  Interpolation of aliased seismic data constitutes a key step in a seismic
processing workflow to obtain high quality velocity models and seismic images.
Building on the idea of describing seismic wavefields as a superposition of
local plane waves, we propose to interpolate seismic data by utilizing a
physics informed neural network (PINN). In the proposed framework, two
feed-forward neural networks are jointly trained using the local plane wave
differential equation as well as the available data as two terms in the
objective function: a primary network assisted by positional encoding is tasked
with reconstructing the seismic data, whilst an auxiliary, smaller network
estimates the associated local slopes. Results on synthetic and field data
validate the effectiveness of the proposed method in handling aliased (coarsely
sampled) data and data with large gaps. Our method compares favorably against a
classic least-squares inversion approach regularized by the local plane-wave
equation as well as a PINN-based approach with a single network and
pre-computed local slopes. We find that introducing a second network to
estimate the local slopes whilst at the same time interpolating the aliased
data enhances the overall reconstruction capabilities and convergence behavior
of the primary network. Moreover, an additional positional encoding layer
embedded as the first layer of the wavefield network confers to the network the
ability to converge faster improving the accuracy of the data term.
