We will solve a Poisson equation:
\Delta u = 2, \qquad x \in [-1, 1],
with the Neumman boundary conditions on the right boundary
\left.\dfrac{du}{dx}\right|_{x=1} =4
and Dirichlet boundary conditions on the left boundary
u(-1)=0.
The exact solution is u(x) = (x+1)^2.
This description goes through the implementation of a solver for the above described Poisson equation step-by-step. First, modules setting is the same as Possion equation in 1D with Dirichlet boundary conditions. More details can be found in this page.
The DeepXDE and TensorFlow (tf
) modules are imported:
import deepxde as dde
from deepxde.backend import tf
We begin by defining a computational geometry. We can use a built-in class Interval
as follows
geom = dde.geometry.Interval(-1, 1)
Next, we express the PDE residual of the Poisson equation:
def pde(x, y):
dy_xx = dde.grad.hessian(y, x)
return dy_xx - 2
The first argument to pde
is the network input, i.e., the x-coordinate. The second argument is the network output, i.e., the solution u(x), but here we use y
as the name of the variable.
Next, we consider the Neumann boundary condition and Dirichlet boundary condition respectively.
The Neumann boundary condition is defined by a simple Python function. The function should return True
for those points satisfying x=1 and False
otherwise (Note that because of rounding-off errors, it is often wise to use dde.utils.isclose
to test whether two floating point values are equivalent). In this function, the argument x
to boundary
is the network input and is a d-dim vector, where d is the dimension and d=1 in this case. Then a boolean on_boundary
is used as the second argument. If the point x
(the first argument) is on the boundary of the geometry, in this case Neumann boundary when it reaches the right endpoint of the interval, then on_boundary
is True
, otherwise, on_boundary
is False
.
def boundary_r(x, on_boundary):
return on_boundary and dde.utils.isclose(x[0], 1)
The Dirichlet boundary condition is defined in a similar way that the function should return True
for those points satisfying x=-1 and False
otherwise. The arguments in this function are similar to boundary_r
, and the only difference is that in this case Dirichlet boundary condition is used when it reaches the left endpoint of the interval.
def boundary_l(x, on_boundary):
return on_boundary and dde.utils.isclose(x[0], -1)
Next, we define a function to return the value of u(x) for the points x on the Dirichlet boundary. In this case, it is u(x)=0. For example, (x+1)^2 is 0 on the boundary, and thus we can also use
def func(x):
return (x + 1) ** 2
Then, the Dirichlet boundary condition is
bc_l = dde.icbc.DirichletBC(geom, func, boundary_l)
For Neumann boundary condition, rather than define a function to return the value of u(x) on the boundary, we use a lambda function and the Neumann boundary condition is defined
bc_r = dde.icbc.NeumannBC(geom, lambda X: 2*(X+1), boundary_r)
Now, we have specified the geometry, PDE residual, Dirichlet boundary condition and Neumann boundary condition. We then define the PDE problem as
data = dde.data.PDE(geom, pde, [bc_l, bc_r], 16, 2, solution=func, num_test=100)
The number 16 is the number of training residual points sampled inside the domain, and the number 2 is the number of training points sampled on the boundary. The argument solution=func
is the reference solution to compute the error of our solution, and can be ignored if we don't have a reference solution. We use 100 residual points for testing the PDE residual.
Next, we choose the network. Here, we use a fully connected neural network of depth 4 (i.e., 3 hidden layers) and width 50:
layer_size = [1] + [50] * 3 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.nn.FNN(layer_size, activation, initializer)
Now, we have the PDE problem and the network. We bulid a Model
and choose the optimizer and learning rate:
model = dde.Model(data, net)
model.compile("adam", lr=0.001, metrics=["l2 relative error"])
We also compute the L^2 relative error as a metric during training.
We then train the model for 10000 iterations:
losshistory, train_state = model.train(iterations=10000)
.. literalinclude:: ../../../examples/pinn_forward/Poisson_Neumann_1d.py :language: python