Investigation of the Performance of Machine Learning Algorithms of Casson-Micropolar Nano Particles Around a Cylindrical Shape: ANN and PINN

Fluids that do not behave like water are everywhere in the modern world, from the thick, flowing chocolate in a factory to the complex movement of blood within our bodies. Unlike simple liquids that flow smoothly under any force, these special fluids, known as non-Newtonian fluids, can change their resistance depending on how hard they are pushed or pulled. Some act like solids until a specific force is applied, at which point they suddenly begin to flow. When these fluids also contain tiny particles that can spin or rotate on their own, the physics becomes even more intricate. Engineers and scientists need to understand exactly how these materials move and transfer heat, especially when they are forced through porous materials like sponges or filters, or when they are exposed to magnetic fields and intense heat. Getting these calculations right is vital for designing better industrial processes, improving thermal insulation, and creating more efficient energy systems. However, solving the complex mathematical equations that describe these behaviors is notoriously difficult, often requiring immense computing power and time.

A team of researchers from Quaid-I-Azam University in Pakistan and the Islamic University of Madinah in Saudi Arabia has tackled this challenge by testing a new way to solve these problems using artificial intelligence. They focused on a specific type of fluid that combines the properties of a Casson fluid, which behaves like a solid until pushed, with a micropolar fluid, where the tiny particles inside can rotate. They modeled how this fluid moves around a cylindrical shape, such as a pipe, while passing through a porous medium. Their model included several real-world factors that complicate the flow, including the heating caused by electrical currents, the effects of thermal radiation, chemical reactions, and the way heat is generated or absorbed within the fluid itself. Instead of relying solely on traditional, slow numerical methods to predict the fluid's speed, temperature, and concentration, the researchers trained two different types of machine learning algorithms to learn the physics of the system.

The first algorithm they used was an artificial neural network, a computer system designed to mimic the way the human brain processes information by finding patterns in data. The second, and more advanced, approach was a physics-informed neural network. This system is unique because it does not just look at data; it is built with the actual laws of physics written directly into its code. This means the computer is forced to obey the rules of fluid motion, heat transfer, and mass conservation as it learns, rather than just guessing based on examples. The researchers set up a rigorous test to see how well these intelligent systems could predict the behavior of the fluid compared to a standard, highly trusted numerical method known as BVP4c, which is a conventional tool for solving such equations. They ran their simulations over a range of iterations, starting with a small number of learning cycles and increasing the count significantly to see how the accuracy improved.

The results showed that the machine learning approaches were not only capable but highly effective. When the neural networks were given only a thousand learning cycles, their predictions were somewhat off, failing to capture the precise details of the fluid's movement and temperature distribution. However, as the researchers increased the number of iterations to twenty thousand, the picture changed dramatically. The predictions from both the standard artificial neural network and the physics-informed version began to overlap almost perfectly with the results from the traditional numerical method. The error between the machine's predictions and the established reference solutions dropped to extremely low levels, indicating that the models had learned the underlying physics with high fidelity. The study confirmed that the physics-informed network, in particular, could generate stable and authentic solutions that matched the complex reality of the fluid flow without the need for repetitive, time-consuming calculations.

The researchers also examined how specific physical factors influenced the fluid. They found that as the strength of the magnetic field increased, the fluid's velocity slowed down, a result of the magnetic force pushing against the motion of the fluid particles. Similarly, they observed that increasing thermal radiation caused the temperature of the fluid to rise, while stronger chemical reactions caused the concentration of the fluid's components to decrease. Throughout these tests, the machine learning models remained robust, accurately tracking these changes and maintaining stability even as the conditions became more complex. The study concluded that these intelligent algorithms offer a reliable and efficient alternative to conventional numerical techniques for solving the nonlinear equations that govern such complex fluid systems. By proving that these networks can converge to accurate solutions with minimal error, the work suggests a powerful new tool for engineers and scientists who need to simulate and understand the behavior of advanced fluids in industrial and biological applications.