# Data-driven PINN inference of oocyte dynamics in fish ovaries

Louis Fostier, Manon Lesage, Violette Thermes, Frédérique Clément, Romain Yvinec

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Louis Fostier, Manon Lesage, Violette Thermes, Frédérique Clément, Romain Yvinec. Data-driven PINN inference of oocyte dynamics in fish ovaries. 2026. hal-05536280

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# Data-driven PINN inference of oocyte dynamics in fish ovaries

Louis Fostier¹,²,*, ¤a, Manon Lesage ³,¤b, Violette Thermes³, Frédérique Clément ², Romain Yvinec¹,²

1 PRC, INRAE, CNRS, Université de Tours, 37380, Nouzilly, France

2 Université Paris-Saclay, Inria, Centre Inria de Saclay, 91120, Palaiseau, France

3 INRAE, Fish Physiology and Genomics Institute, UR1037, LPGP, Rennes, France

¤a Current Address: Sorbonne Université, Inria Paris - MUSCLEES project team, Université de Paris, CNRS, Laboratoire Jacques-Louis Lions UMR7598, Paris, France

¤b Institut Curie, Université PSL, CNRS UMR3215, INSERM U934, Paris, France

* louis.fostier@inria.fr

## Abstract

Early oogenesis in juvenile fish establishes the ovarian reserve and thus 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 the dynamics of precursor germ cells and growing oocytes in ovary, incorporating an AMH feedback exerted on the precursor germ cells renewal. 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 the size-dependent oocyte growth rate, the AMH-regulated renewal rate of precursor cells, and the recruitment rate of new oocytes. The proposed framework enables flexible, data-driven identification of biological rates under minimal prior assumptions. Once calibrated, the model provides in silico access to key unobservable quantities, including cell transit times, the impact of AMH perturbations such as invalidation conditions, and the mechanisms underlying inter-individual variability in the establishment of the pool of small oocytes. This work presents a novel application of PINNs to inverse problems for size-structured partial differential equation models with nonlocal interaction terms, and establishes a quantitative framework for studying early oogenesis in juvenile fish.

## Author summary

The earliest stages of female germ cell development play a key role in shaping the future reproductive capacity of fish. In juvenile ovaries, precursor cells proliferate and differentiate into growing oocytes, a process regulated by Anti-Müllerian Hormone (AMH), whose precise role remains difficult to quantify experimentally. Using oocyte size distributions in fish ovaries of different ages together with a size-structured population model, we apply a Physics-Informed Neural Networks framework to infer oocyte recruitment and growth rates, as well as the intensity of AMH feedback exerted by the oocyte population on precursor cell dynamics. Furthermore, by providing a quantitative and predictive description of early oocyte development, our work helps explain the differences observed between individual fish in the establishment of the pool of small oocytes, and provides new tools to investigate how hormonal disturbances can affect fish reproduction.
# Introduction

## Context, modeling and biological contributions

The biological motivation for this study lies in developmental and reproductive biology. We focus on oogenesis, the process by which germ cells *grow* and mature [1], with particular attention to juvenile fish and the establishment of the pool of small germ cells. In fish ovaries, precursor germ cells first *proliferate* and then *differentiate* into oocytes, a process referred to as *recruitment*. Once recruited, oocytes stop proliferating and continue their development within somatic structures called ovarian follicles. The somatic cells surrounding the oocytes play a key regulatory role by producing hormones that influence oogenesis. In juvenile fish, the main hormone controlling the process is Anti-Müllerian Hormone (AMH). AMH is synthesized locally in the ovary by the somatic cells within follicles, and its production level depends on the maturity stage of the follicles [2,3], which is closely related to oocyte size. Recent studies in medaka and zebrafish suggest that AMH control the precursor germ cells dynamics [4–6].

Understanding these early stages of oogenesis is essential for several reasons. First, the establishment of the pool of small oocytes during the juvenile phase can have long-lasting consequences on the reproductive capacity of fish throughout adulthood, analogously to the formation of the ovarian reserve in mammals [7]. In [8], we have highlighted that the dynamics of small germ cells may impact the fish spawning cycle dynamics and therefore the reproductive capacity.

Moreover, due to their aquatic environment and physiology, fish are highly exposed to chemicals and environmental hormonal disturbances. Therefore, quantifying the hormonal regulations is crucial and allows to study how such disturbances affect oogenesis and the reproductive function. A major objective of this work is to quantify the regulatory influence of Anti-Müllerian Hormone (AMH) on the establishment and renewal of the ovarian oocyte pool.

Developing mathematical models capable of capturing these processes is therefore crucial both to improve our understanding of ovarian biology and to provide quantitative tools for interpreting experimental data. As shown in [8,9], the formalism of partial differential equation model of size-structured population is well suited for representing the oogenesis process and the hormonal controls exerted onto oogenesis, which emanate directly from the oocyte population. In this work, we introduce a semilinear size-structured population model of oocytes in juvenile fish ovaries, with a nonlocal interaction term in the renewal rate of precursor cells to take into account the AMH feedback exerted by the oocyte population.

From observed oocyte size distributions in ovaries from fish of different ages, we develop a Physics-Informed Neural Networks (PINNs) methodology to infer the size-dependent growth rate of oocytes, the AMH-dependent renewal rate of precursor cells and the recruitment rate of new oocytes from the pool of precursor cells. Once the relevant biological rates have been recovered, the model makes it possible to explore in silico aspects of oocyte dynamics that cannot be measured directly without targeted and often invasive biological experiments. This includes simulation of AMH invalidation conditions. We also investigate the origin of the pronounced inter-individual variability observed between oocyte populations in different juvenile fish ovaries. Furthermore, using counts of precursor germ cells from [10], we also estimate the size of the pool of precursor cell with respect to juvenile fish age as well as a differentiation rate into oocyte. Finally, by extending simulations over longer time scales, the framework can provide first insights of the oocyte dynamics during the fish reproductive life.
Inverse problem and methodological contribution

Inverse problems for structured population models aim to identify vital rates (growth, death, recruitment, birth, etc.) from observations such as age or size distributions at multiple times. Several approaches have been proposed for this task.

Finite-difference-based methods combined with least-squares estimation have been used for age-structured models [11–14] and for size-structured models [15, 16]. Another line of work relies on spline approximations of the PDE solution and of the unknown vital rates, as in [17], which is conceptually close to the methodology developed here. Our approach follows this last family of methods, replacing spline approximations with neural networks. There are several reasons for choosing neural networks over splines. Multilayer Perceptrons (MLPs) have shown better performance for data smoothing and numerical differentiation in the context of noisy biological data [18]. Unlike splines, neural networks do not require explicit node placement or preprocessing to handle sparse data. Their architecture is highly modular, and they support automatic differentiation, which facilitates the computation of derivatives and the inclusion in PINNs framework. Finally, their integration into modern deep learning frameworks, such as PyTorch, enables efficient training and GPU acceleration.

Physics-Informed Neural Networks (PINNs) were introduced in their modern form by Raissi et al. [19] to solve both forward and inverse problems involving differential equations, building on earlier pioneering work on neural networks and differential equations [20]. The core idea is to approximate the solution of the differential equations using a neural network, trained to minimize a loss function that penalizes both the discrepancy with data and the residual of the governing equations. This approach offers flexibility in balancing model fidelity and data accuracy, increasing also robustness when the inverse problem might be ill-posed. PINNs are now widely applied to inverse problems in various scientific domains, including biology and medicine. Notable applications include inferring dynamics in biological systems modeled by ODEs [21], by reaction-diffusion PDEs [22], by aggregation-diffusion PDEs [23], or estimating parameters in tumor growth models [24, 25], and epidemiological models [26, 27]. A key advantage of PINNs is their ability to integrate prior physical or biological knowledge of different nature, by penalizing the residual of governing equation, but also enforce other properties (positivity, monotonicity) by encoding them directly in network architectures, or adding soft penalty constraints by adding terms in the loss function.

A key challenge in inverse problems for structured population models lies in identifiability, especially when the available data are challenging (sparse in time and cross-sectional in our application case). In this work, to investigate the practical identifiability of the proposed model and the inference framework, we first generate synthetic dataset from the model using known growth and recruitment functions, and then apply the PINN-based methodology to recover these functions. This in silico validation step allows us to evaluate the robustness of the inference and to assess which components of the model can be reliably identified from the type of data considered.

