We are conducting a proper-motion survey for young brown dwarfs in the Taurus-Auriga molecular cloud based on the Pan-STARRS1 3$pi$ Survey. Our search uses multi-band photometry and astrometry to select candidates, and is wider (370 deg$^{2}$) and deeper (down to $approx$3 M$_{rm Jup}$) than previous searches. We present here our search methods and spectroscopic follow-up of our high-priority candidates. Since extinction complicates spectral classification, we have developed a new approach using low-resolution ($R approx 100$) near-infrared spectra to quantify reddening-free spectral types, extinctions, and gravity classifications for mid-M to late-L ultracool dwarfs ($approx 100-3$ M$_{rm Jup}$ in Taurus). We have discovered 25 low-gravity (VL-G) and the first 11 intermediate-gravity (INT-G) substellar (M6-L1) members of Taurus, constituting the largest single increase of Taurus brown dwarfs to date. We have also discovered 1 new Pleiades member and 13 new members of the Perseus OB2 association, including a candidate very wide separation (58 kAU) binary. We homogeneously reclassify the spectral types and extinctions of all previously known Taurus brown dwarfs. Altogether our discoveries have thus far increased the substellar census in Taurus by $approx 40%$ and added three more L-type members ($approx 5-10$ M$_{rm Jup}$). Most notably, our discoveries reveal an older ($>$10 Myr) low-mass population in Taurus, in accord with recent studies of the higher-mass stellar members. The mass function appears to differ between the younger and older Taurus populations, possibly due to incompleteness of the older stellar members or different star formation processes.
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