# Fourier Neural Operator for Fluid Flow in Small-Shape 2D Simulated Porous Media Dataset
## 1 Introduction
The resolution was \(64\times 64\) for both training and testing. Based on the results, the FNO-3D had the best performance in the case of available sufficient data. When the amount of data was not sufficient, the FNO-2D achieved the lowest error. An improved FNO-based DL model, U-FNO, was developed to solve a CO\({}_{2}\)-water multiphase flow problem over a wide range of rock-fluid properties, such as permeability, anisotropy, and flow rate [20]. The predictions generated for gas saturation and pressure buildup confirmed the superiority of the U-FNO model compared to FNO and CNN models applied to the same dataset. Additionally, U-FNO requires few training data to match the prediction accuracy of CNN. As a substitute for physics-based solvers, an FNO framework was employed to learn/map certain mechanical responses of 2D composites [21]. The FNO models, trained with few data, were able to predict high-resolution stress/strain tensor fields for geometrically complex microstructures. Additionally, the models exhibited zero-shot generalization with high precision on random geometries previously unseen by the trained FNO model. Moreover, the trained FNO models were able to predict high-resolution stress/strain fields when presented with low-resolution inputs. An FNO model has also been applied to a Large Eddy Simulation (LES) of a 3D turbulence system [22]. Filtered direct numerical simulation flow-field of isotropic turbulence recorded at different times comprised the training data. In the a posteriori test of LES, The FNO model outperformed dynamic Smagorinsky and dynamic mixed models to predict the velocity spectrum, probability density functions of vorticity and velocity increments, and the instantaneous flow structures. In addition to solving parametric PDEs, FNO has been successfully applied to solve other problems. For example, FNO was used to classify images contained in the CIFAR-10 image database comprised of 60,000 distinct samples [23]. The input samples were color images of ten different classes. A total of 83% of the total dataset was used to train the FNO, and the remaining images were used to test the trained model. According to the different evaluation criteria, the FNO performed slightly better than ResNet20. However, the FNO model was computationally more costly. As described, some research has been conducted applying FNO models to various existing datasets. However, much of that research addresses problems involving big-shape data (e.g., \(421\times 421\) and \(256\times 256\)). Thus, there is a critical lack of analysis regarding the performance of FNO models on small-shape data. Consequently, the major contribution of this paper is to apply and evaluate an FNO model to predict pressure distribution in small-shape data (\(30\times 30\)). Specifically, the study answers the following questions:  
1. Can FNO models perform accurately on small-shape data problems in terms of the prediction error metrics? 2. How do mode and width affect the performance of FNO models? 3. Does downsampling have a positive or negative effect on FNO model performance when applied to small-shape data? 4. Can FNO models satisfy the pattern applicable to porous media problems? 5. How does the performance of FNO models compare to that of CNN? A \(30\times 30\) uniform mesh problem from the domain of petroleum engineering is evaluated as a relevant topic with a suitable dataset to address the research questions identified. Continued constraints on the ability of the available energy supply to meet global energy demand make it important to improve our understanding of subsurface oil and gas reservoirs to improve production and resource recovery. The problem evaluated is designed to do that. Comparing the FNO model performance with that of a CNN model applied to the same dataset is justified because CNNs are proven in their applications to 2D arrays and the mathematical basis underpinning convolutional-filter functionality is well established.