PINN-based functional inference in size-structured population models: Application to fish oogenesis

Early oogenesis in juvenile fish establishes the ovarian reserve, which conditions lifelong reproductive capacity. This process is regulated by local hormonal feedback mechanisms, mainly involving Anti-Müllerian Hormone (AMH). In this work, we develop a mechanistic size-structured population model describing an unstructured germ-cell compartment and a population of growing oocytes, with AMH feedback on the renewal of the precursor cells compartment. Using repeated cross-sectional observations of oocyte size distributions in fish ovaries, we formulate and solve a nonlinear inverse problem with Physics-Informed Neural Networks (PINNs) to infer three demographic functions: the size-dependent oocyte growth rate, the AMH-regulated net renewal rate of the precursor cells compartment, and the recruitment rate of new oocytes. We introduce an auxiliary PINN formulation to handle the nonlocal interaction term and jointly infer unobserved state variables and demographic functions. Our inference strategy combines non-parametric PINNs with a data-guided model-selection step: demographic functions are first inferred without prescribing their functional forms, and the learned profiles are then used to select parsimonious parametric laws, whose parameters are subsequently estimated within a parametric PINN. We investigate identifiability at both the functional and parametric levels, combining structural identifiability analysis with synthetic-data experiments assessing the practical recovery of functions and parameters. The inference procedure is embedded within an individual-level bootstrap framework to assess sensitivity to repeated-cross-sectional sampling and stochastic training. Applied to juvenile fish oogenesis, the framework recovers interpretable growth and renewal dynamics, provides access to unobserved quantities, and enables conditional in silico investigation. More broadly, this work provides a framework for functional and parametric inference in size-structured population models with nonlocal interactions from sparse cross-sectional data.