To the best of our knowledge, this work constitutes the first application of Physics-Informed Neural Networks to inverse problems in size-structured population dynamics with nonlocal interaction terms. We demonstrate the potential of this approach for data-driven analysis of structured population models by inferring key biological rates from observed cell size distributions at multiple time points. These rates may depend on cell size and/or on nonlocal quantities, such as weighted total population sizes. Rather than prescribing explicit parametric forms, we first approximate these functions using neural networks, which allows for flexible, data-driven inference under minimal prior knowledge. The resulting estimates are then used to guide the selection of compact and biologically interpretable parametric forms, which are subsequently calibrated using a parametric PINN formulation.
Alternative interesting approaches have recently been proposed to infer vital rates in an interpretable manner, notably through extensions of the Weak SINDy framework [28], which rely on selecting parsimonious linear combinations from predefined libraries of biologically plausible functions.

## Outline

We first introduce the size-structured population model of early oogenesis and describe the associated inverse problem. We then present the Physics-Informed Neural Networks framework developed. The practical identifiability of the inverse problem is assessed using synthetic dataset. Finally, we apply the method to experimental oocyte size distribution data in juvenile fish ovaries and discuss the biological insights and predictive capabilities of the calibrated model.

# Materials and methods

## Experimental data and protocol

Experimental data consist of oocyte size distributions measured in the ovaries of juvenile medaka fish (*Oryzias latipes*) at four different ages, following the experimental protocol described in [29, 30]. The data originate from the doctoral work reported in [31]. Briefly, ovaries are dissected, transparized, imaged, and oocyte sizes are quantified by 3D segmentation of ovary images to reconstruct oocyte size distributions. The dataset is composed of repeated cross-sectional (RCS) observations, as each individual is sacrificed at the time of measurement. Consequently, longitudinal trajectories at the individual level are not available. The corresponding datasets, with experimental protocol details, are publicly available in a national research data repository (see Data Availability Statement).

## Model of oocyte population dynamics

<figure><img src="image_2.png" /><figcaption>**Fig 1.** Schematic representation of the early oogenesis process. The model focuses on precursor germ cells and small oocytes. Anti-Müllerian Hormone (AMH) is the only hormonal regulator considered in the model. Oocytes are represented in green, with color intensity reflecting their level of AMH secretion (darker green indicates higher AMH production). AMH exerts a feedback on the renewal of precursor cells, thereby regulating the supply of newly recruited oocytes.</figcaption></figure>

We now introduce the model. We note $\rho(t, x)$ the density of oocytes at time $t$ (age of the fish expressed in day) with respect to size $x$ (diameter of the cell expressed in $\mu$m),
and $S(t)$ the number of precursor cells at time $t$. We note $P_{AMH}(t)$ the quantity of AMH in the fish ovary at time $t$ (expressed in arbitrary unit). The main modeling assumptions are the following:

* Oocytes grow at a size-dependent growth rate $f(x) > 0$ (expressed in $\mu m/day$), from the minimal size $x_r$ to the maximal size $x_m$.

* The quantity of AMH depends dynamically on the oocyte population state, and is given by $P_{AMH}(t) = \int_{x_r}^{x_m} \omega_{AMH}(x)\rho(t, x)dx$, where $\omega_{AMH}$ represents the size-dependent oocyte AMH expression level.

* Precursor cells self-renew at a AMH-dependent rate $\beta(P_{AMH})$ and can differentiate into oocytes at a constant rate $\lambda_0$.

* Oocytes are recruited from the pool of precursor cells at the constant size $x_r$, and at the time-dependent rate $r(t) := \lambda_0 S(t)$.

The oocyte population dynamics are then described by the following semilinear size-structured population model:

$$ 
\left\{
\begin{array}{l}
\partial_t \rho(t, x) + \partial_x (f(x)\rho(t, x)) = 0, \quad \forall t > t_0, x \in (x_r, x_m) \\
f(x_r)\rho(t, x_r) = \lambda_0 S(t), \quad \forall t > t_0 \\
S'(t) = \beta(P_{AMH}(t))S(t) - \lambda_0 S(t), \quad \forall t > t_0 \\
P_{AMH}(t) = \int_{x_r}^{x_m} \omega_{AMH}(x)\rho(t, x)dx, \quad \forall t > t_0 \\
\rho(t_0, x) = \rho_0(x), \quad \forall x \in (x_r, x_m) \\
S(t_0) = S_0.
\end{array}
\right.
\quad (1) $$

With $r(t) := \lambda_0 S(t)$, and by denoting $\alpha(P_{AMH}) := \beta(P_{AMH}) - \lambda_0$ the net precursor cell renewal rate, we can rewrite (1) as:

$$ 
\left\{
\begin{array}{l}
\partial_t \rho(t, x) + \partial_x (f(x)\rho(t, x)) = 0, \quad \forall t > t_0, x \in (x_r, x_m) \\
f(x_r)\rho(t, x_r) = r(t), \quad \forall t > t_0 \\
r'(t) = \alpha(P_{AMH}(t))r(t), \quad \forall t > t_0 \\
P_{AMH}(t) = \int_{x_r}^{x_m} \omega_{AMH}(x)\rho(t, x)dx, \quad \forall t > t_0 \\
\rho(t_0, x) = \rho_0(x), \quad \forall x \in (x_r, x_m) \\
r(t_0) = \lambda_0 S_0.
\end{array}
\right.
\quad (2) $$

## Inverse problem and identifiability

### Inverse problem

From the observed oocyte size distributions at several time points in the observation windows $[t_0, t_m]$, we aim to recover the following key biological quantities of model (2):

* the size-dependent oocyte growth rate $f \in C^0([x_r, x_m], \mathbb{R}^+)$,

* the time-dependent oocyte recruitment rate $r(t) \in C^1([t_0, t_m], \mathbb{R}^+)$,

* the AMH-dependent net precursor cell renewal rate $\alpha(P_{AMH}) \in C^0(\mathbb{R}^+, \mathbb{R})$.

We fix a priori the weight function $\omega_{AMH}$, roughly based on the literature on AMH expression in the fish ovary (see [2]). Note that $\omega_{AMH}$ only represents relative expression along oocyte sizes, as any absolute scaling is absorbed into the free function $\alpha$. We also fix the minimal oocyte size $x_r$ and the maximal observed oocyte size $x_m$, based on the experimental data and prior knowledge on germ cell development in medaka, see [32] (our model only considers the previtellogenic maturation stage). The numerical values of all fixed parameters are given in the Appendix S5 - Implementation details, parameters and hyperparameters.
## Identifiability

Before addressing inference from experimental data, we assess the identifiability properties of the proposed size-structured population model. We first investigate theoretically the structural identifiability of the model assuming idealized data conditions. More precisely, we assume perfect knowledge of the oocyte size distribution $\rho(t, x)$ on the domain $[x_r, x_m]$ over a large time window $[t_0, t_m]$. However, in experimental settings, the data consist of repeated cross-sectional measurements, with one observation per individual at a given age, substantial inter-individual variability, and sparse temporal sampling. Therefore, to assess the practical identifiability of the model and to validate our inference methodology, we generated synthetic dataset that closely mimic the experimental observation, by assuming a Gompertz model for the growth rate (see (4)), and a decreasing sigmoid for the net renewal rate of precursor cells (see (5)). In particular, this dataset incorporate realistic inter-individual variability in growth rates, renewal rates, and initial oocyte size distributions (see Appendix S2 - Synthetic dataset generation). We then apply our Parametric-PINN framework to these synthetic data in order to evaluate its ability to accurately recover the underlying biological functions and parameters.

## PINN for size-structured oocyte population model

In the context of size-structured oocyte population model, each state variable is approximated by a dedicated neural network, while the key vital rates, namely the AMH-dependent precursor cell net renewal rate $\alpha$ and the size-dependent oocyte growth rate $f$ in our model, can be represented in two distinct ways. In a parametric approach, these functions are described using explicit functional forms involving a small number of parameters, which are estimated from the data. This representation provides structural constraints that can improve identifiability and facilitates biological interpretation, constituting a step toward a more mechanistic model. Alternatively, in a non-parametric formulation, no specific functional form is prescribed and the regulatory functions are directly approximated by neural networks, allowing for flexible inference under minimal prior assumptions. In both cases, the training process consists in jointly optimizing all the parameters of the model and the neural networks by minimizing the total loss.

In this work, we start with the non-parametric PINNs (NP-PINNs) approach to flexibly infer the biological functions $\alpha$ and $f$ in a data-driven manner. The reconstructed functions then guide the selection of compact and biologically interpretable parametric forms, which are subsequently calibrated within a parametric PINNs (P-PINNs) formulation.

