How to use a point Neumann type boundary condition in PINN forward problem?

[geo-hy]

Hi Lu,
I have a problem writing down the boundary conditions in the PINN forward problem for a 2-D heat equation. The upper boundary condition is a Neumann-type boundary condition, which is shown as follows:

Where K_z is a function of the gradient of the output and input. But something went wrong when I tried to set up the upper BC.

***This is my code. In this case, the H_net is assumed to be 1. I only try to make it work now.
g = 9.81
rho_0 = 998.2
area = 580000000
rho_w = 1000
c_p = 4181

geom = dde.geometry.Interval(Lmin, Lmax)
timedomain = dde.geometry.TimeDomain(0, T)
geomtime = dde.geometry.GeometryXTime(geom, timedomain)

def calc_dens(wtemp):
dens = (999.842594 + (6.793952 * 1e-2 * wtemp) - (9.095290 * 1e-3 wtemp2) +
(1.001685 * 1e-4 * wtemp3) - (1.120083 * 1e-6 wtemp4) +
(6.536336 * 1e-9 * wtemp5))
return dens

def eddy_diffusivity(rho, x):
drho_x = dde.grad.jacobian(rho, x, i=0, j=0)
buoy = abs(drho_x) * g / rho_0
buoy = tf.where(buoy < 7e-5, buoy, 7e-5)
ak = 0.00706 *(area/1E6)(0.56)
kz = ak * (buoy)(-0.43)
return kz

def pde(x, y):
rho = calc_dens(y)
diff = eddy_diffusivity(rho, x) / 86400
dy_t = dde.grad.jacobian(y, x, i=0, j=1)
dy_x = dde.grad.jacobian(y, x, i=0, j=0)
dy_xx = dde.grad.jacobian(diff * dy_x, x, i=0, j=0)
return dy_t - dy_xx

def func_lb(x, y):
return 1/dde.jacobian(y, x, i=0, j=0)c_prho_w

def func_ub(x):
return np.zeros((len(x),1), dtype=np.float64)

def boundary_u(x, _):
return np.isclose(x[0], Lmin)
ub = dde.icbc.NeumannBC(geom, func_ub, boundary_u)

def boundary_l(x, _):
return np.isclose(x[0], Lmax)
lb = dde.icbc.NeumannBC(geom, func_lb, boundary_l)

data = dde.data.TimePDE(
geomtime,
pde,
[ub, lb, ],
num_domain=2540,
num_boundary=200,
num_initial=160,
num_test=2540,
)

net = dde.nn.FNN([2] + [20] * 3 + [1], "tanh", "Glorot normal")
model = dde.Model(data, net)
model.compile("adam", lr=1e-3)
losshistory, train_state = model.train(iterations=50000)
model.compile("L-BFGS-B")
losshistory, train_state = model.train()
model.save('PINNmodel')

So, what should I do to solve this problem?