A Model-Constrained Tangent Slope Learning Approach for Dynamical Systems
This work addresses the need for efficient solutions in engineering and science applications, such as digital twins, but appears incremental as it builds on existing neural network and model-constrained methods.
The paper tackles the challenge of achieving real-time accurate solutions for large-scale dynamical systems by introducing the model-constrained tangent slope learning (mcTangent) approach, which combines neural network speed with model constraints and sequential learning to enhance stability and accuracy, as demonstrated through numerical results on equations like the Navier-Stokes equation.
Real-time accurate solutions of large-scale complex dynamical systems are in critical need for control, optimization, uncertainty quantification, and decision-making in practical engineering and science applications, especially digital twin applications. This paper contributes in this direction a model-constrained tangent slope learning (mcTangent) approach. At the heart of mcTangent is the synergy of several desirable strategies: i) a tangent slope learning to take advantage of the neural network speed and the time-accurate nature of the method of lines; ii) a model-constrained approach to encode the neural network tangent slope with the underlying governing equations; iii) sequential learning strategies to promote long-time stability and accuracy; and iv) data randomization approach to implicitly enforce the smoothness of the neural network tangent slope and its likeliness to the truth tangent slope up second order derivatives in order to further enhance the stability and accuracy of mcTangent solutions. Rigorous results are provided to analyze and justify the proposed approach. Several numerical results for the transport equation, viscous Burgers equation, and Navier-Stokes equation are presented to study and demonstrate the robustness and long-time accuracy of the proposed mcTangent learning approach.