Predicting nonlinear dynamic systems by causal physics-informed neural networks with ResNet blocks

With the continuous advancement of data computational science, the prediction of nonlinear systems has provided effective support for investigating complex problems in the field of natural sciences. Physics-Informed Neural Networks (PINNs) are playing an increasingly prominent role in nonlinear system prediction. Although PINNs have been widely applied across various engineering domains, their utilization in chaotic system prediction remains notably scarce. This paper proposes a novel causal PINNs framework integrated with ResNet blocks. On the one hand, the framework incorporates temporal weighting into the residual loss, utilizing maximum temporal weight as the training termination criterion. Additionally, an annealing strategy is adopted to adaptively adjust the causal parameters, ensuring that the model adheres to physical causality constraints throughout the training process. On the other hand, the framework employs a ResNet-block-based network, which transforms identity mappings into residual mappings. This architectural design significantly enhances training stability when utilizing deep networks. To validate the performance of the proposed method, numerical experiments are conducted on the Lorenz system, Dadras system, and Kuramoto-Sivashinsky equation. The results demonstrate that the causal PINNs with ResNet blocks significantly outperform conventional PINNs in predicting chaotic systems.