  We study the possibility of probing new physics accounting for $(g-2)_\mu$
anomaly and gravitational waves with pulsar timing array measurements. The
model we consider is either a light gauge boson or neutral scalar interacting
with muons. We show that the parameter spaces of dark $U(1)$ model with kinetic
mixing explaining $(g-2)_\mu$ anomaly can realize a first-order phase
transition, and the yield-produced gravitational wave may address the common
red noise observed in the NANOGrav 12.5-yr dataset.
