4.1 KiB
Deepritz
This section repeats the Deepritz method presented by Weinan E and Bing Yu.
Consider the 2d Poisson's equation such as the following:
\begin{equation}
\begin{aligned}
-\Delta u=f, & \text { in } \Omega \\
u=0, & \text { on } \partial \Omega
\end{aligned}
\end{equation}
Based on the scattering theorem, its weak form is that both sides are multiplied by$ v \in H_0^1$(which can be interpreted as some function bounded by 0),to get
\int f v=-\int v \Delta u=(\nabla u, \nabla v)
The above equation holds for any v \in H_0^1. The bilinear part of the right-hand side of the equation with respect to u,v is symmetric and yields the bilinear term:
a(u, v)=\int \nabla u \cdot \nabla v
By the Poincaré inequality, the a(\cdot, \cdot) is a positive definite operator. By positive definite, we mean that there exists \alpha >0, such that
a(u, u) \geq \alpha\|u\|^2, \quad \forall u \in H_0^1
The remaining term is a linear generalization of v, which is l(v), which yields the equation:
a(u, v) = l(v)
For this equation, by discretizing u,v in the same finite dimensional subspace, we can obtain a symmetric positive definite system of equations, which is the family of Galerkin methods, or we can transform it into a polarization problem to solve it.
To find u satisfies
a(u, v) = l(v), \quad \forall v \in H_0^1
For a symmetric positive definite a , which is equivalent to solving the variational minimization problem, that is, finding u, such that holds, where
J(u) = \frac{1}{2} a(u, u) - l(u)
Specifically
\min _{u \in H_0^1} J(u)=\frac{1}{2} \int\|\nabla u\|_2^2-\int f v
The DeepRitz method is similar to the PINN approach, replacing the neural network with u, and after sampling the region, just solve it with a solver like Adam. Written as
\begin{equation}
\min _{\left.\hat{u}\right|_{\partial \Omega}=0} \hat{J}(\hat{u})=\frac{1}{2} \frac{S_{\Omega}}{N_{\Omega}} \sum\left\|\nabla \hat{u}\left(x_i, y_i\right)\right\|_2^2-\frac{S_{\Omega}}{N_{\partial \Omega}} \sum f\left(x_i, y_i\right) \hat{u}\left(x_i, y_i\right)
\end{equation}
Note that the original u \in H_0^1, which is zero on the boundary, is transformed into an unconstrained problem by adding the penalty function term:
\begin{equation}
\begin{gathered}
\min \hat{J}(\hat{u})=\frac{1}{2} \frac{S_{\Omega}}{N_{\Omega}} \sum\left\|\nabla \hat{u}\left(x_i, y_i\right)\right\|_2^2-\frac{S_{\Omega}}{N_{\Omega}} \sum f\left(x_i, y_i\right) \hat{u}\left(x_i, y_i\right)+\beta \frac{S_{\partial \Omega}}{N_{\partial \Omega}} \\
\sum \hat{u}^2\left(x_i, y_i\right)
\end{gathered}
\end{equation}
Consider the 2d Poisson's equation defined on \Omega=[-1,1]\times[-1,1], which satisfies f=2 \pi^2 \sin (\pi x) \sin (\pi y).
Define Sampling Methods and Constraints
For the problem, boundary condition and PDE constraint are presented and use the Identity loss.
@sc.datanode(sigma=1000.0)
class Boundary(sc.SampleDomain):
def __init__(self):
self.points = geo.sample_boundary(100,)
self.constraints = {"u": 0.}
def sampling(self, *args, **kwargs):
return self.points, self.constraints
@sc.datanode(loss_fn="Identity")
class Interior(sc.SampleDomain):
def __init__(self):
self.points = geo.sample_interior(1000)
self.constraints = {"integral_dxdy": 0,}
def sampling(self, *args, **kwargs):
return self.points, self.constraints
Define Neural Networks and PDEs
In the PDE definition section, based on the DeepRitz method we add two types of PDE nodes:
def f(x, y):
return 2 * sp.pi ** 2 * sp.sin(sp.pi * x) * sp.sin(sp.pi * y)
dx_exp = sc.ExpressionNode(
expression=0.5*(u.diff(x) ** 2 + u.diff(y) ** 2) - u * f(x, y), name="dxdy"
)
net = sc.get_net_node(inputs=("x", "y"), outputs=("u",), name="net", arch=sc.Arch.mlp)
integral = sc.ICNode("dxdy", dim=2, time=False)
The result is shown as follows:
