Fixing format.
"\n",
"### Poisson's equation\n",
"\n",
- "Poisson's equation is a second-order linear PDE. It can be used to describe the potential field caused by a given charge \n",
- "or mass density distribution. In two dimensional Cartesian coordinates, it takes the form:\n",
+ "Poisson's equation is a second-order linear PDE. It can be used to describe the potential field caused by a given charge or mass density distribution. In two dimensional Cartesian coordinates, it takes the form:\n",
+ "\n",
"$$\n",
"\\left(\\frac{\\partial^2}{\\partial x^2} + \\frac{\\partial^2}{\\partial y^2}\\right)u(x, y) = f(x, y).\n",
"$$\n",
+ "\n",
"We solve the equation in the domain $(x, y) \\in (0, 1) \\times (0, 1)$ with homogeneous Dirichlet boundary conditions, \n",
+ "\n",
"$$\n",
"u(x, 0) = u(x, 1) = u(0, y) = u(1, y) = 0.\n",
"$$\n",
+ "\n",
"With $f(x, y) = 2x(y-1)(y-2x+xy+2)e^{x-y}$ the analytical solution is\n",
+ "\n",
"$$\n",
"u(x, y) = x(1-x)y(1-y)e^{x-y}.\n",
"$$\n",
- "Figure 2 presents contours of the neural network solution (left), the analytical solution (middle), and the error between the\n",
- "analytical and neural network solution (right). The largest error in the neural network solution is around $6\\cdot 10^{-5}$.\n",
+ "\n",
+ "Figure 2 presents contours of the neural network solution (left), the analytical solution (middle), and the error between the analytical and neural network solution (right). The largest error in the neural network solution is around $6\\cdot 10^{-5}$.\n",
"\n",
"![case2](case2.png)\n",
"*Figure 2: Comparing analytical and ANN-based solutions of Poisson's equation.*"
