Chaos-inspired active learning for physics-informed neural networks to assess the reliability of multi-state systems

Abstract

Multi-state systems (MSS) are widely used for modeling the behavior of engineering applications, where the system and its components can have more than two distinct states. Physics-Informed Neural Networks (PINNs) offer a viable solution for characterizing the dynamic state evolution of MSS. However, existing methods predominantly rely on uniformly sampled collocation points across the problem domain when training PINNs. Although some residual-based active learning methods exist, they are inherently static and local, and often fail to capture a crucial aspect of PINN training: identification and accurate modeling of the “critical transition regions” within the problem domain. To address this fundamental challenge, we treat PINN as a dynamic system and introduce a novel active learning method grounded in chaos theory to identify regions within the problem domain that are highly sensitive to initial conditions. Specifically, our method quantifies the degree of chaos at candidate collocation points by introducing small perturbations and using PINN’s forward propagation to simulate the dynamic evolution of both the original and perturbed collocation points. Collocation points that exhibit pronounced chaotic behavior—where evolutionary trajectories diverge rapidly following perturbation—are identified as the system’s most unstable and valuable regions for PINN training. By prioritizing these dynamically unstable points, our method directs PINN to focus its learning on accurately delineating the boundaries of state transitions, thereby significantly enhancing the accuracy of reliability analysis. Experimental results on multiple benchmark partial differential equations (PDEs) and several MSSs demonstrate that, compared to other PINN learning schemes, our method shows superior accuracy and computational efficiency in MSS reliability assessment.