Iacca, G. | bioRxiv Science

bioRxiv2025

A novel approach to quantify out-of-distribution uncertainty in Neural and Universal Differential Equations

Dynamical systems play a central role across the quantitative sciences, offering a powerful mathematical framework to describe, analyze, and predict the evolution of complex processes over time. In systems biology, dynamical systems provide a foundation for modeling and predicting the intricate behaviors of biological systems. Recent advances in data-driven approaches, such as Neural Ordinary Differential Equations (NODEs) and Universal Differential Equations (UDEs), have enabled the development of models that are either fully or partially data-driven. Integrating data-driven components into dynamical systems amplifies the challenge of generalization beyond training data, highlighting the need for robust methods to quantify uncertainty in out-of-distribution (OOD) scenarios--i.e., conditions not encountered during training. In this work, we investigate the reliability of uncertainty quantification (UQ) based on ensembles of models in the reconstruction of dynamical systems. We show that standard ensembles (i.e., models trained independently with different random initializations) risk producing overconfident predictions in previously unseen scenarios, as the models in the ensemble tend to exhibit similar behaviors. To address this issue, we propose a novel ensemble construction method for NODEs and UDEs that fosters diversity in the reconstructed vector field across models within specific regions of the state space, while maintaining explicit control over the fit on the training set. We first evaluate our method on numerical test cases derived from three models commonly used as benchmarks for data-driven reconstruction of dynamical systems: the Lotka-Volterra model, the damped oscillator, and the Lorenz system. We then apply the method to a biologically motivated model of cell apoptosis, considering more realistic conditions such as partial observability of the system outputs and noise in the training dataset. Overall, our results show that the proposed method improves the reliability of UQ in previously unseen scenarios compared with standard ensembles, especially where the latter exhibit overconfidence.

systems biology↗

bioRxiv2024

Robust parameter estimation and identifiability analysis with Hybrid Neural Ordinary Differential Equations in Computational Biology

Parameter estimation is one of the central problems in computational modeling of biological systems. Typically, scientists must fully specify the mathematical structure of the model, often expressed as a system of ordinary differential equations, to estimate the parameters. This process poses significant challenges due to the necessity for a detailed understanding of the underlying biological mechanisms. In this paper, we present an approach for estimating model parameters and assessing their identifiability in situations where only partial knowledge of the system structure is available. The partially known model is extended into a system of Hybrid Neural Ordinary Differential Equations, which captures the unknown portions of the system using neural networks. Integrating neural networks into the model structure introduces two primary challenges for parameter estimation: the need to globally explore the search space while employing gradient-based optimization, and the assessment of parameter identifiability, which may be hindered by the expressive nature of neural networks. To overcome the first issue, we treat biological parameters as hyperparameters in the extended model, exploring the parameter search space during hyperparameter tuning. The second issue is then addressed by an a posteriori analysis of parameter identifiability, computed by introducing a variant of a well-established approach for mechanistic models. These two components are integrated into an end-to-end pipeline that is thoroughly described in the paper. We assess the effectiveness of the proposed workflow on test cases derived from three different benchmark models. These test cases have been designed to mimic real-world conditions, including the presence of noise in the training data and various levels of data availability for the system variables. Author summaryParameter estimation is a central challenge in modeling biological systems. Typically, scientists calibrate the parameters by aligning model predictions with measured data once the model structure is defined. Our paper introduces a workflow that leverages the integration between mechanistic modeling and machine learning to estimate model parameters when the model structure is not fully known. We focus mainly on analyzing the identifiability of the model parameters, which measures how confident we can be in the parameter estimates given the available experimental data and partial mechanistic understanding of the system. We assessed the effectiveness of our approach in various in silico scenarios. Our workflow represents a first step to adapting traditional methods used in fully mechanistic models to the scenario of hybrid modeling.

systems biology↗