# Fourier Neural Operator for Fluid Flow in Small-Shape 2D Simulated Porous Media Dataset
## 1 Introduction  
A wide range of phenomena/processes in science and engineering are described via measurable/estimable quantities that rely on independent variables. As an example, in subsurface fluid flow, pressure and temperature are typically measured based on the time and location variables. Given the available fundamental laws, it is feasible to determine the relationships among the rates of change of these physical quantities. The mathematical correlations typically used to do this are Ordinary and/or Partial Differential Equations (ODEs/PDEs). In ODEs, the derivatives of the dependent variable(s) are taken with respect to only one independent variable. On the other hand, partial derivatives are required in PDEs when there are two or more independent variables involved. Theories, methods, and tools available in scientific computing (also called computational science) make it possible to solve mathematical models of physical phenomena described in terms of ODEs and/or PDEs [1]. The theories and methods are together called numerical analysis/numerical mathematics, and tools refer to computer systems on which codes are run. The more complex the mathematical models, the more advanced the computational hardware requirements are to solve them. There are various numerical methods available that can provide approximate solutions to such problems. These include the finite difference method [2], Finite Element Method (FEM) [3], finite volume method [4],spectral method [5], and meshless method [6]. Such methods are usually time-consuming to apply. One way to mitigate the problem of the high computational cost in numerical calculations required to determine complex systems is to apply Machine Learning (ML) and/or Deep Learning (DL) techniques. Machine learning methods such as Neural Network (NN), adaptive-neuro-fuzzy-inference system, support-vector machine, and decision tree are widely employed to find and predict relevant patterns within datasets. Machine learning is now used to great advantage in various fields [7; 8; 9; 10]. Deep learning methods represent more complex extensions of existing ML methods, particularly neural networks, and have demonstrated improved performance, particularly when applied to large datasets [11; 12; 13; 14; 15]. There are various DL algorithms, including Convolutional Neural Network (CNN), deep auto-encoder, deep-belief network, recurrent neural network, and generative adversarial network. There are some differences between ML and DL. For instance, manual feature engineering tends to be performed with ML algorithms, sometimes requiring domain knowledge about a given problem. To make the point clear, consider 'filling missing values'. A dataset can include missing values due to the difficulty of collecting complete data. Missing values can be manually filled in based on expert knowledge, which is sometimes a tedious process. However, with DL algorithms this is more often not required, being performed automatically. Classical neural networks concentrate on learning mappings between finite-dimensional spaces. This makes such networks, when configured, confined to a particular discretization (i.e., they are mesh-dependent). Mesh-independent networks have been developed to reduce such constraints. In this regard, the Fourier Neural Operator (FNO) has recently been proposed to learn a continuous function via parameterizing the model in its function space [16]. This makes it possible for FNO to be trained on one mesh and subsequently evaluated on another. Unlike standard feed-forward networks that use activation functions (e.g., sigmoid, tanh, relu), training an FNO model using the Fourier transform to find optimum weights and biases is performed by employing sines and cosines as activation functions [17; 18; 19]. The Fourier neural operator has demonstrated its capabilities in solving parametric PDEs. Different models were developed to solve the Navier-Stokes equation for a viscous, incompressible fluid in vorticity form on the unit torus [16]. The viscosity was set to 0.001, 0.0001, and 0.00001.