Table 2.

Case study 2. Learned DE models for the BDM process for various numbers of ABM simulations. We fixed $P_p=0.01,P_d=0.005\hspace{0.17em}\mathrm{and}\hspace{0.17em}{P}_m=1$ in each scenario and averaged ABM output over the given value of N ABM simulations ten separate times to investigate the EQL method’s performance in the presence of stochastic ABM fluctuations. The presented learned DE models depicts the final learned equation whose coefficients were averaged over all the learned DE models for each realization of 〈C_ABM(t)〉. The right-most column corresponds to the MSE between successive $\hat{\xi }$ estimates: e.g. for N = 5, we compute $\parallel \hspace{0.17em}{\hat{\xi }}^5-\hspace{0.17em}{\hat{\xi }}^1{\parallel }_2.$

N	mean-field model (MSE)	learned model (MSE)	$\hspace{0.17em}\hat{\xi }$ MSE
1	dC/dt = 0.005C − 0.01C2 (0.0037)	dC/dt = 0.00568C − 0.01953C2 + 0.01624C3 (0.0025)	—
5	dC/dt = 0.005C − 0.01C2 (0.0031)	dC/dt = 0.00482C − 0.01299C2 + 0.00622C3 (0.0012)	0.012
10	dC/dt = 0.005C − 0.01C2 (0.0028)	dC/dt = 0.00472C − 0.01193C2 + 0.00439C3 (0.0008)	0.002
25	dC/dt = 0.005C − 0.01C2 (0.0027)	dC/dt = 0.00453C − 0.01054C2 + 0.00232C3 (0.0005)	0.003