> Are there actual NN methods that can PDEs without depending on the initial conditions?

The initial condition needs to be known (but we can actually have some noisy initial condition, like measurements corrupted by noise [1]), but NN based models can efficiently solve some parametric PDEs faster than traditional solvers. [2]

There is also a lot of work in training NNs on data generated from traditional methods, and this can be combined jointly with the above method to solve a whole class of problems at once. [3]

Solving a whole parametric family of PDEs (i.e. a parameterized family of initial conditions) and handling complicated geometries will be the next avenue of this specific field IMO. Actually it is being actively worked on.

[1] https://arxiv.org/abs/2205.07331

[2] https://arxiv.org/abs/2110.13361

[3] https://arxiv.org/abs/2111.03794