<![CDATA[Reservoir simulation requires solving large, sparse systems of nonlinear equations, where iterative Krylov subspace solvers such as the conjugate gradient (CG), stabilized conjugate gradient (BiCG-STAB), and generalized minimal residual (GMRES) are widely applied. However, these methods often have limitations in terms of their stability and accuracy in nonlinear systems. This paper introduces a hybrid probabilistic backpropagation neural network (Prob-BPNN) solver that integrates neural-network-based initialization with probabilistic inference to improve robustness. The solver was benchmarked against CG, BiCG-STAB, and GMRES using two synthetic reservoir models with the GMRES solution at a tolerance of 10-10, serving as the reference solution. The results show that Prob-BPNN consistently achieved production profiles closely matching the reference solution, with errors of MAE ≤ 0.066, RMSE ≤ 0.071, MAPE ≤ 2.04%, and R2 ≥ 0.945. In contrast, CG and BiCG-STAB produced unstable and nonphysical results, with errors exceeding 292% and negative R2 values. In terms of computational performance, Prob- BPNN required 9.96 s in Case 1 and 45.90 s in Case 2, compared to 2.85 s and 1.53 s for GMRES, respectively. Although more computationally expensive, Prob-BPNN delivered convergence on the same residual order of magnitude (below 10-3) as GMRES while avoiding the severe instabilities observed in CG and BiCG-STAB. These findings indicate that the Prob-BPNN is preferable in applications where solver robustness and accuracy are critical, even at the expense of a higher execution time. Future research should focus on reducing computational overhead through parallelization and hybridization strategies to enhance the scalability of large-scale reservoir models.]]>
