Non-Dimensionalizing a Traffic Flow PDE for Physics Informed Neural Network Issue

Question

I'm working on analyzing a traffic flow model described by the following partial differential equation (PDE):

$V_{\max} \left(1 - \frac{2\rho}{\rho_{\max}}\right) \frac{\partial \rho}{\partial x} + \frac{\partial \rho}{\partial t} = 0$

here:

• p(x,t) represents the vehicle density as a function of position
• $V_{max}$ represents the maximum velocity in km/h
• $\rho_{max}$ represents the maximum density in veh/km

My goal is to non-dimensionalize this PDE to facilitate its application within a Physics Informed Neural Network (PINN), but since I'm new to this area, I am seeking validation or advice on my methodology. I have introduced the following dimensionless variables:

1. $\hat{x} = \frac{x}{L}$
2. $\hat{t} = \frac{t}{T}$, where $T = \frac{L}{V_{max}}$
3. $\hat{\rho} = \frac{\rho}{\rho_{max}}$

Substituting these into the PDE yields: $(1-2\hat{\rho}) \frac{\partial{\hat{\rho}}} {\partial{\hat{x}}} + \frac{\partial{\hat{\rho}}} {\partial{\hat{t}}} = 0$

I am uncertain if my approach is correct. The PINN model struggles to train, leading me to suspect the issue might be rooted in my non-dimensionalization process. Considering that L is in km, and $\rho_{max}$ is in veh/km, I am pondering whether there's a more appropriate way to establish this relationship or if there's an oversight in my non-dimensionalization affecting model training.

It's crucial for the effectiveness of backpropagation that all variables, including x, t, and $\rho$, are kept within small value ranges. Any insights, corrections, or suggestions, especially from those familiar with PINNs and non-dimensionalization in this context, would be immensely appreciated.

Answer 1

The non-dimentional form seems correct : $$(1-2\hat{\rho}) \frac{\partial{\hat{\rho}}} {\partial{\hat{x}}} + \frac{\partial{\hat{\rho}}} {\partial{\hat{t}}} = 0\quad\text{is OK.}$$ The analytic solution (expressed on the form of an implicit equation) is : $$\hat{\rho}=F\bigg(\hat{x}-(1-2\hat{\rho})\hat{t}\bigg)$$ $F$ is an artbitrary function. This means that they are infinity many solutions (if no initial condition is given).

In the wording of the question no initial condition is given. So one cannot go further.