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Gravitational acceleration fields can be deduced from the collisionless Boltzmann equation, once the distribution function is known. This can be constructed via the method of normalizing flows from data sets of the positions and velocities of stars. Here, we consider application of this technique to the solar neighbourhood. We construct mock data from a linear superposition of multiple 'quasi-isothermal' distribution functions, representing stellar populations in the equilibrium Milky Way disc. We show that given a mock data set comprising a million stars within 1 kpc of the Sun, the underlying acceleration field can be measured with excellent, sub-per cent level accuracy, even in the face of realistic errors and missing line-of-sight velocities. The effects of disequilibrium can lead to bias in the inferred acceleration field. This can be diagnosed by the presence of a phase space spiral, which can be extracted simply and cleanly from the learned distribution function. We carry out a comparison with two other popular methods of finding the local acceleration field (Jeans analysis and 1D distribution function fitting). We show our method most accurately measures accelerations from a given mock data set, particularly in the presence of disequilibria. [ABSTRACT FROM AUTHOR]

The advent of data sets of stars in the Milky Way with 6D phase-space information makes it possible to construct empirically the distribution function (DF). Here, we show that the accelerations can be uniquely determined from the DF using the collisionless Boltzmann equation, providing the Hessian determinant of the DF with respect to the velocities is non-vanishing. We illustrate this procedure and requirement with some analytic examples. Methods to extract the potential from data sets of discrete positions and velocities of stars are then discussed. Following Green & Ting, we advocate the use of normalizing flows on a sample of observed phase-space positions to obtain a differentiable approximation of the DF. To then derive gravitational accelerations, we outline a semi-analytic method involving direct solutions of the overconstrained linear equations provided by the collisionless Boltzmann equation. Testing our algorithm on mock data sets derived from isotropic and anisotropic Hernquist models, we obtain excellent accuracies even with added noise. Our method represents a new, flexible, and robust means of extracting the underlying gravitational accelerations from snapshots of 6D stellar kinematics of an equilibrium system. [ABSTRACT FROM AUTHOR]