We present a systematic search for wide-separation (with Einstein radius theta(E)greater than or similar to 1.5 ''), galaxy-scale strong lenses in the 30 000 deg(2) of the Pan-STARRS 3 pi survey on the Northern sky. With long time delays of a few days to weeks, these types of systems are particularly well-suited for catching strongly lensed supernovae with spatially-resolved multiple images and offer new insights on early-phase supernova spectroscopy and cosmography. We produced a set of realistic simulations by painting lensed COSMOS sources on Pan-STARRS image cutouts of lens luminous red galaxies (LRGs) with redshift and velocity dispersion known from the sloan digital sky survey (SDSS). First, we computed the photometry of mock lenses in gri bands and applied a simple catalog-level neural network to identify a sample of 1 050 207 galaxies with similar colors and magnitudes as the mocks. Second, we trained a convolutional neural network (CNN) on Pan-STARRS gri image cutouts to classify this sample and obtain sets of 105 760 and 12 382 lens candidates with scores of p(CNN)>0.5 and > 0.9, respectively. Extensive tests showed that CNN performances rely heavily on the design of lens simulations and the choice of negative examples for training, but little on the network architecture. The CNN correctly classified 14 out of 16 test lenses, which are previously confirmed lens systems above the detection limit of Pan-STARRS. Finally, we visually inspected all galaxies with p(CNN)>0.9 to assemble a final set of 330 high-quality newly-discovered lens candidates while recovering 23 published systems. For a subset, SDSS spectroscopy on the lens central regions proves that our method correctly identifies lens LRGs at z similar to 0.1-0.7. Five spectra also show robust signatures of high-redshift background sources, and Pan-STARRS imaging confirms one of them as a quadruply-imaged red source at z(s)=1.185, which is likely a recently quenched galaxy strongly lensed by a foreground LRG at z(d)=0.3155. In the future, high-resolution imaging and spectroscopic follow-up will be required to validate Pan-STARRS lens candidates and derive strong lensing models. We also expect that the efficient and automated two-step classification method presented in this paper will be applicable to the similar to 4 mag deeper gri stacks from the Rubin Observatory Legacy Survey of Space and Time (LSST) with minor adjustments