Meta-Learning-Based Physics-Informed Neural Network: Numerical Simulations of Initial Value Problems of Nonlinear Dynamical Systems Without Labeled Data and Correlation Analyses (Journal of Nonlinear Modeling and Analysis, 2024)
There are several main challenges in solving nonlinear differentialequations with artificial neural networks (ANNs), such as a nonlinear system’ssensitivity to its initial values, discretization, and strategies for incorporatingphysics-based information into ANNs. As for the first issue, this paper addresses the initial value problems of nonlinear dynamical systems (a Duffing oscillator and a Burger’s equation), which cause large global truncation errors insub-domains with a significant reduction in the influence of initial constraints,using meta-learning-based physics-informed neural networks (MPINNs). TheMPINNs with dual learners outperform physics-informed neural networks witha single learner (no fine reinitialization capability). As a result, the former approach improves solution convergence by 98.83% in the sub-time domain (III)of a Duffing oscillator, and by 85.89% at $t = 45$ in a Burger’s equation problem, compared to the latter one. Model accuracy is highly dependent on theadaptability of the initial parameters in the first hidden layers of the meta-models. From correlation analyses, it is obvious that the parameters becomeless (the Duffing oscillator) or more (the Burger’s equation) correlated during fine reinitialization, as the update manner differs or is similar to the oneused in pre-initialization. In the first example, the MPINN achieves both themitigation of model sensitivity to its output and the improvement of modelaccuracy. Conversely, the second example shows that the proposed approachis not enough to solve both issues simultaneously, as increased model sensitivity to its output leads to higher model accuracy. The application of transferlearning reduces the number of iterative pre-meta-trainings.
Keywords: Physics-informed neural network; meta-learning; fine reinitialization; transfer learning; initial value problem; nonlinear dynamical systems; correlation analyses.