Physics informed neural network with Fourier feature for natural convection problems

Abstract

In this paper, we investigate the application of deep neural networks to solve the Navier-Stokes and heat equations within the framework of modeling the natural convection phenomenon. The main objective is to reconstruct the velocity and temperature fields in a differentially heated rectangular cavity while adhering to the imposed boundary conditions. Two main architectures are compared: the Fully Connected Neural Network and the Fourier Features Neural Network. A hyper-parameter tuning process was carried out to optimize the network performances. This tuning led to a final architecture composed of 6 layers, each with 128 neurons, and 64 Fourier frequencies, with the Mish activation function selected after testing several alternatives. Both architectures were trained on four cases, where the Rayleigh number ranges from 10⁴ to 10⁷, with quasi-randomly sampled points. The network predictions were then compared to the results obtained from numerical simulations of the Navier-Stokes and heat equations. The results show that for low Rayleigh numbers (10⁴ and 10⁵), both architectures converge quickly, producing smooth profiles dominated by low frequencies. However, for higher Rayleigh numbers (10⁶), the Fourier Features Neural Network outperforms the Fully Connected Neural Network by better capturing the complex and localized variations, thanks to its explicit integration of periodic components, which makes it particularly well-suited for multi-scale problems.

This study highlights the potential of deep neural networks to solve partial differential equations in complex configurations, offering a promising alternative to traditional methods. It also emphasizes the importance of choosing an architecture that fits the specific characteristics of the problem at hand, especially in cases where the solutions exhibit multi-scale variations or high frequencies.