No abstract

In this paper, we propose a physical informed neural network approach for designing the electromagnetic metamaterial. The approach can be used to deal with various practical problems such as cloaking, rotators, concentrators, etc. The advantage of this approach is the flexibility that we can deal with not only the continuous parameters but also the piecewise constants. As our best knowledge, there is no other faster and much efficient method to deal with these problems. As a byproduct, we propose a method to solve high frequency Helmholtz equation, which is widely used in physics and engineering. Some benchmark problems have been solved in numerical tests to verify our method.

Partial differential equations (PDEs) on surfaces are ubiquitous in all the nature science. Many traditional mathematical methods has been developed to solve surfaces PDEs. However, almost all of these methods have obvious drawbacks and complicate in general problems. As the fast growth of machine learning area, we show an algorithm by using the physics-informed neural networks (PINNs) to solve surface PDEs. To deal with the surfaces, our algorithm only need a set of points and their corresponding normal, while the traditional methods need a partition or a grid on the surface. This is a big advantage for real computation. A variety of numerical experiments have been shown to verify our algorithm.

Accurate dynamic modeling is critical for autonomous racing vehicles, especially during high-speed and agile maneuvers where precise motion prediction is essential for safety. Traditional parameter estimation methods face limitations such as reliance on initial guesses, labor-intensive fitting procedures, and complex testing setups. On the other hand, purely datadriven machine learning methods struggle to capture inherent physical constraints and typically require large datasets for optimal performance. To address these challenges, this paper introduces the Fine-Tuning Hybrid Dynamics (FTHD) method, which integrates supervised and unsupervised Physics-Informed Neural Networks (PINNs), combining physics-based modeling with data-driven techniques. FTHD fine-tunes a pre-trained Deep Dynamics Model (DDM) using a smaller training dataset, delivering superior performance compared to state-of-the-art methods such as the Deep Pacejka Model (DPM) and outperforming the original DDM. Furthermore, an Extended Kalman Filter (EKF) is embedded within FTHD (EKF-FTHD) to effectively manage noisy real-world data, ensuring accurate denoising while preserving the vehicle's essential physical characteristics. The proposed FTHD framework is validated through scaled simulations using the BayesRace Physics-based Simulator and full-scale realworld experiments from the Indy Autonomous Challenge. Results demonstrate that the hybrid approach significantly improves parameter estimation accuracy, even with reduced data, and outperforms existing models. EKF-FTHD enhances robustness by denoising real-world data while maintaining physical insights, representing a notable advancement in vehicle dynamics modeling for high-speed autonomous racing.

No abstract