Physics-Embedded Networks: Improving Convergence and Precision of Physics-Informed Neural Networks for Real-Time Applications

This article introduces the physics-embedded neural network (PENN), an enhanced physics-informed neural network (PINN) architecture tailored for visual servoing applications of multirotors. Classical PINNs, while interpretable and data-efficient due to their incorporation of physical laws in the training loss function, often suffer from poor convergence and sensitivity to network initialization and activation functions (AFs). To overcome these challenges, this work proposes two improved architectures: the layer-wise PENN (L-PENN) and the neuron-wise PENN (N-PENN). These architectures embed nominal physical dynamics directly into the structure of the network, thereby improving both training efficiency and predictive accuracy. A spectral analysis of the Hessian matrix is conducted to rigorously demonstrate the enhanced convergence behavior of the proposed architectures compared to traditional PINNs. The proposed methods are experimentally validated on a visual servoing task using a multirotor platform, with performance evaluated in terms of tracking performance and training time. The results are also benchmarked against existing literature, confirming that both L-PENN and N-PENN significantly outperform classical PINNs and other learning-based control strategies. The article concludes by outlining selection criteria for choosing between the two architectures based on specific characteristics of the application.