## Model reformulation

To avoid explicit discretization of the nonlocal interaction term describing AMH quantity in ovary, we reformulate the model (2) using an auxiliary variable $n$ defined as $n(t, x) = \int_{x_r}^{x} \omega_{AMH}(y)\rho(t, y)dy$ and such that $P_{AMH}(t) = n(t, x_r)$, following an approach similar to that proposed in [33] for integro-differential equations:

$$ \left\{ \begin{array}{l} \partial_t \rho(t, x) + \partial_x (f(x)\rho(t, x)) = 0, \forall t > t_0, x \in (x_r, x_m), \\ \partial_x n(t, x) = \omega_{AMH}(x)\rho(t, x), \forall t > t_0, x \in (x_r, x_m), \\ r'(t) = \alpha(n(t, x_r))r(t), \forall t > t_0, \\ f(x_r)\rho(t, x_r) = r(t), \forall t > t_0, \\ n(t, x_r) = 0, \forall t > t_0, \end{array} \right. \qquad (3) $$
The training process consists in minimizing a composite loss function with respect to two sets of parameters: (i) $\Theta$ for the neural networks representing the state variables, namely $\rho$, $n$, $r$, and (ii) $\gamma$ for the biological functions to be inferred, namely $\alpha$ and $f$, which can be either explicitly parametrized or approximated by neural networks. The total loss reads:

$$ \mathcal{L}(\Theta, \gamma) = w_{pde1} \mathcal{L}_{\text{PDE}}(\Theta, \gamma) + w_{pde2} \mathcal{L}_{\text{PDE}}^{\text{aux}}(\Theta, \gamma) + w_{ode} \mathcal{L}_{\text{ODE}}(\Theta, \gamma) + w_{bc} \mathcal{L}_{\text{BC}}(\Theta, \gamma) + w_{d1} \mathcal{L}_{\text{DATAmse}}(\Theta) + w_{d2} \mathcal{L}_{\text{DATAlog}}(\Theta). $$

The PDE residual terms enforces the dynamics of the state variables:

$$ \mathcal{L}_{\text{PDE}} = \frac{1}{N_{\text{PDE}}} \sum_{i=1}^{N_{\text{PDE}}} (\partial_t \rho_{\Theta}(t_i, x_i) + \partial_x (f_{\gamma}(x_i) \rho_{\Theta}(t_i, x_i)))^2, $$

$$ \mathcal{L}_{\text{PDE}}^{\text{aux}} = \frac{1}{N_{\text{PDE}}} \sum_{i=1}^{N_{\text{PDE}}} (\partial_x n_{\Theta}(t_i, x_i) - \omega_{\text{AMH}}(x_i) \rho_{\Theta}(t_i, x_i))^2. $$

The ODE residual term enforces the recruitment dynamics:

$$ \mathcal{L}_{\text{ODE}} = \frac{1}{N_{\text{ODE}}} \sum_{i=1}^{N_{\text{ODE}}} (r'_{\Theta}(t_i) - \alpha_{\gamma}(n_{\Theta}(t_i, x_m)) r_{\Theta}(t_i))^2. $$

The boundary condition term penalizes deviations from the model boundary conditions:

$$ \mathcal{L}_{\text{BC}} = \frac{1}{N_{\text{BC}}} \sum_{i=1}^{N_{\text{BC}}} (f_{\gamma}(x_r) \rho_{\Theta}(t_i, x_r) - r_{\Theta}(t_i))^2. $$

Finally, the data losses compare the predicted size density $\rho_{\Theta}$ with the observed oocyte size distributions $\hat{\rho}$:

$$ \mathcal{L}_{\text{DATAlog}} = \frac{1}{N_x N_j} \sum_{j=1}^{N_j} \sum_{i=1}^{N_x} [\log(\rho_{\Theta}(t_j, y_i) + 10^{-3}) - \log(\hat{\rho}_j(y_i) + 10^{-3})]^2, $$

$$ \mathcal{L}_{\text{DATAmse}} = \frac{1}{N_x N_j} \sum_{j=1}^{N_j} \sum_{i=1}^{N_x} (\rho_{\Theta}(t_j, y_i) - \hat{\rho}_j(y_i))^2, $$

where $(t_i, x_i)$ denote collocation points, and $(y_i)_{1 \le i \le N_x}$ discretizes the size domain at observation times $(t_j)_{1 \le j \le N_j}$ corresponding to individual fish ovaries (more details in S5 - Implementation details, parameters and hyperparameters).

The joint optimization of $(\Theta, \gamma)$ allows simultaneous inference of the state dynamics and the underlying biological rates, whether represented parametrically or via neural networks.

Positivity of the state variables $\rho_{\Theta}$, $n_{\Theta}$ and $r_{\Theta}$, as well as of the biological functions and $f_{\gamma}$ in the non-parametric approach, is enforced via **SoftPlus** activation functions. Similarly, the Dirichlet boundary condition $n(t, x_r) = 0$ is incorporated through the network architecture and activation choice. The overall PINN architecture is illustrated in figure 2.

Neural networks are trained using the Adam optimizer [34], followed by a few epochs of stochastic gradient descent to improve stability, as introduced in [35]. At each epoch,
data are divided into mini-batches for gradient computation and parameter updates. A learning rate scheduler reduces the learning rate when the loss plateaus. To balance the multiple components of the loss, we apply the SoftAdapt algorithm [36], which dynamically adjusts the weights ($w_{pde1}$, $w_{pde2}$, $w_{ode}$, $w_{bc}$, $w_{d1}$, $w_{d2}$) during training, preventing domination by any single term. Additionally, the weights of the PDE and boundary loss terms are gradually increased to prioritize fitting the data in early training stages. Details on the network implementation, hyperparameters, and training procedure are provided in the Appendix S5 - Implementation details, parameters and hyperparameters, and the whole code will be available on GitHub.

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<figure><img src="image_3.png" /><figcaption>**Fig 2.** Physics-Informed Neural Networks framework used to infer oocyte growth and recruitment dynamics in juvenile fish ovaries. The network combines experimental oocyte size distribution data with a mechanistic population PDE model. Both parametric (P-PINNs) and non-parametric (NP-PINNs) formulations are considered, allowing either the inference of explicit functional forms or flexible data-driven representations of growth and renewal rates, through the optimization of parameters $\gamma$.</figcaption></figure>

# Results

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## Method validation and identifiability analysis

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We first investigate the structural identifiability properties of the model under ideal observation assumptions, i.e. by assuming perfect knowledge of the oocyte size distribution $\rho(t, x)$ on the domain $[x_r, x_m]$ over a large time window $[t_0, t_m]$. In this setting, we show (see Appendix S1 - Structural identifiability analysis) that both the time-dependent recruitment rate $r(t)$ and the size-dependent growth rate $f(x)$ are structurally identifiable, implying in particular the identifiability of the AMH-dependent net renewal rate $\alpha(P_{AMH}(t)) = \frac{r'(t)}{r(t)}$ for the observed values of $P_{AMH}$. However, the explicit reconstruction formulas derived in this perfect-data framework rely on the estimation of time derivatives and on dense longitudinal observations. In practice, the data are highly sparse in time and consist of repeated cross-sectional measurements, with only one observation per individual and a large inter-individual variability. This

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may strongly affects the practical identifiability of the time-dependent recruitment rate and therefore of the precursor cell dynamics, whereas the time-independent growth rate is less directly impacted by temporal sparsity. In addition, the reconstruction of the growth rate becomes sensitive to uncertainty in spatial regions where the oocyte density is low. Although measurement noise is assumed negligible, inter-individual variability remains the dominant source of uncertainty. As a result, the explicit reconstruction formulas derived for perfect data cannot be applied directly, which motivates the use of a Physics-Informed Neural Network approach to deal with sparse and cross-sectional data.

The results of the fit on the synthetic dataset show that parametric and non-parametric PINN approaches yield very similar inferences, as expected given that the synthetic datasets were generated using the parametric forms (see table 1). Regarding the recovered functional rates profiles, growth rates of small oocytes are better recovered than those of larger oocytes, which was expected as the oocyte density is lower for large oocytes, and the renewal rate exhibit a slightly better recovery for low values of $P_{AMH}$ (see figures 10 and 11 in appendix S3 - NP-PINN Method validation on synthetic dataset and S4 - P-PINN Method validation on synthetic dataset). Overall, the recovery is really good, considering the high inter-individual variability and the sparsity of temporal observations. Focusing on the parametric fit, all parameters are recovered within the correct order of magnitude, with generally better identifiability for growth rate parameters than for renewal rate parameters, except for the growth parameter B (see table 2) Importantly, the main source of difficulty in parameter recovery here stems from the imposed inter-individual variability, such that the synthetic dataset does not correspond to a single set of reference parameters. Exact recovery of the nominal reference values is not expected. Nevertheless, the proposed approach captures well the correct population-level trends, supporting the ability to reliably estimate reference population growth and renewal rates from real experimental data.

