Abstract
<title>Abstract</title> <p>While Physics-Informed Neural Networks (PINNs) offer a mesh-free approach for solving the Navier-Stokes (N-S) equations, traditional models struggle with spectral bias, mass leakage, and high computational costs. To address these issues, this paper proposes a novel framework integrating Proper Orthogonal Decomposition (POD) with a Fourier-gradient PINN (FPINN). The POD algorithm reduces high-dimensional flow features to extract core energy modes, lowering computational complexity. Meanwhile, FPINN combines Fourier feature mapping to capture high-frequency multi-scale structures with selective spatial gradient penalties to ensure mass conservation. Validated on benchmark problems including Kovasznay flow, lid-driven cavity flow up to $Re = 5000$, and flow past a circular cylinder at $Re = 3900$, the results demonstrate that the POD-assisted FPINN minimizes prediction errors and achieves high accuracy in velocity and pressure field reconstructions, particularly in wake regions. Coupled with transfer learning across Reynolds numbers, this approach provides an efficient and physically consistent reduced-order framework for fluid dynamics.</p>