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    Meta-Learning-Based Physics-Informed Neural Network: Numerical Simulations of Initial Value Problems of Nonlinear Dynamical Systems without Labeled Data and Correlation Analyses
    (2024-06-01)
    Duanyai, Worrawat
    ;
    Song, Weon Keun
    ;
    Chitthamlerd, Thanadol
    ;
    Kumar, Girish
    There are several main challenges in solving nonlinear differential equations with artificial neural networks (ANNs), such as a nonlinear system’s sensitivity to its initial values, discretization, and strategies for incorporating physics-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 in sub-domains with a significant reduction in the influence of initial constraints, using meta-learning-based physics-informed neural networks (MPINNs). The MPINNs with dual learners outperform physics-informed neural networks with a 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 the adaptability of the initial parameters in the first hidden layers of the meta-models. From correlation analyses, it is obvious that the parameters become less (the Duffing oscillator) or more (the Burger’s equation) correlated during fine reinitialization, as the update manner differs or is similar to the one used in pre-initialization. In the first example, the MPINN achieves both the mitigation of model sensitivity to its output and the improvement of model accuracy. Conversely, the second example shows that the proposed approach is not enough to solve both issues simultaneously, as increased model sensitivity to its output leads to higher model accuracy. The application of transfer learning reduces the number of iterative pre-meta-trainings.
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    Event-triggered model reference adaptive control system design for SISO plants using meta-learning-based physics-informed neural networks without labeled data and transfer learning
    (2024-04-01)
    Duanyai, Worrawat
    ;
    Song, Weon Keun
    ;
    Konghuayrob, Poom
    ;
    Parnichkun, Manukid
    This paper examines the controllability of a novel Lyapunov-based model reference adaptive control (MRAC) system designed with a meta-learning-based physics-informed neural network (MLPINN) for linear and nonlinear single-input and single-output (SISO) plants without labeled data (MLPINN-MRAC system). It is devised with the benefits of several techniques: the integration of the identification process in a training mode into an online control mode (straightforward design); no labeled data generation for online identification by a physics-informed neural network; the prevention of degradation in tracking performance by meta-learning, as the system triggers a meta-learning process only when an error threshold detects the deterioration (high efficiency and the reduction of computation cost); quick adaptation to new inputs and an updated control input for each sub-time span by transfer learning. It is worth noting that the frequency of the meta-learning event detection significantly affects the step response stability of the nonlinear plant. To achieve a better quality of the stabilization, more frequent event detection is necessary for both beginning and end intervals in control. Sixteen triggering events are enough to shape the acceptable step response of the nonlinear plant, and 44 triggering events achieve the plant's desired step response with its minor modeling error. While, as for the linear plant, a single triggering event is sufficient to attain its tolerable step response and modeling error, implying that intensive event detection is not critical in the identification. It is obvious that the MLPINN-MRAC system functions well and is more beneficial and efficient for the nonlinear plant.