  While numerical simulations have been playing a key role in the studies of
planet-disk interaction, testing numerical results against observations has
been limited so far. With the two directly imaged protoplanets embedded in its
circumstellar disk, PDS 70 offers an ideal testbed for planet-disk interaction
studies. Using two-dimensional hydrodynamic simulations we show that the
observed features can be well explained with the two planets in formation,
providing strong evidence that previously proposed theories of planet-disk
interaction are in action, including resonant migration, particle trapping,
size segregation, and filtration. Our simulations suggest that the two planets
are likely in 2:1 mean motion resonance and can remain dynamically stable over
million-year timescales. The growth of the planets at $10^{-8}-10^{-7}~M_{\rm
Jup}~{\rm yr}^{-1}$, rates comparable to the estimates from H$\alpha$
observations, does not destabilize the resonant configuration. Large grains are
filtered at the gap edge and only small, (sub-)$\mu$m grains can flow to the
circumplanetary disks and the inner circumstellar disk. With the sub-millimeter
continuum ring observed outward of the two directly imaged planets, PDS 70
provides the first observational evidence of particle filtration by gap-opening
planets. The observed sub-millimeter continuum emission at the vicinity of the
planets can be reproduced when (sub-)$\mu$m grains survive over multiple
circumplanetary disk gas viscous timescales and accumulate therein. One such
possibility is if (sub-)$\mu$m grains grow in size and remain trapped in
pressure bumps, similar to what we find happening in circumstellar disks. We
discuss potential implications to planet formation in the solar system and
mature extrasolar planetary systems.
