We present a wave-equation inversion method that inverts skeletonized data for the subsurface velocity model. The skeletonized representation of the seismic traces consists of the low-rank latent-space variables predicted by a well-trained autoencoder neural network. The input to the autoencoder is seismic traces, and the implicit function theorem is used to determine the Frchet derivative, i.e. the perturbation of the skeletonized data with respect to the velocity perturbation. The gradient is

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... computed by migrating the shiftedobserved traces weighted by the skeletonized data residual, and the final velocity model is the one that best predicts the observed latent-space parameters. We denote this hybrid inversion method as inversion by Newtonian machine learning (NML) because it invertsfor the model parameters by combining the forward and backward modeling of Newtonian wave propagation with the statistical dimension reduction of machine learning. Empirical results suggest that inversion by Newtonian machine learning can sometimes mitigate the cycle-skipping problem of conventional full-waveform inversion (FWI). Numerical tests with both synthetic and field data demonstrate the success of NML inversion method in recovering a low-wavenumber approximation to the subsurface velocity model. The advantage of this method over other skeletonized data methods is that no manual picking of important features is required because the skeletal data are automatically selected by the autoencoder. The disadvantage is that the inverted velocity model has less resolution compared to the FWI result, but it can be a good initial model for FWI. The most significant contribution of this paper is that it provides a general framework for using wave-equation inversion to invert skeletal data generated by any type of neural networks. In other words, we have combined the deterministic modeling of Newtonian physics and the pattern matching capabilities of machine learning to invert seismic data by Newtonian machine learning.