Full waveform inversion (FWI) is pivotal for exploring subsurface structures and physical parameters. However, classical FWI methods often experience challenges like cycle skipping and non-linearity, necessitating accurate initial velocity models. Pure data-driven approaches based on deep learning are constrained by limited real labeled data and inadequate generalization, hindering practical applications. To address these issues, we propose an unsupervised physics-informed machine learning FWI with low-rank implicit neural representation (termed LR-IFWI), which utilizes a low-rank matrix factorization parameterized by coordinate-based neural networks, continuously and compactly representing the velocity model by implicitly encoding low-rank properties and smoothness. Our method has three crucial advantages: (i) LR-IFWI considerably improves inversion accuracy and shows steadier inversion convergence behavior; (ii) LR-IFWI can reduce the number of iterations for comparable inversion accuracy, improving efficiency attributed to the compact low-rank representation; (iii) LR-IFWI has better robustness, alleviating the dependence on the initial models and improving noise resistance due to the physical constraints and low-rank neural representation. Numerical tests on the two-dimensional Marmousi model demonstrate that LR-IFWI achieves efficient and accurate inversion with fewer iterations and greater precision when utilizing smooth, linear, and random initial velocity models. Further experiments with noisy seismic data, missing low-frequency components, and more challenging velocity models, such as 2D SEG/EAGE Salt and Overthrust models, highlight its robustness and generalization.