In the following pages, we will tackle the one-dimensional Poisson equation, a fundamental elliptic partial differential equation that describes many steady-state physical phenomena, such as heat conduction, electrostatics, and ideal fluid flow.

Our specific problem is to find the solution \( u(x) \) that satisfies:

Governing Equation:

Boundary Conditions (BCs):

This is a boundary value problem with Dirichlet conditions (i.e., the value of the solution is specified at the boundaries).

Analytical Solution: For validation purposes, this problem has a known exact solution, which can be found by integrating the equation twice and applying the boundary conditions:

Our goal is to train a Physics-Informed Neural Network (PINN) to discover this solution without being given the analytical form, using only the governing equation and its boundary conditions. We will explore two common methods for enforcing the boundary conditions:

1. Soft Constraints: Adding a penalty term to the loss function for boundary violations.
2. Hard Constraints: Modifying the network architecture to satisfy the boundary conditions by construction.