Neural Network ODE Solver

This app demonstrates how a Physics-Informed Neural Network (PINN) can approximate the solution to the differential equation: d²y/dx² + y = 0.

The neural network `N(x; θ)` is used to construct a trial solution `y_trial(x) = y(0) + x*y'(0) + x<sup>2</sup>*N(x; θ)`. This form inherently satisfies the boundary conditions `y(0)` and `y'(0)`. The network is then trained to minimize the PDE residual: `(d<sup>2</sup>y_trial/dx<sup>2</sup> + y_trial)<sup>2</sup>` across several "collocation points" in the domain.

Adjust neural network parameters, boundary conditions, and training settings. Then, click 'Train' to see the network learn.

• Blue line: Neural network's predicted solution `y_trial(x)`.
• Green line: Actual analytical solution.
• Red dot & tangent: Visualization of boundary conditions at x=0.