small PINN loss doesn't mean accurate

[stevengogogo]

Question

Why smaller PINN loss does not lead to accurate solution?

PINN Loss	Prediction	Method
1.5129e-03		With BC hard constraint #4 (comment)
1.8789e-01		with BC soft constraint #4 (comment)

Does problem 1 actually have 1 solution?

1D PDE problem:

$-u_{xx} + \gamma u = f$

and homogeneous boundary conditions (BC)

The analytical solution is

$u(x) = \sum_k c_k \sin(w_k \pi x)$
and
$f = \sum_k c_k (w_k^2 \pi^2 + \gamma) \sin(w_k \pi x)$

• Is $w_k$ an integer to guarantee $f(1)=0$

Originally posted by @stevengogogo in #4 (comment)

[stevengogogo]

This might be a well-known issue for PINN training. As described in this paper

https://neurips.cc/virtual/2024/poster/94109