PINGS-X: Physics-Informed Normalized Gaussian Splatting with Axes Alignment for Efficient Super-Resolution of 4D Flow MRI

4D flow magnetic resonance imaging (MRI) is a reliable, non-invasive approach for estimating blood flow velocities, vital for cardiovascular diagnostics. Unlike conventional MRI focused on anatomical structures, 4D flow MRI requires high spatiotemporal resolution for early detection of critical conditions such as stenosis or aneurysms. However, achieving such resolution typically results in prolonged scan times, creating a trade-off between acquisition speed and prediction accuracy. Recent studies have leveraged physics-informed neural networks (PINNs) for super-resolution of MRI data, but their practical applicability is limited as the prohibitively slow training process must be performed for each patient. To overcome this limitation, we propose PINGS-X, a novel framework modeling high-resolution flow velocities using axes-aligned spatiotemporal Gaussian representations. Inspired by the effectiveness of 3D Gaussian splatting (3DGS) in novel view synthesis, PINGS-X extends this concept through several non-trivial novel innovations: (i) normalized Gaussian splatting with a formal convergence guarantee, (ii) axes-aligned Gaussians that simplify training for high-dimensional data while preserving accuracy and the convergence guarantee, and (iii) a Gaussian merging procedure to prevent degenerate solutions and boost computational efficiency. Experimental results on computational fluid dynamics (CFD) and real 4D flow MRI datasets demonstrate that PINGS-X substantially reduces training time while achieving superior super-resolution accuracy. Our code and datasets are available at https://github.com/SpatialAILab/PINGS-X.