Inverse analysis of the continuity equation for accretion-driven black hole mass evolution

Supermassive black holes (SMBHs) are ubiquitous in massive galaxies, and their cosmic growth is commonly modeled with continuity equations describing the evolution of the black hole mass function (BHMF). We investigate the inverse problem associated with an accretion-driven cosmological continuity equation, focusing on the stability and identifiability of reconstructing earlier mass distributions from present-day observational information. We formulate and implement a numerical inversion framework to estimate the ancient SMBH population using synthetic present-day BHMFs contaminated with different levels of observational noise. Our analysis considers an idealized scenario in which black hole growth occurs exclusively through smooth mass accretion, without mergers or stochastic variability. The inverse framework consistently reproduces the present-day BHMF with good accuracy across all tested noise levels, while the reconstruction of the initial condition is significantly more sensitive to observational perturbations. Although the recovered initial distributions generally preserve the global structure of the reference solution, the associated errors exhibit substantial dispersion and non-monotonic behavior driven by nonlinearities, moderate parameter degeneracies, and local minima in the optimization landscape. Monte Carlo analyses further indicate that the inverse reconstruction remains numerically stable under different stochastic noise realizations, despite occasional outlier solutions. These results highlight the intrinsic ill-posedness of inverse mass-transport problems subject to single-redshift constraints. Although restricted to an accretion-only framework, the proposed methodology provides a controlled computational basis for studying inverse cosmological mass transport and establishes a benchmark case for future extensions that incorporate merger-driven growth and multi-redshift observational constraints.