Innovative Neural Network Optimization for Dynamic Systems Under Differential Equation Constraints

This study proposes a dynamic modeling and control framework based on physical constraint neural network for dynamic system modeling and optimization under differential equation constraints. The algorithm is based on a differential neural network. From a theoretical perspective, by introducing the time dimension and state continuity of system evolution, the differential equation form in traditional dynamic modelling and the nonlinear adaptation ability in deep learning are integrated. This model can achieve high-precision prediction and modelling of complex systems. This operation improves the robustness and generalisation performance of the model. At the same time, this study introduces a structural similarity modelling mechanism. This mechanism completes the consistency identification and parameter migration of the dynamic response behaviour by constructing a system-like parameter space through highdimensional embedding and geometric constraints. In terms of specific applications, this study built an enhanced PID system based on neural network optimization. The system was used to verify the adaptability and stability of the method in multivariable, nonlinear, and time-varying system control. The experiment was compared in multiple dynamic system tasks. The results show that the proposed method is superior to the existing mainstream models in terms of prediction accuracy, convergence speed and control response time.