<table><caption>**Table 1.** Training losses on the synthetic dataset for the parametric and non-parametric PINN. Results are reported as mean ± standard deviation over $N = 5$ independent trainings performed on the same synthetic dataset.</caption><thead><tr><th>Loss term</th><th>Parametric PINN</th><th>Non-parametric PINN</th></tr></thead><tbody><tr><td>L<sub>PDE</sub></td><td>8.0 × 10<sup>-6</sup> ± 0.0 × 10<sup>-6</sup></td><td>8.0 × 10<sup>-6</sup> ± 1.0 × 10<sup>-6</sup></td></tr><tr><td>L<sup>aux</sup><sub>PDE</sub></td><td>1.0 × 10<sup>-6</sup> ± 0.0 × 10<sup>-6</sup></td><td>1.0 × 10<sup>-6</sup> ± 0.0 × 10<sup>-6</sup></td></tr><tr><td>L<sub>BC</sub></td><td>1.4 × 10<sup>-5</sup> ± 5.0 × 10<sup>-6</sup></td><td>1.3 × 10<sup>-5</sup> ± 4.0 × 10<sup>-6</sup></td></tr><tr><td>L<sub>ODE</sub></td><td>&lt; 1.0 × 10<sup>-6</sup></td><td>&lt; 1.0 × 10<sup>-6</sup></td></tr><tr><td>L<sub>DATAmse</sub></td><td>1.18 × 10<sup>-1</sup> ± 7.3 × 10<sup>-5</sup></td><td>1.18 × 10<sup>-1</sup> ± 1.4 × 10<sup>-5</sup></td></tr><tr><td>L<sub>DATAlog</sub></td><td>1.61 × 10<sup>-2</sup> ± 5.4 × 10<sup>-5</sup></td><td>1.61 × 10<sup>-2</sup> ± 4.0 × 10<sup>-5</sup></td></tr></tbody></table>

<table><caption>**Table 2.** Parameter recovery on the synthetic dataset using the parametric PINN. Results are reported as mean ± standard deviation over $N = 5$ independent trainings on the same synthetic dataset. Relative errors are computed with respect to the reference parameter values used to generate the synthetic data.</caption><thead><tr><th>Parameter</th><th>Reference value</th><th>Inferred value</th><th>Mean relative difference</th></tr></thead><tbody><tr><td>α<sub>min</sub></td><td>-0.22</td><td>-0.28 ± 0.26</td><td>0.27</td></tr><tr><td>α<sub>max</sub></td><td>1.94</td><td>2.54 ± 0.23</td><td>0.31</td></tr><tr><td>P<sub>m</sub></td><td>309.0</td><td>395.9 ± 10.28</td><td>0.28</td></tr><tr><td>k</td><td>1.57</td><td>2.25 ± 0.77</td><td>0.43</td></tr><tr><td>A</td><td>35.85</td><td>30.91 ± 0.99</td><td>0.14</td></tr><tr><td>B</td><td>13.36</td><td>9.16 ± 0.15</td><td>0.31</td></tr><tr><td>C</td><td>2.54</td><td>2.69 ± 0.02</td><td>0.06</td></tr><tr><td>f<sub>0</sub></td><td>0.57</td><td>0.53 ± 0.00</td><td>0.06</td></tr></tbody></table>
PINNs inference of oocyte dynamics

We first assess the performance of the non-parametric PINN approach on experimental data. Overall, the model provides a good fit of the observed oocyte size distributions at all observation times and across the whole range of oocyte sizes (see figure 3 and table 3). In particular, the reconstructed distributions consistently interpolate the data points, capturing both small and large oocyte classes. Moreover, the results are robust with respect to the stochastic optimization procedure: repeated training runs lead to very similar reconstructed distributions. Some variability is nevertheless observed in the inferred growth rate for large oocyte sizes, as well as a moderate variability in the inferred renewal rate. These variations across optimization runs are consistent with the identifiability limitations discussed in subsection Method validation and identifiability analysis, reflecting the reduced information content of the data in regions of low oocyte density and under sparse temporal sampling.

Based on the shapes of the functions inferred from this initial non-parametric fit, see figure 3, we impose the following parametric forms. For the oocyte growth function, we use a Gompertz model:

$$f(x) = f_0 + A e^{-B e^{-C \frac{x-x_p}{x_m-x_r}}}, \quad (4)$$

where $f_0, A, B, C$ are parameters to be determined. The Gompertz model is chosen because it accurately captures the inferred growth rate shape with a small number of parameters, making it well-suited for compact parametric inference. For the AMH-dependent net renewal rate of precursor cells, we adopt a sigmoidal form:

$$\alpha(P_{AMH}) = \alpha_{min} + \frac{\alpha_{max} - \alpha_{min}}{1 + \left(\frac{P_{AMH}}{P_m}\right)^k}, \quad (5)$$

with parameters $\alpha_{min}, \alpha_{max}, P_m, k$ to be determined. This sigmoidal form reflects a saturating negative feedback. It is consistent with previous models of hormone-dependent feedback (see e.g. [37,38]) and provides a flexible yet interpretable representation of the AMH effect on precursor cell renewal.

From Table 3, we see that the parametric PINN achieves training losses comparable to the non-parametric PINN on the experimental dataset. This indicates that choosing compact parametric representations of growth and renewal rates does not reduce the expressivity of the model, as the observed dynamics and experimental data are captured equally well. Exploring even simpler functional forms with comparable performance could further improve parameter identifiability, in the spirit of sparse model discovery approaches such as SINDy method [39]. This is out of the scope of this work.

<table><caption>**Table 3.** Training losses on the experimental dataset for the parametric and non-parametric PINN. Results are reported as mean ± standard deviation over $N = 5$ independent trainings performed on the same dataset.</caption><thead><tr><th>Loss term</th><th>Parametric PINN</th><th>Non-parametric PINN</th></tr></thead><tbody><tr><td>L<sub>PDE</sub></td><td>4.1 × 10<sup>-5</sup> ± 1.0 × 10<sup>-6</sup></td><td>4.1 × 10<sup>-5</sup> ± 3.0 × 10<sup>-6</sup></td></tr><tr><td>L<sub>aux PDE</sub></td><td>2.0 × 10<sup>-6</sup> ± 1.0 × 10<sup>-6</sup></td><td>3.0 × 10<sup>-6</sup> ± 2.0 × 10<sup>-6</sup></td></tr><tr><td>L<sub>BC</sub></td><td>2.4 × 10<sup>-5</sup> ± 1.0 × 10<sup>-5</sup></td><td>2.7 × 10<sup>-5</sup> ± 1.4 × 10<sup>-5</sup></td></tr><tr><td>L<sub>ODE</sub></td><td>&lt; 1.0 × 10<sup>-6</sup></td><td>1.0 × 10<sup>-6</sup> ± 1.0 × 10<sup>-6</sup></td></tr><tr><td>L<sub>DATAmse</sub></td><td>6.77 × 10<sup>-1</sup> ± 7.17 × 10<sup>-4</sup></td><td>6.76 × 10<sup>-1</sup> ± 8.62 × 10<sup>-4</sup></td></tr><tr><td>L<sub>DATAlog</sub></td><td>1.02 × 10<sup>-1</sup> ± 7.0 × 10<sup>-5</sup></td><td>1.02 × 10<sup>-1</sup> ± 4.0 × 10<sup>-4</sup></td></tr></tbody></table>

We now consider the parametric PINN formulation and focus on the biological interpretation of the inferred vital rates. The estimated recruitment rate increases from approximately 20 to 50 oocytes per day during early stages, before reaching a plateau,
corresponding to a stabilization of the net renewal rate around zero. This behavior suggests that, toward the end of the juvenile period, the ovary has accumulated a sufficient pool of oocytes, and the renewal of precursor germ cells progressively ceases. An alternative interpretation could involve continued proliferation of precursor germ cells accompanied by a reduction in differentiation. However, such a mechanism is not captured in the present model, which assumes a constant differentiation rate $\lambda_0$. In addition, the inferred AMH-dependent regulation indicates an inhibitory effect of AMH on precursor cell proliferation, in agreement with deductions from experimental observations reported in AMH invalidation studies (see, e.g., [5, 10]). Finally, the inferred growth rate strongly increases with oocyte size, as reflected by the estimated transit times through successive maturation stages (see table 5).

