Ghazal Farhani

Ghazal Farhani, Alexander Kazachek, Boyu Wang

Physics-informed neural network (PINN) algorithms have shown promising results in solving a wide range of problems involving partial differential equations (PDEs). However, they often fail to converge to desirable solutions when the target function contains high-frequency features, due to a phenomenon known as spectral bias. In the present work, we exploit neural tangent kernels (NTKs) to investigate the training dynamics of PINNs evolving under stochastic gradient descent with momentum (SGDM). This demonstrates SGDM significantly reduces the effect of spectral bias. We have also examined why training a model via the Adam optimizer can accelerate the convergence while reducing the spectral bias. Moreover, our numerical experiments have confirmed that wide-enough networks using SGDM still converge to desirable solutions, even in the presence of high-frequency features. In fact, we show that the width of a network plays a critical role in convergence.

Nima Hosseini Dashtbayaz, Ghazal Farhani, Boyu Wang et al.

The residual loss in Physics-Informed Neural Networks (PINNs) alters the simple recursive relation of layers in a feed-forward neural network by applying a differential operator, resulting in a loss landscape that is inherently different from those of common supervised problems. Therefore, relying on the existing theory leads to unjustified design choices and suboptimal performance. In this work, we analyze the residual loss by studying its characteristics at critical points to find the conditions that result in effective training of PINNs. Specifically, we first show that under certain conditions, the residual loss of PINNs can be globally minimized by a wide neural network. Furthermore, our analysis also reveals that an activation function with well-behaved high-order derivatives plays a crucial role in minimizing the residual loss. In particular, to solve a $k$-th order PDE, the $k$-th derivative of the activation function should be bijective. The established theory paves the way for designing and choosing effective activation functions for PINNs and explains why periodic activations have shown promising performance in certain cases. Finally, we verify our findings by conducting a set of experiments on several PDEs. Our code is publicly available at https://github.com/nimahsn/pinns_tf2.

Ghazal Farhani, Taufiq Rahman, Syed Mostaquim Ali et al.

Because 3D structure of a roadway environment can be characterized directly by a Light Detection and Ranging (LiDAR) sensors, they can be used to obtain exceptional situational awareness for assitive and autonomous driving systems. Although LiDARs demonstrate good performance in clean and clear weather conditions, their performance significantly deteriorates in adverse weather conditions such as those involving atmospheric precipitation. This may render perception capabilities of autonomous systems that use LiDAR data in learning based models to perform object detection and ranging ineffective. While efforts have been made to enhance the accuracy of these models, the extent of signal degradation under various weather conditions remains largely not quantified. In this study, we focus on the performance of an automotive grade LiDAR in snowy conditions in order to develop a physics-based model that examines failure modes of a LiDAR sensor. Specifically, we investigated how the LiDAR signal attenuates with different snowfall rates and how snow particles near the source serve as small but efficient reflectors. Utilizing our model, we transform data from clear conditions to simulate snowy scenarios, enabling a comparison of our synthetic data with actual snowy conditions. Furthermore, we employ this synthetic data, representative of different snowfall rates, to explore the impact on a pre-trained object detection model, assessing its performance under varying levels of snowfall

Ghazal Farhani, Taufiq Rahman, Dominique Charlebois

Rainy weather significantly increases the risk of road accidents due to reduced visibility and vehicle traction. Understanding how experienced drivers adapt their visual perception through gaze behavior under such conditions is critical for designing robust driver monitoring systems (DMS) and for informing advanced driver assistance systems (ADAS). This case study investigates the eye gaze behavior of a driver operating the same highway route under both clear and rainy conditions. To this end, gaze behavior was analyzed by a two-step clustering approach: first, clustering gaze points within 10-second intervals, and then aggregating cluster centroids into meta-clusters. This, along with Markov transition matrices and metrics such as fixation duration, gaze elevation, and azimuth distributions, reveals meaningful behavioral shifts. While the overall gaze behavior focused on the road with occasional mirror checks remains consistent, rainy conditions lead to more frequent dashboard glances, longer fixation durations, and higher gaze elevation, indicating increased cognitive focus. These findings offer valuable insight into visual attention patterns under adverse conditions and highlight the potential of leveraging gaze modeling to aid in the design of more robust ADAS and DMS.