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A novel Fourier neural operator framework for classification of multi-sized images: Application to three dimensional digital porous media

Ali Kashefi, Tapan Mukerji

TL;DR

This paper addresses classifying images of varying sizes by leveraging Fourier neural operators (FNOs) that are inherently size-agnostic. It introduces a novel static max pooling strategy in the Fourier-channel width to connect FNOs to an MLP classifier, avoiding the pitfalls of adaptive pooling in 3D space. Applied to predicting 3D digital porous media permeability, the approach achieves high $R^2$ scores across multiple sizes, demonstrates excellent generalization to unseen sizes, and delivers substantial speedups over traditional lattice Boltzmann simulations. The work advances multi-size image classification with FNOs and suggests extensions to broader 2D/3D classification tasks and real-world porous-media data.

Abstract

Fourier neural operators (FNOs) are invariant with respect to the size of input images, and thus images with any size can be fed into FNO-based frameworks without any modification of network architectures, in contrast to traditional convolutional neural networks (CNNs). Leveraging the advantage of FNOs, we propose a novel deep-learning framework for classifying images with varying sizes. Particularly, we simultaneously train the proposed network on multi-sized images. As a practical application, we consider the problem of predicting the label (e.g., permeability) of three-dimensional digital porous media. To construct the framework, an intuitive approach is to connect FNO layers to a classifier using adaptive max pooling. First, we show that this approach is only effective for porous media with fixed sizes, whereas it fails for porous media of varying sizes. To overcome this limitation, we introduce our approach: instead of using adaptive max pooling, we use static max pooling with the size of channel width of FNO layers. Since the channel width of the FNO layers is independent of input image size, the introduced framework can handle multi-sized images during training. We show the effectiveness of the introduced framework and compare its performance with the intuitive approach through the example of the classification of three-dimensional digital porous media of varying sizes.

A novel Fourier neural operator framework for classification of multi-sized images: Application to three dimensional digital porous media

TL;DR

This paper addresses classifying images of varying sizes by leveraging Fourier neural operators (FNOs) that are inherently size-agnostic. It introduces a novel static max pooling strategy in the Fourier-channel width to connect FNOs to an MLP classifier, avoiding the pitfalls of adaptive pooling in 3D space. Applied to predicting 3D digital porous media permeability, the approach achieves high scores across multiple sizes, demonstrates excellent generalization to unseen sizes, and delivers substantial speedups over traditional lattice Boltzmann simulations. The work advances multi-size image classification with FNOs and suggests extensions to broader 2D/3D classification tasks and real-world porous-media data.

Abstract

Fourier neural operators (FNOs) are invariant with respect to the size of input images, and thus images with any size can be fed into FNO-based frameworks without any modification of network architectures, in contrast to traditional convolutional neural networks (CNNs). Leveraging the advantage of FNOs, we propose a novel deep-learning framework for classifying images with varying sizes. Particularly, we simultaneously train the proposed network on multi-sized images. As a practical application, we consider the problem of predicting the label (e.g., permeability) of three-dimensional digital porous media. To construct the framework, an intuitive approach is to connect FNO layers to a classifier using adaptive max pooling. First, we show that this approach is only effective for porous media with fixed sizes, whereas it fails for porous media of varying sizes. To overcome this limitation, we introduce our approach: instead of using adaptive max pooling, we use static max pooling with the size of channel width of FNO layers. Since the channel width of the FNO layers is independent of input image size, the introduced framework can handle multi-sized images during training. We show the effectiveness of the introduced framework and compare its performance with the intuitive approach through the example of the classification of three-dimensional digital porous media of varying sizes.
Paper Structure (19 sections, 15 equations, 11 figures, 1 table)

This paper contains 19 sections, 15 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction and motivation
  2. Fourier neural operators for image classification
  3. Our novel approach: Static max pooling in channel width of FNO layers
  4. Intuitive approach: Adaptive max pooling in 3D spatial space
  5. A brief review of theoretical aspects of Fourier neural operators
  6. Data generation
  7. Training
  8. Results and discussion
  9. General analysis
  10. Number of Fourier modes in each dimension
  11. Channel width of FNOs
  12. Number of FNO units
  13. Activation functions
  14. Static max pooling versus static average pooling
  15. Generalizability
  16. ...and 4 more sections

Figures (11)

  • Figure 1: Schematic of the proposed FNO-based framework for multi-size image classification
  • Figure 2: Schematic of the intuitive FNO-based framework for multi-size image classification
  • Figure 3: A few examples of synthetically generated three-dimensional digital porous media for training the proposed neural network; a an image of size $40^3$, b an image of size $48^3$, and c an image of size $56^3$. Blue represents grain space, while red indicates pore space.
  • Figure 4: $R^2$ plots for the test set (375 data) using the proposed approach for classification of multi-sized images
  • Figure 5: $R^2$ plots for the test set (375 data) using the proposed approach for the classification of multi-sized images. The results are individually shown for a images of size $40^3$ (125 data), b images of size $48^3$ (125 data), and c images of size $56^3$ (125 data)
  • ...and 6 more figures