<table><caption>**Table 4.** Parametric PINN inference of growth rate and net renewal rate parameters on the experimental dataset: mean ± standard deviation over $N = 5$ independent trainings performed on the same dataset.</caption><thead><tr><th colspan="2">Oocyte growth rate (f)</th><th colspan="2">AMH-dependent net renewal rate (α)</th></tr><tr><th>Parameter</th><th>Estimate</th><th>Parameter</th><th>Estimate</th></tr></thead><tbody><tr><td>A</td><td>35.85 ± 0.60</td><td>α<sub>min</sub></td><td>-0.22 ± 0.22</td></tr><tr><td>B</td><td>13.36 ± 0.18</td><td>α<sub>max</sub></td><td>1.94 ± 0.14</td></tr><tr><td>C</td><td>2.54 ± 0.03</td><td>P<sub>m</sub></td><td>553.8 ± 44.8</td></tr><tr><td>f<sub>0</sub></td><td>0.57 ± 0.002</td><td>k</td><td>3.95 ± 0.98</td></tr></tbody></table>

<table><caption>**Table 5.** Estimated transit times of oocytes in each oocyte maturation stage, from the parametric PINN fit</caption><thead><tr><th>Stage</th><th>Transit time (days)</th><th>Percentage of total (%)</th></tr></thead><tbody><tr><td>Total</td><td>101.3</td><td>100.0</td></tr><tr><td>Stage I (33.7 – 60µm)</td><td>45.9</td><td>45.3</td></tr><tr><td>Stage II (60 – 90µm)</td><td>39.6</td><td>39.2</td></tr><tr><td>Stage III (90 – 120µm)</td><td>12.2</td><td>12.0</td></tr><tr><td>Stage IV (120 – 150µm)</td><td>3.6</td><td>3.5</td></tr></tbody></table>
<figure><img src="image_4.png" /><figcaption>**Fig 3.** Results of the non-parametric PINN applied to the experimental dataset. Top panels: data fitting results obtained from *N* = 5 independent trainings of the non-parametric PINN. Light blue dots correspond to experimental data, dashed red lines to individual PINN fits, and the dashed black line to the average fit across trainings. Top left: comparison between experimental measurements and PINN predictions for the number of oocytes per size class, as well as for the AMH-related proxy *P*<sub>AMH</sub>. Top right: fits of the oocyte size distributions at the four observation times (one panel per time point). In the PINN formulation, the data loss explicitly enforces the minimization of the discrepancy between the predicted and observed distributions. Bottom panels: inferred model components, from left to right: the size-dependent oocyte growth velocity, the AMH-dependent renewal rate of precursor cells, and the time-dependent oocyte recruitment rate. Dashed red lines correspond to individual training runs, the dashed black line to the mean inference, and the light gray shaded area to ±3 standard deviations across trainings, highlighting the variability induced by stochastic optimization.</figcaption></figure>

Estimation of the precursor cell number and differentiation rate

Counts of precursor germ cells (oogonia) in juvenile medaka ovaries are available from
Satoh and Egami (1972) [10] for fish at five distinct ages between 30 and 50 days

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<figure><img src="image_5.png" /><figcaption>**Fig 4.** Results of the parametric PINN applied to the experimental dataset. Top panels: data fitting results obtained from $N = 5$ independent trainings of the parametric PINN. Light blue dots correspond to experimental data, dashed red lines to individual PINN fits, and the dashed black line to the average fit across trainings. Top left: comparison between experimental measurements and PINN predictions for the number of oocytes per size class, as well as for the AMH-related proxy $P_{\text{AMH}}$. Top right: fits of the oocyte size distributions at the four observation times (one panel per time point). In the PINN formulation, the data loss explicitly enforces the minimization of the discrepancy between the predicted and observed distributions. Bottom panels: inferred model components, from left to right: the size-dependent oocyte growth velocity, the AMH-dependent renewal rate of precursor cells, and the time-dependent oocyte recruitment rate. Dashed red lines correspond to individual training runs, the dashed black line to the mean inference, and the light gray shaded area to $\pm 3$ standard deviations across trainings, highlighting variability induced by stochastic optimization.</figcaption></figure>

post-fertilization. In our model, the recruitment rate satisfies $r(t) = \lambda_0 S(t)$, where $S(t)$ denotes the number of precursor cells and $\lambda_0$ the differentiation rate. Using the recruitment rate $r(t)$ estimated by the Parametric-PINN and obtained by direct simulation of the model at the corresponding ages, we estimate the differentiation rate $\lambda_0$. Assuming $\lambda_0$ to be constant in time, the mean and standard deviation across the five

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age points yield $\lambda_0 = 0.082 \pm 0.023$. This corresponds to a characteristic differentiation timescale of approximately 12.2 days, with a plausible range from 9.6 to 16.8 days.

The number of precursor cells as a function of age is then reconstructed as $S(t) = r(t)/\lambda_0$ and is shown in Fig. 5. The dispersion of experimental data points around the inferred trajectory reflects the strong inter-individual variability reported between juvenile medaka ovaries.

Let us remark that rather than incorporating these data directly into the PINN framework, we used them in a second step to estimate $\lambda_0$. This choice reflects the higher confidence in the oocyte size distribution data, which are obtained from exhaustive 3D imaging and automated segmentation (instead of manual counting on sampled 2D ovary slices), and avoids potential inconsistencies due to differences in experimental protocols between datasets. This strategy ensures that the recruitment dynamics are inferred only from the size distribution data.

<figure><img src="image_6.png" /><figcaption>**Fig 5.** The dashed red line corresponds to the number of precursor cells $S(t)$, which is reconstructed from the P-PINN inferred recruitment rate $r(t)$ using the relation $S(t) = r(t)/\lambda_0$, where $\lambda_0$ is the estimated differentiation rate from the precursor cells counts (skyblue points) reported in [10].</figcaption></figure>

## Origins of inter-individual variability

We investigate the origin of the pronounced inter-individual variability observed in juvenile fish, with a particular focus on late juvenile fish (112 days post fertilization, dpf), preceding the transition to adulthood, where large differences in oocyte populations are observed (see data reported in table 6).

We first assess whether variability in the early ovarian stock, quantified by the number of small oocytes at 28.5 dpf, is sufficient to explain the variability observed at later ages. To this end, we perform numerical simulations of the model (see left panel of figure 6) in which the oocyte growth rate and recruitment rate are fixed to the values inferred by the parametric PINN trained on pooled data (table 4), while the inferred initial oocyte distribution is scaled by a multiplicative factor $m_{\rho_0}$. This factor is varied over a broad range informed by the variability observed in the youngest fish. These simulations fail to reproduce the experimentally observed variability in oocyte size distributions at 112 dpf, indicating that differences in the initial oocyte stock alone cannot account for the observed inter-individual variability.

To further disentangle the respective roles of model parameters, we have performed a sensitivity analysis of the model outputs at 112 dpf using Sobol indices (right panel of figure 6). The model parameters for the growth rate, renewal rate and scaling factor $m_{\rho_0}$ were varied simultaneously within ±30% of the values inferred by the parametric PINN. For each parameter set, six relevant outputs were computed at 112 dpf: the total
number of oocytes, the number of oocytes in stages I–IV, and the ratio of total oocytes to stage I oocytes. This analysis reveals distinct roles of model parameters in shaping oocyte numbers at 112 dpf. The number of small oocytes (stage I) is sensitive to the AMH-dependent recruitment parameters, particularly the maximum renewal rate ($\alpha_{max}$) and the inflection point ($P_m$), and to the minimal growth rate $f_0$. In contrast, the number of large oocytes (stages III–IV) is predominantly controlled by growth rate parameters. The scaling factor of the initial oocyte distribution has a moderate influence across all stages. Importantly, the ratio of total oocytes to stage I oocytes, which exhibits substantial inter-individual variability in experimental data (see table 6), is strongly dependent on growth rate parameters. This suggests that differences in oocyte growth dynamics can largely explain the contrast between individual F17 (low ratio) and individuals F18, F19, and F21 (higher similar ratio). Differences in the total number of oocytes between F21 (low oocyte number) and the group F17–F18–F19 (higher oocyte numbers) are consistent with combined variations in both the initial oocyte stock and the renewal rate of precursor cells.

To further support these conclusions, we performed individual-level fits using the parametric PINN framework. Specifically, for each of the four fish measured at 112 dpf, a separate model was trained using the same hyperparameters as in the pooled analysis, but relying only on the full set of measurements at 28.5 dpf together with the single observation available at 112 dpf for the corresponding individual. This approach allows us to isolate individual-specific estimates of oocyte growth and precursor cell renewal rates.

