A Model-Constrained Tangent Slope Learning Approach for Dynamical Systems

TL;DR

This paper introduces mcTangent, a neural network-based method that combines model constraints, tangent slope learning, and data randomization to achieve accurate, stable, and efficient solutions for large-scale dynamical systems in real time.

Contribution

The paper proposes a novel model-constrained tangent slope learning approach that enhances neural network solutions for dynamical systems by integrating governing equations and stability strategies.

Findings

Demonstrates robustness on transport, Burgers, and Navier-Stokes equations.

Achieves long-time stability and accuracy in numerical experiments.

Outperforms traditional methods in efficiency and precision.

Abstract

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.