This paper proposes a class of physics-informed neural networks called noise robust physics-informed generative adversarial networks (NR-PIGANs) to solve stochastic differential equations in the presence of noisy measurements. In these scenarios, while the governing equations are known, only a limited number of sensor measurements of the system parameters are available, and some may contain significant measurement errors. To address this, NR-PIGAN incorporates an additional noise generator with specific distribution constraints into a physics-informed generative adversarial network framework. The noise generator is trained alongside the clean data generators in an end-to-end manner, enabling the model to effectively capture both clean and noisy data distributions under the given physical constraints. Numerical experiments demonstrate that NR-PIGAN excels in handling forward and inverse problems under diverse noise perturbations, and its advantage becomes more pronounced as the noise level increases.