FIG. 13.

Application of the BINODE to empirical biodegradation data. (a) Time evolution of the state variables $x_1(t)$ (biomass) and $x_2(t)$ (toluene), with units of kg/m³. Black dots indicate empirical observations, and red lines show the corresponding BINODE trajectories. (b–d) Red wireframes show the learned process contributions $w_{11}{\mathrm{N}\mathrm{N}\mathrm{P}}_1\left({\hat{x}}_1,{\hat{x}}_2;{\theta }_1\right)$, $-w_{12}{\mathrm{N}\mathrm{N}\mathrm{P}}_2\left({\hat{x}}_1;{\theta }_2\right)$, and $-w_{21}{\mathrm{N}\mathrm{N}\mathrm{P}}_1\left({\hat{x}}_1,{\hat{x}}_2;{\theta }_1\right)$.