  Newtonian machine learning (NML) is a wave-equation inversion method that
inverts single-dimensional latent space (LS) features of the seismic data for
retrieving the subsurface background velocity model. The single-dimensional LS
features mainly contain the kinematic information of the seismic data, which
are automatically extracted from the seismic signal by using an autoencoder
network. Because its LS feature dimension is too small to preserve the dynamic
information, such as the waveform variations, of the seismic data. Therefore
the NML inversion is not able to recover the high-wavenumber velocity details.
To mitigate this problem, we propose to invert multi-dimensional LS features,
which can fully represent the entire characters of the seismic data. We denote
this method as multi-dimensional Newtonian machine learning (MNML). In MNML, we
define a new multi-variable connective function that works together with the
multi-variable implicit function theorem to connect the velocity perturbations
to the multi-dimensional LS feature perturbations. Numerical tests show that
(1) the multi-dimensional LS features can preserve more data information than
the single-dimensional LS features; (2) a higher resolution velocity model can
be recovered by inverting the multi-dimensional LS features, and the inversion
quality is comparable to that of FWI; (3) the MNML method requires a much
smaller storage space than conventional FWI because only the low-dimensional
representations of the high-dimensional seismic data are needed to be stored.
The disadvantage of MNML is that it can more easily get stuck in local minima
compared to the NML method. So we suggest a multiscale inversion approach that
inverts for higher dimensional LS features as the iteration count increase.
