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An analytical comparison of seismic inversion with several multivariate predictive techniques is made. Statistical data reduction techniques are examined that incorporate various machine learning algorithms, such as linear regression, alternating conditional expectation regression, random forest, and neural network. Seismic and well-log data are combined to estimate petrophysical or petroelastic properties, like bulk density. Currently, spatial distribution and estimation of reservoir properties is leveraged by inverting 3D seismic data calibrated to elastic properties (VP, VS, and bulk density) obtained from well-log data. Most commercial seismic inversions are based on linear convolution, i.e., one-dimensional models that involve a simplified plane-parallel medium. However, in cases that are geophysically more complex, such as fractured and/or fluid-rich layers, the conventional straightforward prediction relationship breaks down. This is because linear convolution operators no longer adequately describe seismic wavefield propagation due to nonlinear energy absorption. Such nonlinearity is also suggested by the seismic nonstationarity phenomenon, expressed by vertical and horizontal changes in the shape of the seismic wavelet (amplitude and frequency variations). The nonlinear predictive operator, extracted by machine learning algorithms, makes it possible in certain cases to estimate petrophysical reservoir properties more accurately and with less influence of interpretational bias.

The paper deals with a combined approach to approximation of velocity fields and minimization the objective functional in solving viscous fluids flow problems. The mathematical formulation of the problem is presented in the form of a generalized Lagrange functional. The flow function is designed using a feed forward artificial neural network with one hidden layer and with logistic activation function. The boundary values of the flow function are determined using the fluid flow rate. Thus, the problem of determination the velocity field is reduced to the problem of finding the network weights by minimizing the generalized Lagrange functional.

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