As shown in figure 7, the inferred kinetic rates exhibit marked inter-individual variability. In particular, individuals F18 and F19 display nearly identical inferred parameters, the growth rate of F17 is substantially lower than those of F18, F19, and F21, and the renewal rate of F21 is markedly lower than that of the other individuals. These individual-level inferences are fully consistent with the conclusions drawn from the sensitivity analysis and the initial condition study. Together, these results demonstrate that the inter-individual variability observed at 112 dpf cannot be explained solely by differences in the initial oocyte size distribution, but instead primarily reflects variability in individual-specific oocyte growth and precursor cells renewal dynamics.

<table><caption>**Table 6.** Experimental data. Oocyte numbers per maturation stage, total numbers of oocytes, and ratio of total to stage I oocyte numbers for the four fish ovaries observed at 112 dpf.</caption><thead><tr><th>Label</th><th>Stage I</th><th>Stage II</th><th>Stage III</th><th>Stage IV</th><th>Total</th><th>Total / Stage I</th></tr></thead><tbody><tr><td>F17</td><td>4707</td><td>1105</td><td>99</td><td>4</td><td>5915</td><td>1.26</td></tr><tr><td>F18</td><td>2638</td><td>1808</td><td>322</td><td>101</td><td>4869</td><td>1.85</td></tr><tr><td>F19</td><td>2497</td><td>2116</td><td>405</td><td>126</td><td>5144</td><td>2.06</td></tr><tr><td>F21</td><td>1135</td><td>758</td><td>243</td><td>105</td><td>2241</td><td>1.98</td></tr></tbody></table>
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>**Fig 6.** Sensitivity analysis of the oogenesis model at 112 days post-fertilization (dpf). Left panel: Model response to variations in the initial condition $\rho_0$. Red dashed lines show simulations obtained by scaling the PINN-inferred initial condition, with all other parameters fixed; blue points denote experimental data. Results are shown as the number of stage I oocytes versus the total number of stages II–IV oocytes. Right panel: Total-order Sobol sensitivity indices ($S_T$) of selected model outputs at 112 dpf for growth rate, renewal rate, and initial condition scaling factor $m_{\rho_0}$ input parameters. Color intensity reflects each parameter's contribution to output variability, including interaction effects.</figcaption></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>**Fig 7.** Comparison of Parametric-PINN fits for the four individuals measured at 112 dpf. For each fish, the model is trained using the same hyperparameters as in the fit performed on the full dataset, but only a reduced subset of observations is retained: all the data at the first time point (28.5 dpf) and a single observation at the final time point (112 dpf). Left panel: inferred oocyte growth rates. Right panel: inferred precursor cell renewal rates.</figcaption></figure>

## In silico prediction in AMH invalidation conditions

We have performed forward simulations of the inferred model with the parametric PINN in AMH invalidation scenario, until 4.5 months post fertilization (135 dpf). The results are shown in figure 8. The comparison between the reference simulation and the AMH invalidation scenario reveals a marked increase in the number of oocytes under AMH deficiency. In particular, the model predicts a substantial accumulation of stage I oocytes after approximately 3 months post fertilization, while later stages are comparatively less affected in this time window. This qualitative and quantitative behavior is consistent with experimental observations in zebrafish AMH invalidation

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models, where the number of previtellogenic oocytes is reported to be increased by a factor of approximately two compared to wild-type individuals after 4.5 months (see Fig. 2 in [5]). The agreement in orders of magnitude between model predictions and experimental measurements supports the biological relevance of the inferred AMH-dependent renewal regulation and further validates the parametric PINN framework.

<figure><img src="image_11.png" /><figcaption>**Fig 8.** Comparison between the model simulations and experimental data under normal and AMH invalidation conditions. Dashed red lines show direct simulations of the model inferred using the parametric PINN approach, while green lines correspond to simulations of the same model under AMH invalidation conditions, i.e. by fixing the renewal rate to its maximal value $\alpha_{max}$. Light blue dots represent experimental data. Left panel: oocyte counts per maturity stage as a function of age. Right panel: total oocyte number as a function of age.</figcaption></figure>

## Predicted dynamics over the reproductive life span

We first investigate theoretically the long-time behavior of the model to gain insight into how AMH-mediated feedback shapes oocyte population dynamics over the reproductive life of the fish. Non-trivial stationary states exist only when the AMH-dependent net renewal rate of precursor cells satisfies

$$\alpha(\bar{P}_{AMH}) = 0,$$

where $\bar{P}_{AMH}$ denotes the AMH quantity generated by the oocyte population at stationary state (see Appendix S6 - Model stationary states for details). In our case, the inferred net renewal rate of precursor cells $\alpha$ is monotonically decreasing and has one zero (Fig. 4), which implies the existence of a unique non-trivial stationary oocyte size distribution. While a full analytical study of the stability and attractiveness of this stationary state is beyond the scope of this work, the model can be explored numerically to assess its long-time behavior.

Using the parameters inferred by the parametric PINN from juvenile fish data (up to 112 days post fertilization), we performed forward simulations over an extended time horizon of two years (730 days), well beyond the temporal window covered by the experimental observations. The resulting dynamics are summarized in Fig. 9. These long-term simulations should be interpreted with caution, as the available data span only a short period and do not capture potential physiological changes associated with sexual maturation and adulthood.

Despite these limitations, the predicted dynamics exhibit coherent and biologically plausible trends. The model predicts a fast increase followed by a progressive decrease in
<figure><img src="image_12.png" /><figcaption>**Fig 9.** Long-term simulation of oocyte recruitment dynamics based on parameters inferred from juvenile data. Model simulations are performed over a period of two years (730 days), using the parametric PINN calibrated on juvenile fish data up to 112 days post fertilization. Left panel: predicted number of oocytes per maturity stage as a function of fish age (dashed red lines), compared to available experimental data (light blue dots). The light blue shaded area indicates the temporal window over which data are available, highlighting that model trajectories beyond this interval correspond to long-term predictions. Right panel: predicted daily recruitment rate of new oocytes (top) and number of precursor cells (bottom) as a function of fish age.</figcaption></figure>

the number of oocytes across all maturity stages, accompanied by a decline in the daily
recruitment of new oocytes, until a stationary regime is reached. Around two years of
age, the total number of oocytes stabilizes at approximately 2000, while the recruitment
rate converges toward values close to 20 oocytes per day. Interestingly, this order of
magnitude is comparable to reported daily spawning outputs in adult medaka [40],
suggesting a balance between oocyte recruitment and spawning in mature individuals.

Although substantial inter-individual variability in ovarian content has been
reported in adult medaka, a total oocyte number on the order of two thousand is
consistent with available experimental observations [40]. Overall, these results should be
viewed as first-order predictions derived from juvenile data. Future work will aim to
incorporate data from older individuals to refine the model calibration and to assess the
consistency of these long-term predictions across the full reproductive lifespan.
In this work, we have developed a quantitative model of early oogenesis in juvenile fish that accurately reproduces experimentally observed oocyte size distributions across multiple developmental stages. By coupling a size-structured population model with data-driven parameter inference, we were able to extract biologically meaningful dynamics from static experimental snapshots.

A central result of the model is the identification of an increasing oocyte recruitment rate that reaches a plateau during juvenile development. Mechanistically, this can be understood as a consequence of AMH negative feedback: initially, in the absence of a significant population of AMH-secreting oocytes, recruitment occurs near its maximal rate. As the oocyte population grows and AMH secretion increases, this feedback gradually reduces the renewal of precursor cells, leading to the observed plateau in the recruitment of new oocytes. While such a regulation has been proposed qualitatively, our approach provides a quantitative description of this process and shows that it is sufficient to explain the observed temporal evolution of oocyte populations.

Beyond reproducing observed population-level data, the model allows inference of quantities at the single-oocyte scale that are experimentally inaccessible. In particular, we inferred the dynamics of oocyte recruitment and a reference growth velocity as a function of oocyte size. While the data reflect population-level oocyte distributions, our approach leverages these measurements to reconstruct oocyte-level dynamics, providing a quantitative baseline for oocyte growth under normal developmental conditions.

Using parameters inferred from experimental data, we then employed the model to generate predictions in perturbed conditions, including AMH invalidation, and over longer timescales extending into the reproductive life of the fish. The resulting dynamics are in agreement with independent experimental studies, supporting both the robustness of the model and the relevance of the underlying mechanisms. This predictive capability highlights the value of mechanistic models calibrated on developmental data for exploring conditions that are difficult or costly to probe experimentally.

