A comprehensive PINN method with hybrid Fourier feature for high-precision natural convection solution

Natural convection plays a vital role in various natural and industrial scenarios, where accurate modeling is essential. Physics-Informed Neural Networks (PINNs) have recently emerged as a promising mesh-free and data-free approach for solving differential equations governing such processes. However, existing models encounter performance bottleneck and inefficiency in simulating air natural convection at high Rayleigh (Ra) number reaching 107 that shows high-frequency features. This paper proposes a novel PINN with hybrid Fourier feature architecture to enable accurate and efficient simulations of natural convection across broad Ra ranges. Combining with a newly hybrid adaptive sampling strategy and an adaptive loss weight scheme, a comprehensively improved PINN method is formed. Evaluated using the benchmark problem of thermally driven cavity flow, the proposed model significantly outperforms both a conventional PINN model and three PINN variants incorporating different Fourier features, achieving average accuracies exceeding 99.95% for velocity and temperature fields at Ra = 107. Furthermore, tests across Ra from 103 to 107 confirm that the model delivers consistently accurate predictions, making it feasible for simulating natural convection over a broad parameter range. Ablation studies highlight the critical contribution of the hybrid Fourier feature in performance improvement. Finally, transfer learning is successfully applied to realize fast convergence at higher Ra values. Overall, this work presents a new PINN framework for high-fidelity natural convection simulations, which also holds great potential for other physical problems involving high-frequency information.