# Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook
## 5 Conclusions and Future Work
### Outlook
#### 5.2.1 Physics Representation  
What physics model is selected depends on the data fidelity, model fidelity, and available computational resources.  
While there are a significant amount of models to describe traffic dynamics on micro- [100; 101], meso- [102; 103], and macro-scales [42; 43; 46; 47], the future is in the discovery of a multiscale traffic model (i.e., a mapping from multiscale measurements to traffic states on different scales) that collectively infers traffic states with various measurements. Analytical multiscale models have been thoroughly studied in fields such as biological [13] and materials science [104], but remain under-exploited in transportation modeling. The drivers of developing a multiscale traffic model are two-fold: (1) Sensor data are at different scales. The collected traffic data include high-resolution individual trajectories and low-resolution aggregate information, both at various spatiotemporal granularity. Traffic measurements on different scales essentially measure the same system and phenomenon. Accordingly, a multiscale model, capable of integrating measurements of various scales, could characterize traffic dynamics more accurately with a smaller dataset [105]. (2) Traffic optimization and management strategies need to rely on diverse models and data of different scales. Individual traces are normally modeled using ODEs while aggregate traffic patterns are modeled with PDEs. An integrated ODE-PDE physics model can potentially accommodate these measurements at both the micro- and macro-scales [106]. Multiscale traffic models could also help to reduce model complexity and speed up simulation for real-time applications.  
A multiscale traffic model, however, presents computational challenges due to the curse of dimensionality (i.e., high-dimensional input-output pairs). It is thus important to utilize reduced modeling techniques and multi-fidelity modeling [107; 108; 109].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**PUNN-PICG Topology** & **PUNN Hierarchy** & **Shared** & **Separate** \\ \hline Sequential & [25; 27; 28; 30; 65; 91; 96; 97; 98] & [26; 31; 65; 78; 90; 95; 99] \\ \hline Parallel & [9; 10] & - \\ \hline \hline \end{tabular} Note: PUNN-PICG topology refers to the layout between PUNN and PICG, which can be either “sequential” (i.e., the output of PUNN is the input into PICG, so PICG follows PUNN) or “parallel” (i.e., PUNN and PICG output predictions side-by-side to compute a loss). PUNN hierarchy refers to the layout of the NNs that are used to predict \(\rho\) and \(u\). “Shared” means that only one NN is used to output both \(\rho\) and \(u\), while “separate” means that two NNs are used to output \(\rho\) and \(u\), respectively.  
\end{table}
Table 5: Configuration of physics in the physics-based component.  
An emerging direction to improve physics representation is to consider proxy models. For example, symbolic regression has been demonstrated to learn relatively simple physical forms to describe complex physical systems [110; 111]. Traffic engineers have spent decades discovering various physical laws to describe human behavior models, such as car-following, lane-change, and other driving scenarios and tasks. Is it possible to develop a systematic method to select mathematical models? For example, the selection of a model can be reformulated as finding an optimal path over the PICG from inputs to outputs [112]. Accordingly, model selection is converted to an optimization problem. Machine learning methods, such as neural architecture search [113] and automatic modularization of network architecture [114], could enable the automatic architecture design of PICGs and PUNNs.