A hybrid MVDA-neural network model with auxiliary learning for pharmaceutical quality by design under small-sample constraints

Quality by Design (QbD) is a systematic approach that builds quality into pharmaceutical products from early development. Within this framework, multivariate data analysis (MVDA), implemented here as a second-order polynomial response-surface model, has long served as the core modeling technique. However, this fixed functional form is limited in capturing complex nonlinear behavior. Artificial neural networks (ANNs) offer a nonlinear alternative. In the small-sample regime typical of DoE-based pharmaceutical development, however, ANNs are prone to overfitting, and generalization to independent test data is not reliably ensured. To address both limitations, the Autoregressive Auxiliary Layer Network (AXLN) and a Hybrid MVDA-AXLN Model are proposed. AXLN employs a decoupled training-inference scheme in which inter-response correlations are exploited as an auxiliary loss during training. The Hybrid Model learns AXLN-based residuals on top of an MVDA backbone. Four models were compared on two pharmaceutical DoE datasets (Dataset A, n = 32; Dataset B, n = 15). The Hybrid Model achieved the lowest test MAE across all responses (0.551-2.514), significantly outperforming MVDA, AXLN, and ANN (p < 0.05). In the design space comparison, the Hybrid Model correctly excluded an infeasible point that MVDA had erroneously included. The resulting boundary was more reliable from a quality-assurance standpoint. These results show that the proposed Hybrid MVDA-AXLN Model improves both predictive precision and design space reliability within the pharmaceutical QbD framework.