The model also offers a framework to interpret inter-individual variability in oogenesis. By disentangling the respective contributions of the early-established oocyte stock, oocyte growth dynamics, and AMH-dependent precursor cell renewal, we provide potential explanations for the diversity of ovarian trajectories observed between individuals at the same age.

From a methodological perspective, this study demonstrates the suitability of physics-informed neural networks for inferring size-structured population models governed by partial differential equations. The proposed framework is flexible and can be readily adapted to other structured population systems.

Several extensions of this work naturally emerge. First, the representation of precursor cell dynamics could be refined to explicitly account for additional regulatory pathways or cell states, an extension that is increasingly feasible given recent advances in the experimental characterization of precursor cell populations. Second, integrating data from older fish would allow the model to bridge juvenile development and adult ovarian function more directly.

From a methodological point of view, given the strong variability observed in the experimental data, it would be particularly relevant to incorporate uncertainty quantification techniques into the inference framework. Methods such as rPINN [41] or B-PINN [42] could already improve the robustness and interpretability of the inferred dynamics by providing confidence intervals on the reconstructed solutions and parameters. However, considering the repeated cross-sectional nature of the data, an other pertinent approach could consist in adapting the methodology proposed in [43]. This promising approach might allow the estimation of full parameter distributions across the fish population, thereby capturing better the biological heterogeneity. Such
an extension would offer a richer biological interpretation of the model but would also come at the cost of a significantly increased computational cost.

## Supporting information

S1 Text. Structural identifiability analysis

S2 Text. Synthetic dataset generation.

S3 Fig. NP-PINN method validation on synthetic dataset.

S4 Fig. P-PINN method validation on synthetic dataset.

S5 Text. Implementation details, parameters and hyperparameters.

S6 Text. Model stationary states

## Acknowledgments

This work was funded by INRAE Metaprogramme DIGIT-BIO (Digital biology to explore and predict living organisms in their environment), as well as from ANSES, PNR EST 2020, Project GINFIZ (Anses 2020/01/133), and from the French National research Agency (ANR) through the OVOPAUSE project (ANR-22-CE45-0017).

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## Data Availability Statement

All experimental datasets used in this study are publicly available from the French national research data repository *Recherche Data Gouv*, see [44]. Oocyte size distribution data at 28.5, 41, 103, and 112 days post fertilization are accessible at the following persistent URLs:
* 28.5 dpf:
  https://entrepot.recherche.data.gouv.fr/file.xhtml?persistentId=doi:10.57745/YQHWFA&version=1.0

* 41 dpf:
  https://entrepot.recherche.data.gouv.fr/file.xhtml?persistentId=doi:10.57745/DU3KOI&version=1.0

* 103 dpf:
  https://entrepot.recherche.data.gouv.fr/file.xhtml?persistentId=doi:10.57745/TWEQRH&version=1.0

* 112 dpf:
  https://entrepot.recherche.data.gouv.fr/file.xhtml?persistentId=doi:10.57745/ZMU9BM&version=1.0

All the code of this work is implemented in Python, using the PyTorch library. It will be available soon in a GitHub repository.

# Appendix

## S1 - Structural identifiability analysis

We study the structural identifiability of the model under the assumption that perfect longitudinal observations of the oocyte size distribution $\rho(t, x)$ are available for all $x \in [x_r, x_m]$ and $t \in [t_0, t_m]$. We define $x_c := \sup\{x \in [x_r, x_m] : \rho(t_0, x) \neq 0\}$, the right boundary of the support of the initial density. We assume that the observation window satisfies $t_m > t_c := \int_{x_c}^{x_m} \frac{1}{f(x)} dx > t_0$, so that the first time $t_c$ at which an oocyte reaches the maximal size $x_m$ is observed. We introduce the cumulative density

$$m(t, x) := \int_{x_r}^{x} \rho(t, y) dy,$$

which represents the number of oocytes with size smaller than $x$ at time $t$. If $\rho$ satisfies Eq. (2), then $m$ satisfies the transport equation:

$$\begin{cases} \partial_t m(t, x) + f(x) \partial_x m(t, x) - r(t) = 0, & t > t_0, \ x \in (x_r, x_m), \\ m(t_0, x) = m_0(x), & x \in (x_r, x_m), \\ m(t, x_r) = 0, & t > t_0 \end{cases}$$

Evaluating this equation at $x = x_m$ yields

$$r(t) = f(x_m) \partial_x m(t, x_m) + \partial_t m(t, x_m), \quad t \neq t_c. \qquad (6)$$

By definition of $t_c$, we have $\partial_x m(t, x_m) = 0$ for $t < t_c$, so that $r(t) = \partial_t m(t, x_m)$ for $t < t_c$. Moreover, for all $x \in [x_r, x_m]$ such that $\rho(t, x) > 0$, the growth rate is given by

$$f(x) = \frac{r(t) - \partial_t m(t, x)}{\partial_x m(t, x)} = \frac{\partial_t m(t, x_m) - \partial_t m(t, x)}{\partial_x m(t, x)}.$$

As $t \to t_c$, the right boundary of the support of $\rho(t, \cdot)$ converges to $x_m$, which allows the identification of $f$ on any interval $[x_r, x_m - \varepsilon]$, $\varepsilon > 0$. By continuity, $f$ is therefore identifiable on the whole interval $[x_r, x_m]$. As a consequence, using Eq. (6), the recruitment rate $r(t)$ is also identifiable on the whole observation window $[t_0, t_m]$. Since $\beta(P_{AMH}(t)) - \lambda_0 = r'(t)/r(t)$, the net renewal rate $\alpha = \beta - \lambda_0$ is identifiable for the observed values of $P_{AMH}$.
## S2 - Synthetic dataset generation

To generate synthetic datasets reproducing the cross-sectional dataset with inter-individual variability between juvenile fish, we simulate the juvenile oocyte population model independently for each individual, introducing variability in the model parameters and in the initial oocyte size distribution.

**Growth rate parameters** The oocyte growth rate is described by a Gompertz function (4), that we recall here:

$$f(x) = f_0 + Ae^{-B}e^{-C \frac{x-x_r}{x_m-x_r}}$$

parameterized by four positive parameters ($x_r$ and $x_m$ are set to 33.7 and 150, as explained in PINNs inference of oocyte dynamics). The reference parameter values are set to $A = 35.85$, $B = 13.36$, $C = 2.54$, $f_0 = 0.57$. Individual-specific growth parameters are independently sampled from lognormal distributions centered on these reference values, with standard deviations $\sigma_A = 0.1$, $\sigma_B = 0.1$, $\sigma_C = 0.04$, $\sigma_{f_0} = 0.2$. This sampling strategy ensures positivity of the parameters while introducing biologically plausible inter-individual variability in growth dynamics.

**Renewal rate parameters** The precursor cellrenewal rate is modeled by a decreasing sigmoidal function (5), that we recall here:

$$\alpha(P_{AMH}) = \alpha_{min} + \frac{\alpha_{max} - \alpha_{min}}{1 + \left(\frac{P_{AMH}}{P_m}\right)^k}$$

with reference parameter values $\alpha_{min} = 0.22$, $\alpha_{max} = 1.94$, $P_m = 0.095$, $k = 3.95$. Each renewal parameter is independently sampled from a lognormal distribution with identical standard deviation, $\sigma_{\alpha_{min}} = \sigma_{\alpha_{max}} = \sigma_{P_m} = \sigma_k = 0.1$.

**Initial oocyte size distribution** Inter-individual variability in the initial oocyte size distribution is introduced by perturbing a reference initial density $\rho_0(x)$. For each individual, the initial condition is defined as

$$\rho_0(x) = \rho_{\text{mean}}(x) A_i (1 + \eta_i(x)),$$

where $\rho_{\text{mean}}(x)$ is a fixed positive concave-convex reference profile supported on $x \le 90\mu m$, $A_i$ is an individual-specific positive amplitude sampled from a lognormal distribution, and $\eta_i(x)$ is a smooth zero-mean random modulation.

