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.
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