Parameterized Poisson

We consider an extended problem of Simple Poisson.

$$ \begin{array}{l} -\Delta u=1\ \frac{\partial u(x, -1)}{\partial n}=\frac{\partial u(x, 1)}{\partial n}=0 \ u(-1,y)=T_l\ u(1, y)=0, \end{array} $$ where $T_l$ is a design parameter ranging in $(-0.2,0.2)$. The target is to train a surrogate that $u_\theta(x,y,T_l)$ gives the temperature at $(x,y)$ when $T_l$ is provided.

Train A Surrogate

In addition, we define the parameter

temp = sp.Symbol('temp')
temp_range = {temp: (-0.2, 0.2)}

The usage of temp is similar to the time variable in Burgers’ Equation. temp_range should be passed to the argument param_ranges in sampling domains.

The left bound value condition is

@sc.datanode
class Left(sc.SampleDomain):
    # Due to `name` is not specified, Left will be the name of datanode automatically
    def sampling(self, *args, **kwargs):
        points = rec.sample_boundary(1000, sieve=(sp.Eq(x, -1.)), param_ranges=temp_range)
        constraints = sc.Variables({'T': temp})
        return points, constraints

The result is shown as follows:

See examples/parameterized_poisson.