**Synthetic data generation** For each individual, the model was initialized with the sampled growth, renewal, and initial condition parameters and simulated forward in time using an Upwind numerical solver. Synthetic oocyte size distributions were then evaluated on a fixed size grid and at the same observation times as in the experimental dataset. This procedure yields synthetic datasets that reproduce realistic inter-individual variability in oocyte size distributions, while maintaining full control over the underlying sources of variability.
S3 - NP-PINN Method validation on synthetic dataset

<figure><img src="image_13.png" /><figcaption>**Fig 10.** Results of the non-parametric PINN applied to the synthetic dataset. Top panels: data fitting results obtained from *N* = 5 independent trainings of the non-parametric PINN. Orange dots and lines correspond to synthetic data, dashed red lines to individual PINN fits, and the dashed black line to the average fit across trainings. Top left: comparison between synthetic dataset and PINN predictions for the number of oocytes per size class, as well as for the AMH-related proxy *P*<sub>AMH</sub>. Top right: fits of the oocyte size distributions at the four observation times (one panel per time point). In the PINN formulation, the data loss explicitly enforces the minimization of the discrepancy between the predicted and observed distributions. Bottom panels: inferred model components, from left to right: the size-dependent oocyte growth velocity, the AMH-dependent renewal rate of precursor cells, and the time-dependent oocyte recruitment rate. Dashed red lines correspond to individual training runs, the dashed black line to the mean inference, and the light gray shaded area to ±3 standard deviations across trainings. The gray lines in the growth rate and renewal rate panels are the rates used for the synthetic dataset generation (one line per synthetic individual).</figcaption></figure>
S4 - P-PINN Method validation on synthetic dataset

<figure><img src="image_14.png" /><figcaption>**Fig 11.** Results of the parametric PINN applied to the synthetic dataset. Top panels: data fitting results obtained from $N = 5$ independent trainings of the parametric PINN. Orange dots and lines correspond to synthetic data, dashed red lines to individual PINN fits, and the dashed black line to the average fit across trainings. Top left: comparison between synthetic dataset and PINN predictions for the number of oocytes per size class, as well as for the AMH-related proxy $P_{AMH}$. Top right: fits of the oocyte size distributions at the four observation times (one panel per time point). In the PINN formulation, the data loss explicitly enforces the minimization of the discrepancy between the predicted and observed distributions. Bottom panels: inferred model components, from left to right: the size-dependent oocyte growth velocity, the AMH-dependent renewal rate of precursor cells, and the time-dependent oocyte recruitment rate. Dashed red lines correspond to individual training runs, the dashed black line to the mean inference, and the light gray shaded area to ±3 standard deviations across trainings. The gray lines in the growth rate and renewal rate panels are the rates used for the synthetic dataset generation (one line per synthetic individual).</figcaption></figure>
# S5 - Implementation details, parameters and hyperparameters

## Biologically informed model parameters

The minimal oocyte size is fixed to $x_r = 33.7$ µm, corresponding to the lower detection limit of oocyte sizes in the experimental datasets. The maximal oocyte size is fixed to $x_m = 150$ µm, corresponding to the maximal size of previtellogenic oocytes, as reported in [32].

To our knowledge, quantitative measurements of AMH expression as a function of follicle size are not available in medaka. However, experimental data in zebrafish indicate that AMH expression is maximal in late previtellogenic follicles [2]. The functional form of $\omega_{AMH}$ is therefore chosen phenomenologically to capture this qualitative behavior, rather than being fitted to quantitative expression data. We define the weight function

$$ \omega_{AMH}(x) = \exp\left(-\left(\frac{x - \mu_{AMH}}{\sigma_{AMH}}\right)^4\right), $$

with $\mu_{AMH} = 120$ µm and $\sigma_{AMH} = 60$ µm.

## Data preprocessing

**Kernel density estimation** To estimate the experimental oocyte size densities from discrete measurements (individual counts and sizes of oocytes within each ovary), we used an adaptive kernel density estimation (KDE): smaller bandwidths were used in regions of high data concentration, and larger ones where data were sparse. We first computed a standard Gaussian KDE with a fixed base bandwidth $h_0$, which provides an initial density estimate $\hat{\rho}_{init}(x)$. Then, the bandwidth at each data point $x_i$ was adjusted according to the local density:

$$ h_i = h_0 \left( \frac{\hat{\rho}_{init}(x_i)}{Q} \right)^{-1/2}, $$

where $Q$ is a reference value used to normalize the local densities, defined as a low quantile (typically the 5th percentile) of the initial density values $\hat{\rho}_{init}(x_i)$. This choice ensures that points in dense regions get reduced bandwidths while avoiding extremely small values for sparse regions. The final adaptive KDE is then given by:

$$ \hat{\rho}(x) = \frac{1}{n} \sum_{i=1}^{n} \frac{1}{h_i} K \left( \frac{x - x_i}{h_i} \right), $$

where $x_1, \dots, x_n$ are the observed oocyte sizes, and $K$ is the Gaussian kernel $K(u) = \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{u^2}{2}\right)$. This estimation yields a normalized density (i.e., with integral equal to one). To obtain the actual oocyte size density (in terms of number of oocytes per µm), we multiply $\hat{\rho}(x)$ by the total number of oocytes observed in the corresponding ovary.

**Renormalization** Standardizing inputs and outputs is a common practice to stabilize neural network training. To ensure numerical stability during the training of the PINN, we rescaled both the data and the equations as follows: the spatial domain $[x_r, x_m]$ and the time domain are both rescaled to the interval $[0, 1]$, the experimental size densities are rescaled so that their maximum value is of order 1, and the output of the neural networks are multiplied by a constant to ensure that the output remains of order 1 when needed.
## Neural Network architecture

All neural networks in this work are fully connected multilayer perceptrons (MLPs). Between each hidden layer, a Sigmoid activation is applied. Outputs that must remain positive are transformed with a Softplus activation. The networks $\rho_{\Theta}$ (approximating the oocyte density) and $n_{\Theta}$ (auxiliary network) both take two inputs $(t, x)$ and output a positive scalar; in addition, $n_{\Theta}(t, x)$ is multiplied by $(x - x_r)/(x_m - x_r)$ to satisfy $n_{\Theta}(t, x_r) = 0$. The network $r_{\theta}$ takes a scalar input and outputs a positive recruitment rate. Network architectures are as follows:

* $\rho_{\Theta}$: 4 hidden layers, 64 neurons each,

* $n_{\Theta}$: 3 hidden layers, 64 neurons each,

* $r_{\Theta}$: 3 hidden layers, 50 neurons each,

* $f_{\gamma}$ (NP-PINN growth rate network): 3 hidden layers, 50 neurons

* $\alpha_{\gamma}$ (NP-PINN renewal rate network): 3 hidden layers, 32 neurons

## Training hyperparameters

For all simulations, we set the following:

* $N_{BC} = 500$ points randomly sampled in $[x_r, x_m]$ at each epoch for evaluating the boundary conditions,

* $N_{PDE} = 3,000$ points randomly sampled in $[t_0, t_m] \times [x_r, x_m]$ at each epoch for evaluating the PDE residual,

* $N_x = 100$ spatial points in $[x_r, x_m]$ to evaluate the experimental oocyte densities, with a refined discretization over small oocyte sizes to better resolve early-stage oocyte dynamics.

PINNs were trained for a total of 10,000 epochs. During the first 9,500 epochs, training relied on the Adam optimizer with an initial learning rate of 0.003, combined with a ReduceLROnPlateau scheduler (reduction factor 0.98, patience 50 epochs). During the final 500 epochs, optimization was continued using stochastic gradient descent (SGD) with a learning rate of 0.001 and a momentum of 0.9, applied to all trainable parameters. The weights of the PDE and boundary-condition loss terms were progressively increased over the first 500 epochs to prioritize fitting the experimental data at the beginning of training.

## S6 - Model stationary states

Stationary size densities $\bar{\rho}$ of model (2) verify:

$$ 
\left\{ 
\begin{array}{l} 
\frac{d}{dx} (f(x)\bar{\rho}(x)) = 0, \quad \forall x \in (x_r, x_m) \\ 
f(x_r)\bar{\rho}(x_r) = \bar{r}, \\ 
0 = \alpha(\bar{P}_{AMH})\bar{r}, \\ 
\bar{P}_{AMH} = \int_{x_r}^{x_m} \omega_{AMH}(x)\bar{\rho}(x)dx. 
\end{array} 
\right. 
$$

Therefore, the stationary distributions satisfy:

$$ \bar{\rho}(x) = \frac{\bar{r}}{f(x)}, $$
where the stationary recruitment and the stationary renewal rate of precursor cells are given respectively by

$$ \bar{r} = \frac{\bar{P}_{AMH}}{\int_{x_r}^{x_m} \frac{\omega}{f}}, \quad \alpha(\bar{P}_{AMH}) = 0. $$

The number of (non trivial) stationary size densities is then given by the number of solution of the scalar equation $\alpha(\bar{P}_{AMH}) = 0$.