Quality indicators are critical for process monitoring and safety assessment in industrial chemical processes, yet their online measurement remains difficult or cost-prohibitive. This work presents a transparent and accurate soft sensor development framework termed the Dynamic Symbolic Regression Network (DSRN), which explicitly reconstructs symbolic relationships between easy-to-measure process variables and target quality indicators. Unlike mainstream black-box deep learning approaches, the proposed method leverages symbolic regression techniques to derive interpretable mathematical expressions. A genetic algorithm is employed to search for multi-layer optimal symbolic structures, while time-lag terms are incorporated to capture dynamic features, thereby revealing interpretable data-driven relationships within the chemical process. Furthermore, a fully objective feature selection strategy is introduced to determine the optimal subset of auxiliary variables by evaluating the uncertainty of the quality indicator via information entropy, which reduces the dimensionality of the DSRN input space and focuses the symbolic search on informative process variables. The effectiveness of the proposed approach is first demonstrated on a numerical example, where it accurately recovers the true variable relationships and achieves superior prediction performance. A case study on the Dow data challenge problem further validates that DSRN attains competitive prediction accuracy compared with advanced deep learning models, while offering explicit mathematical equations that provide valuable interpretability for operators. These results highlight the great potential of DSRN for practical process monitoring and control applications.