Physics-informed machine learning (PIML) has been proven to be a powerful paradigm that can achieve the high predictive accuracy of modern machine learning while preserving the physical interpretability offered by mechanistic models, which is particularly appealing in transportation applications. In general, a PIML model contains two key components, as illustrated in the left-hand figure: a data-driven part and a physics-driven part. 

The data-driven part is essentially a standard machine learning model. In transportation applications, it is often implemented as a supervised learning model whose goal is to approximate one or several functions over space and time. In traffic flow theory, key variables such as density, flow, and speed vary with both location x and time t. Taking macroscopic traffic flow PIML as an example, the data commonly come from detector data, probe data, or measurements converted from trajectory data. The inputs are position-time pairs, and the outputs are traffic state observations such as density-speed pairs. Therefore, the data-driven part aims to learn continuous mappings from position-time pairs to density-speed pairs, so that density and speed can be predicted at any location x and time t within the domain. In contrast, the physics-driven part encodes domain knowledge, which is often expressed as partial differential equations in macroscopic traffic models, although non PDE physical models can also be embedded, for example in microscopic traffic modeling. As shown in the left figure, automatic differentiation can be used to construct the residual of the classic LWR conservation law within the physics-driven part. For low order models such as LWR, the model parameters typically do not require calibration using real world data. However, for higher order macroscopic models or microscopic car following models, reliable parameter calibration is a prerequisite before enforcing physics constraints, and this step often plays a decisive role in determining whether an effective and successful PIML model can be built.