Physics-informed hybrid neural networks with data constraints: A case study on lorenz63

Physics-Informed Neural Networks (PINNs) are a hybrid modeling approach that integrates physical priors into neural networks. By encoding governing equations as soft constraint terms into the loss function, this method accomplishes organic integration of data-driven approaches with physical laws. PINNs demonstrate significant advantages in complex scenarios including data scarcity, high-dimensional spaces, and multi-physics coupling. However, existing research has identified many challenges that PINNs face when tackling with strongly nonlinear dynamical systems. Specifically, during the numerical prediction of complex systems, the nonlinear interactions between system state variables can lead to error coupling and propagation across the variables, which hinders the effective optimization of physical constraint terms. This accumulation of errors may cause inconsistencies between the gradient directions of data constraints and physical constraints, consequently weakening the actual guiding role of physical constraints in the training process. To mitigate these issues, this paper proposes an Observation-Guided Physics-Informed Neural Networks (OG-PINN). By integrating partial observed data into the physics-driven loss function, this approach effectively reconciles competing optimization objectives between physical constraints and data constraints. This mechanism not only enhances the robustness of physical constraints but also significantly reduces the interference of multivariable forecast errors on physical residual. Consequently, the model captures system dynamics more stably. We applied for the OG-PINN to reconstruct the Lorenz 63 system, a typical nonlinear chaotic system, and conducted a systematic comparison with conventional PINN methods. Results demonstrate that the OG-PINN exhibits superior performance over PINN in multiple training sample experiments, especially in long-term forecasting, reducing errors by up to 60% compared to PINN. These experiments not only validate the effectiveness of the OG-PINN in complex dynamical system modeling but also provide technically feasible pathways for physics-constrained deep learning under data-scarce and strongly nonlinear scenarios. Furthermore, we also analyze the impacts of regularization weight (λ) and training sample size (N) on model convergence and generalization capability.