Improve hard constraint on Boundary Condition

[stevengogogo]

Reference:

Page 5, https://arxiv.org/pdf/2102.04626

Topic about bounday condition and multi-dnn is discussed in #7 (comment)

[stevengogogo]

Exp: No hard-constraint

I reduce high frequency to 8 for experiment the training process.

python pinn_1d.py   --levels 1 --epochs 100000 --lr 1e-3 --activation tanh --sweeps 1 --plot --hidden_dims 256 256 256 256 256 --loss_type 0 --high_freq 8

Iteration  99999/100000, PINN Loss: 1.8789e-01, Err 2-norm:  3.6336e+00, inf-norm: 3.9357e-01

[stevengogogo]

Exp: with BC enforcement

y = g0 + (x-0)/(1-0)*(g1 - g0) + (1 - torch.exp(0-x)) * (1 - torch.exp(x-1)) * y # in domain x in [0, 1]

python pinn_1d.py   --levels 1 --epochs 100000 --lr 1e-3 --activation tanh --sweeps 1 --plot --hidden_dims 256 256 256 256 256 --loss_type 0 --high_freq 8 --enforce_bc

Iteration  99999/100000, PINN Loss: 4.1775e-02, Err 2-norm:  7.6223e+01, inf-norm: 8.2844e+00

[stevengogogo]

python pinn_1d.py   --levels 1 --epochs 100000 --lr 1e-3 --activation tanh --sweeps 1 --plot --hidden_dims 256 256 256 256 256 --loss_type 0 --high_freq 8 --enforce_bc

Iteration  99999/100000, PINN Loss: 4.1775e-02, Err 2-norm:  7.6223e+01, inf-norm: 8.2844e+00

[stevengogogo]

Exp: enforce boundary condition with linear term

 y = g0 * (1 - x) + g1 * x + x * (1 - x) * y

Iteration  99999/100000, PINN Loss: 1.5129e-03, Err 2-norm:  1.7308e+01, inf-norm: 1.5587e+00

[stevengogogo]

Question

Why smaller PINN loss does not lead to accurate solution?

#4 (comment)

compare to

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

[liruipeng]

lines 260 and 261 seems to be the same math to me.

[stevengogogo]

The difference is (1-torch.exp(0-x)) keeps positive during the domain [0,1]

This is followed by the idea in page 5 of this paper:
https://arxiv.org/pdf/2102.04626

• We may need to remove #260

[liruipeng]

OK. I see. A minor question: why you write things like exp(0-x) instead of exp(-x), and also (x-0)/(1-0). Are these necessary? Thanks

[stevengogogo]

No, it is just for understanding the boundary is $[x_0, x_1] = [0,1]$ where $x_0 < x_1$. I can remove it

[liruipeng]

No it is OK. Actually we should update the code use mesh.ax and mesh.bx, to allow intervals that are not [0,1]

[stevengogogo]

I added this as future feature #6 , and will be added after we figure out the bug of training in this PR.

[liruipeng]

Question

Why smaller PINN loss does not lead to accurate solution?

#4 (comment)

compare to

#4 (comment)

Does problem 1 actually have 1 solution?

1D PDE problem:

− u x x + γ u = f

and homogeneous boundary conditions (BC)

The analytical solution is

u ( x ) = ∑ k c k sin ⁡ ( w k π x ) and f = ∑ k c k ( w k 2 π 2 + γ ) sin ⁡ ( w k π x )

Good question. I am not sure. @siuwuncheung ideas?

Refer to: #5

[stevengogogo]

Exp: Single Level PINN without hard contraint on BC

python pinn_1d.py   --levels 1 --epochs 100000 --lr 1e-3 --activation tanh --sweeps 1 --plot --hidden_dims 512 512 512 --loss_type 0 --high_freq 8