Species transport models typically combine partial differential equations (PDEs) with relations from hindered transport theory to quantify electromigrative, convective, and diffusive transport through complex nanoporous systems; however, these formulations are frequently substantial simplifications of the governing dynamics, leading to the poor generalization performance of PDE-based models. Given the growing interest in deep learning methods for the physical sciences, we develop a machine learning-based approach to characterize ion transport across nanoporous membranes. Our proposed framework centers around attention-enhanced neural differential equations that incorporate electroneutrality-based inductive biases to improve generalization performance relative to conventional PDE-based methods. In addition, we study the role of the attention mechanism in illuminating physically-meaningful ion-pairing relationships across diverse mixture compositions. Further, we investigate the importance of pre-training on simulated data from PDE-based models, as well as the performance benefits from hard vs. soft inductive biases. Our results indicate that physics-informed deep learning solutions can outperform their classical PDE-based counterparts and provide promising avenues for modelling complex transport phenomena across diverse applications.

This research contributes to the advancement of traffic state estimation methods by leveraging the benefits of the nonlocal LWR model within a physics-informed deep learning framework. The classical LWR model, while useful, falls short of accurately representing real-world traffic flows. The nonlocal LWR model addresses this limitation by considering the speed as a weighted mean of the downstream traffic density. In this paper, we propose a novel PIDL framework that incorporates the nonlocal LWR model. We introduce both fixed-length and variable-length kernels and develop the required mathematics. The proposed PIDL framework undergoes a comprehensive evaluation, including various convolutional kernels and look-ahead windows, using data from the NGSIM and CitySim datasets. The results demonstrate improvements over the baseline PIDL approach using the local LWR model. The findings highlight the potential of the proposed approach to enhance the accuracy and reliability of traffic state estimation, enabling more effective traffic management strategies.

Development in deep learning (DL) neural networks [1] [2] benefits a wide range of engineering applications. The learning capability of a DL neural network helps practitioners in numerous fields such as transportation engineering and has been widely adopted in projects on object detection, autonomous driving, and estimations of system conditions. Traffic state estimation (TSE) is a crucial task for transportation planners in understanding travel demand and road infrastructure's level of service (LOS). Due to the cost constraints associated with installing sensing devices along freeways and arterial roads, traffic observations can solely be obtained at predetermined locations, leaving areas where traffic conditions are unperceived. Traffic states such as vehicle speed v, density ρ, and flow f in the unobserved regions need to be approximated by using the collected measurements of traffic at sparse locations [3]. Take loop detectors as an example: the signal indicating the passage of a vehicle can only be obtained at predetermined locations where the electrically conducting loops are planted. The number of detectors deployed considerably affects the quantity of traffic data collected from a highway system [4]. The task of TSE is further impeded by issues such as the measurement noise in detectors and data loss due to sensor malfunctions [5] [6] [7]. The inaccuracy in recorded traffic data and the limited data resolution during signal processing contribute to the challenges in precise TSE [8] [9]. Figure 1 illustrates the process of traffic state data acquisition: sensing devices collect information such as speed v and headway T at designated locations and broadcast the information to central cloud infrastructure for data processing and storage.

Numerical methods such as Finite element method [Hug12], Isogeomteric analysis [CHB09], and mesh-free methods [LJZ95, BLG94] are few of the conventional approaches employed in solving the Partial Differential Equations (PDEs) involved in computational solid mechanics problems. However, the ever-increasing sophistication of material models by incorporating more complex physics, such as modeling size-effect [FMAH94, AA20b] or dislocation density evolution [AZA20, Aro19, AA20a, AAA22, JABG20], or advanced materials such as composites and multicomponent alloys with spatially-varying material properties (heterogeneity) and direction dependent behavior (anisotropy) is bringing these numerical solvers to their limits. Hence, it is becoming a formidable task to perform simulations that can resolve the complex physical phenomena occurring at small spatial and temporal scales and accurately predict the macro-scale behavior of materials. Therefore, a cost-effective physicsbased surrogate model that allows the researchers to perform simulations on a coarse mesh without sacrificing accuracy will be highly beneficial for many reasons. First, researchers can choose to run their simulations at a lower resolution (online stage) and later reconstruct the solution back to the target resolution (offline stage). This will significantly reduce the computational expense during the online stage, thus accelerating the process of scientific investigation and discovery. Second, the surrogate model based on data super-resolution can also be used to enhance outputs from experimental techniques for full-field displacement and strain measurement such as Digital Image Correlation (DIC) which would allow researchers to generate and store a small fraction of data. Recent advances in Deep Learning (DL) and Physics-Informed Neural Networks (PINN) [RPK17, RPK19] make it a promising tool to tackle this computational challenge.

Pumping carbon dioxide underground may help combat the warming of the atmosphere but finding appropriate underground sites that could safely serve as reservoirs can be complicated. To address this complexity, a Penn State-led research team combined an artificial intelligence technique with an understanding of physics to develop an efficient, cost-effective predictive modeling approach. They published their results in the Journal of Contaminant Hydrology. "Storing carbon dioxide underground is one environmentally friendly way to reduce the amount of the gas in the atmosphere," said Parisa Shokouhi, associate professor of engineering science and mechanics. "But the geological structure can be unfavorable to carbon dioxide injection. For example, if pressure surpasses a certain limit, there can be fractures, gas leakage and earthquakes, and if you over-inject with too much gas, you can have similar issues."