A Novel Approach for Data-Free, Physics-Informed Neural Networks in Fluid Mechanics Using the Principle of Minimum Pressure Gradient

What is it about?

Fluid dynamics is all about understanding how fluids like water or air move. Imagine you're stirring a cup of coffee - you can see the whirls and swirls as the liquid moves around. Scientists use complicated math equations called the Navier-Stokes equations to figure out how these motions work. But sometimes, these equations are too tricky and expensive to solve, especially when the flows get really complex like in turbulent chaotic flows. Now, imagine you have a new tool - let's call it Physics-Informed Neural Networks (PINNs). These are like super-smart algorithms inspired by how our brains work. They can help us understand physics better without needing to rigorously solve those difficult equations directly. In this study, we used PINNs to tackle fluid flow problems in a new approach. Instead of diving into the nitty-gritty of the Navier-Stokes equations, we focused on a "magic quantity" - pressure gradient, which the fluids tend to always try to minimize naturally. This principle is called Principle of Minimum Pressure Gradient or abbreviated PMPG. By using this idea and some math tricks, we were able to sidestep the tricky parts of the Navier-Stokes equations. We tested this method by simulating the flow of water around a cylinder, and it worked really well! Our model matched up nicely with what we already knew about how fluids behave.

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Photo by Uriel SC on Unsplash

Photo by Uriel SC on Unsplash

Why is it important?

In complex fluid flow scenarios, calculating the governing equations, such as the Navier-Stokes equations, can be incredibly challenging and computationally expensive. These equations describe how fluids move and change over time, taking into account factors like viscosity, pressure, turbulence, gravity, rotation of the earth or any external exerted forces. However, solving them accurately requires dividing the fluid domain into small elements and solving equations for each of these elements, which quickly becomes computationally expensive especially as the complexity of the flow or its velocity increases. For example, when dealing with turbulent flows or scenarios with irregular geometries, the number of equations and variables to solve can grow exponentially. This means that traditional numerical methods for solving the Navier-Stokes equations may require vast amounts of computational resources and time to obtain meaningful results. Furthermore, the complexity of the equations often leads to numerical instabilities and inaccuracies, especially near boundaries or regions with strong gradients in fluid properties. These challenges can hinder the ability to accurately predict real-world fluid behaviors and limit the applicability of traditional computational fluid dynamics approaches. By introducing innovative methods like Physics-Informed Neural Networks (PINNs), we aim to address these challenges by bypassing the need for explicit solutions to the Navier-Stokes equations. Instead, these methods leverage principles from physics and machine learning to learn the underlying patterns and behaviors of fluid flows directly from data. This approach not only reduces the computational burden associated with solving complex fluid dynamics problems but also opens up new avenues for understanding and predicting fluid behavior in scenarios where traditional methods struggle to provide accurate results.

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