The development of deep and ultra-deep underground resources is often constrained by high drilling costs and low operational efficiency, among which the rate of penetration (ROP) serving as a critical indicator of drilling performance. Accurate ROP prediction is therefore essential for drilling optimization and cost reduction. However, existing data-driven models typically neglect domain-specific drilling knowledge and physical constraints, leading to poor generalization capability and physically inconsistent predictions, which limits their reliability in practical engineering applications. To address these challenges, this study develops a physically informed and robust ROP prediction framework by integrating drilling knowledge into deep learning models. Based on field data from seven ultra-deep wells in the Tarim Oilfield, China, a drilling knowledge constrained deep neural network (DCDNN) is proposed, in which both soft constraints (correlation relationships among drilling parameters) and hard constraints (boundary condition constraints) are explicitly embedded into the loss function and network structure. This design enhances the physical consistency and robustness of the model, particularly under boundary and extreme operating conditions. Experimental results demonstrate that the proposed DCDNN model improves ROP prediction accuracy by 12.17% compared with conventional neural networks, while maintaining stable performance under constraint conditions. Furthermore, a multi-objective optimization framework is established to simultaneously maximize ROP and minimize bit mechanical specific energy (BMSE). Under simulated optimized drilling parameters, the model projects a 51.09% increase in ROP in the Paleogene Kumugeliemu Formation, indicating strong optimization potential for drilling strategies characterized by high weight on bit and medium rotary speed. The novelty of this work lies in the systematic fusion of drilling mechanism knowledge with deep neural networks through unified soft–hard constraint incorporation, enabling both high prediction accuracy and physical interpretability. Despite the promising results, challenges remain in terms of data completeness, model transferability across different geological settings, and real-time deployment. Future work will focus on extending the proposed framework to multi-block datasets and integrating it into closed-loop intelligent drilling optimization systems.