due 2/6/2025 before midnight via Learning Suite 25 possible points
We’re going to build on the same problem as the last homework, but now solve a PDE inverse problem. In this case we are given some data (the columns correspond to x, t, and u respectively), and we want to learn something about the governing equations. There are techniques to perhaps learn the underlying equations in a more free-form way, though in this problem we’ll take a simpler approach where we know the governing equation but don’t know the coefficients (e.g., in this case, we’re learning the viscosity corresponding to the data).
In other words, the governing equation is:
\[\frac{\partial u}{\partial t} + \lambda_1 u \frac{\partial u}{\partial x} + \lambda_2 \frac{\partial^2 u}{\partial x^2} = 0\]In the process of developing a neural net that can reproduce the data, we’d also like to learn the parameters \(\lambda_1\) and \(\lambda_2\).
You’ll again formulate this problem as a PINN where you have a data loss and physics loss term, but the data loss comes just from the supplied data. You’ll need to add \(\lambda_1\) and \(\lambda_2\) as design variables in addition to the model parameters.
The true coefficients are \(\lambda_1 = 1\) and \(\lambda_2 = -0.01/\pi\). I just pulled data from my neural net solution to the last homework. It’s not an exact solution to the PDE, so even if your neural net in this assignment was perfect you wouldn’t get exact coefficients. But you should be able to get at least with 0.1 of the first parameter and 0.001 of the second.