  This text describes a method to simultaneously reconstruct flow states and
determine particle properties from Lagrangian particle tracking (LPT) data. LPT
is a popular measurement strategy for fluids in which particles in a flow are
illuminated, imaged (typically with multiple cameras), localized in 3D, and
then tracked across a series of frames. The resultant "tracks" are spatially
sparse, and a reconstruction algorithm is commonly employed to determine dense
Eulerian velocity and pressure fields that are consistent with the data as well
as the equations governing fluid dynamics. Existing LPT reconstruction
algorithms presume that the particles perfectly follow the flow, but this
assumption breaks down for inertial particles, which can exhibit lag or
ballistic motion and may impart significant momentum to the surrounding fluid.
We report an LPT reconstruction strategy that incorporates the transport
physics of both the carrier fluid and particle phases, which may be
parameterized to account for unknown particle properties like size and density.
Our method enables the reconstruction of unsteady flow states and determination
of particle properties from LPT data and the coupled governing equations for
both phases. We use a neural solver to represent flow states and
data-constrained polynomials to represent the tracks (though we note that our
technique is compatible with a variety of solvers). Numerical tests are
performed to demonstrate the reconstruction of forced isotropic turbulence and
a cone-cylinder shock structure from inertial tracks that exhibit significant
lag, streamline crossing, and preferential sampling.
