Overview

Physics-Informed Machine Learning (PIML) is an emerging paradigm that integrates physical laws (often expressed as differential equations) into machine learning models to improve their accuracy, generalizability, and interpretability, especially in domains where data may be scarce or noisy. Instead of training a machine learning model purely on data, PIML incorporates prior knowledge from physics (such as conservation laws, constitutive relations, or governing equations like LWR) directly into the learning process. This is done in one or more of the following ways: 

1. Loss Function Augmentation: Adding physical residuals (like the mismatch from a differential equation) to the loss function. 

2. Soft Constraints: Penalizing violations of physical laws during training. 

3. Hard Constraints: Designing the model architecture to inherently satisfy physical constraints.

Physics-Informed Neural Networks (PINNs) and Physics-Regularized Gaussian Processes (PRGPs) are two representative approaches in the emerging field of PIML. 

• PINNs utilize deep neural networks as function approximators while incorporating physical constraints directly into the loss function. Instead of relying solely on empirical data, the training process of a PINN minimizes a composite loss that combines data fidelity with the residuals of known physical equations. This approach enables the model to respect conservation laws or system dynamics even in regions with limited or no observations. 

• PRGPs bring physical knowledge into a probabilistic learning framework. Gaussian Processes (GPs) are non-parametric models that offer interpretable predictions with quantifiable uncertainty, making them particularly appealing for applications where safety and reliability are critical. PRGPs incorporate physics as soft constraints or regularization terms that guide the learning process, effectively balancing data fit and physical consistency. These models are well-suited for problems where the underlying processes are smooth and governed by known equations. Unlike PINNs, PRGPs inherently provide uncertainty estimates, which are valuable in risk-sensitive decision-making and model validation.

PINNs and PRGPs have recently emerged as powerful tools for enhancing traffic flow modeling. In PINNs, deep neural networks are trained not only to fit traffic data (e.g., speed or density) but also to satisfy governing equations like the Lighthill-Whitham-Richards (LWR) model. By embedding these physical laws into the loss function, PINNs can reconstruct traffic states across space and time even with limited sensor coverage. This makes them effective for traffic state estimation and anomaly detection in freeway networks. PRGPs, on the other hand, embed traffic flow theory as regularization terms in Gaussian Process models. These models offer both predictions and uncertainty estimates, which are especially valuable for travel time estimation, fundamental diagram learning, and risk-aware traffic control. Their probabilistic nature and smooth interpolation capabilities make them well-suited for low-dimensional traffic modeling with sparse data.

Education Module

The education module of the PIMLAP is designed to provide a comprehensive, hands-on learning experience for students, educators, and early-career researchers. It offers structured tutorials, annotated source code, and step-by-step examples that bridge theoretical foundations with practical implementation.

To support active learning, the module includes a series of mini-projects with real or synthetic datasets, enabling users to experiment with PIML models, such as Physics-Informed Neural Networks (PINNs) and Physics-Regularized Gaussian Processes (PRGPs), in diverse application domains, including transportation, physics, and engineering systems. These projects are crafted to build core competencies in problem formulation, model development, and evaluation, while emphasizing the integration of physical laws into machine learning workflows.

By lowering the barrier to entry for PIML, this module serves as a foundational resource for teaching and self-guided study, fostering a deeper understanding of how to develop interpretable, data-efficient, and physically consistent machine learning models.

Enter the Education Module. 

Research Module

TBD

Enter the Research Module.