, as it often requires expensive physical simulations.
To address this issue, the authors propose to leverage multi-resolution data, which combines accurate but expensive high-resolution examples with inaccurate but cheap-to-generate low-resolution examples. Specifically, they extend the FNO architecture to a probabilistic multi-resolution FNO that can capture the influence of the resolution choice on the predictive distribution. They then develop an active learning algorithm that dynamically selects the most valuable input function and resolution at each step to optimize the learning efficiency while minimizing the data cost.
The key contributions are:

Probabilistic multi-resolution FNO: The authors propose a probabilistic FNO that integrates multi-resolution training data by appending a resolution embedding to the input function samples. This allows the model to capture the influence of the resolution choice on the predictive distribution.
Efficient active learning: The authors develop an active learning algorithm that maximizes a utility-cost ratio to select the most valuable input function and resolution at each step. They address two challenges: (1) the computation of the utility function is analytically intractable, and (2) directly maximizing the utility-cost ratio tends to trap the active learning at low-resolution queries due to the significant cost discrepancy between resolutions. The authors propose moment matching and the matrix determinant lemma to enable efficient utility computation, and a novel cost annealing framework to avoid over-penalizing high-resolution examples.
Experimental results: The authors evaluate MRA-FNO on four benchmark operator learning tasks and show that it consistently achieves much better prediction accuracy with the same accumulated data cost compared to other active learning methods and probabilistic FNO variants.

Stats

The authors report the following key statistics:

The relative L2 error and negative log-likelihood (NLL) of MRA-FNO are significantly better than the standard FNO and other probabilistic FNO variants on the Burgers, Darcy, and other benchmark tasks.
Using only low-resolution examples leads to a performance bottleneck, while using only high-resolution examples incurs much higher data cost.
MRA-FNO dynamically selects a mix of low and high-resolution examples, with more high-resolution queries at the early stage and more low-resolution queries later, leading to superior performance.
The choice of the cost annealing schedule (decay rate) in MRA-FNO has a significant impact on the active learning performance. An appropriate decay rate can balance the exploration of high and low-resolution examples.

Quotes

"To reduce the cost, one can consider leveraging multi-resolution data. The low-resolution data is cheap to obtain — typically computed with rough meshes — but the provided output function samples are quite inaccurate (large bias). On the contrary, high-resolution data offers accurate output function samples, yet is much more costly to generate from dense meshes."
"To optimize the learning efficiency while reducing the data cost as much as possible, we maximize the utility-cost ratio to select the best training input and resolution at each step, where the utility is measured by mutual information."
"Directly maximizing the utility-cost ratio as in previous methods, tends to trap the active learning at low-resolution queries and inferior performance. This is due to that when the data is few (at the early stage), the mutual information measurement for examples at different resolutions is close. High-resolution examples are thereby over-penalized by the large cost."

Key Insights Distilled From

Multi-Resolution Active Learning of Fourier Neural Operators

by Shibo Li,Xin... at arxiv.org 04-01-2024

https://arxiv.org/pdf/2309.16971.pdf

Deeper Inquiries

How can the cost annealing framework in MRA-FNO be extended to other multi-fidelity learning and optimization problems beyond operator learning

The cost annealing framework in MRA-FNO can be extended to other multi-fidelity learning and optimization problems by adapting the concept of dynamically adjusting the cost allocation based on the resolution or fidelity of the data. This framework can be applied to various domains where data collection at different resolutions or fidelities is involved. For example:

Scientific Simulations: In computational science and engineering, where simulations are conducted at varying levels of detail or resolution, the cost annealing framework can help optimize the selection of simulation parameters or resolution levels to balance accuracy and computational cost.
Healthcare: In medical imaging, where images are captured at different resolutions or qualities, the cost annealing approach can aid in selecting the most informative images for diagnosis or analysis while minimizing the cost of acquiring high-resolution data.
Financial Modeling: In financial forecasting or risk assessment, where data is available at different levels of granularity or frequency, the cost annealing strategy can assist in selecting the most relevant data points for accurate predictions while managing computational expenses.

By applying the principles of cost annealing to these multi-fidelity learning and optimization problems, it is possible to enhance the efficiency of data collection, model training, and decision-making processes across various domains.

Can the active learning strategy in MRA-FNO be combined with other operator learning architectures beyond FNOs to further improve the data efficiency

The active learning strategy in MRA-FNO can be combined with other operator learning architectures beyond Fourier Neural Operators (FNOs) to further improve data efficiency in various ways:

Graph Neural Operators: By integrating the active learning approach of MRA-FNO with graph neural operators, the model can intelligently select the most informative nodes or edges in a graph for training, optimizing the learning process and reducing data acquisition costs.
Kernel-Based Operators: When applied to kernel-based operator learning models, the active learning strategy can assist in selecting the most relevant kernel functions or parameters to improve model accuracy while minimizing the need for extensive data collection.
DeepONet: Combining the active learning framework of MRA-FNO with DeepONet, a deep operator network, can enhance the selection of input-output pairs for training, leading to more efficient learning and improved predictive performance.

By integrating the active learning strategy of MRA-FNO with diverse operator learning architectures, it is possible to enhance data efficiency, model accuracy, and decision-making capabilities in a wide range of applications.

What are the potential applications of the learned multi-resolution FNO models beyond the benchmark tasks considered in this work, and how can the models be deployed in real-world scenarios

The learned multi-resolution Fourier Neural Operator (FNO) models from MRA-FNO have several potential applications beyond the benchmark tasks considered in the study:

Image Processing: The models can be deployed in image processing tasks such as image super-resolution, denoising, and enhancement, where multi-resolution data is common. The FNO models can learn complex mappings between low-resolution and high-resolution images, improving image quality.
Signal Processing: In signal processing applications like audio denoising or restoration, the multi-resolution FNO models can learn the underlying signal transformations across different resolutions, enabling more accurate signal reconstruction and analysis.
Climate Modeling: Deploying the models in climate modeling can help in predicting weather patterns, climate changes, and natural disasters by learning the complex relationships between multi-resolution climate data variables.
Healthcare Imaging: The models can be utilized in medical imaging tasks for tasks like MRI reconstruction, CT image enhancement, or pathology image analysis, leveraging the learned mappings between low and high-resolution medical images.

These models can be deployed in real-world scenarios to enhance decision-making, improve data analysis, and optimize processes in diverse fields beyond the scope of the benchmark tasks, showcasing the versatility and applicability of multi-resolution FNO models.