Conserve integral in a PINN

[ChrisRackauckas]

@ste

ρ\rho should always be positive as in eqn2 a square-root is applied. I fixed this by using a softmax in the last layer instead of a linear activation function. It feels a bit strange to do it like this. Is there a way to have \rhoρ\rho positive as a constraint?

That’s a good way to do it. Or abs. Just make a positive function approximator and it’s fine.

@ste

Similarly, I want \int_{L/2}^{L/2} \rho dx = 1∫L/2L/2ρdx=1\int_{L/2}^{L/2} \rho dx = 1 i.e. the integral of \rhoρ\rho, a probability density, should be normalized to 1 and then kept that way (thereby also automatically fulfilling 1)) at all times. How can I do that in SciML or something?

Just add that as a 3rd equation.