Dynamic modeling of in vitro ruminal fermentation of agricultural residues using mechanistic and neural network–based approaches under limited experimental information

Abstract The in vitro ruminal fermentation of agricultural residues is commonly assessed through measurements of substrate degradation, volatile fatty acid (VFA) production, and gas generation. However, modeling this process remains challenging due to the limited availability of experimentally measurable variables and the intrinsic complexity of the rumen ecosystem. This study proposes and compares three dynamic modeling approaches to describe in vitro ruminal fermentation using only experimentally accessible data from control treatments, focusing on maize, sorghum, and oat residues. A phenomenological mechanistic simplified model based on Monod-type kinetics, a feedforward neural network (FNN), and a neural network differential equation model (NNODE) were developed, trained, and evaluated. Experimental pressure measurements were converted into gas concentrations using the Soave-Redlich-Kwong equation of state, ensuring consistency across state variables. Savitzky-Golay filtering was applied to smooth experimental data and estimate time derivatives required for NNODE training. Model performance was assessed using mean squared error (MSE), mean absolute error (MAE), and root mean squared error (RMSE) for substrate, VFAs, and gas concentrations. While the NNODE generally exhibited lower prediction errors and improved dynamic consistency (particularly for substrate consumption and gas production) statistical analysis using one-way ANOVA and Tukey’s HSD test revealed no significant differences in global RMSE among the three models ( p > 0.05). Sensitivity analyses with respect to in vitro dry matter digestibility (IVDMD), neutral detergent fiber (NDF), and initial substrate concentration ( S 0 ) highlighted the superior dynamic coherence of the NNODE, although limitations were observed under low substrate concentrations and early VFA production phases. Overall, this study demonstrates that meaningful dynamic models of in vitro ruminal fermentation can be developed under severe data constraints. The results highlight the potential of neural differential equation models as an intermediate approach between mechanistic and data-driven models, while emphasizing the importance of dynamic consistency and careful data preprocessing in complex biological systems.