Chiikaawaa/Araneae ? reverse-engineered prompt

Build me a small C++ project that shows how to do automatic differentiation and a physics informed neural network from scratch, without relying on ready made machine learning libraries.

I want it to include a simple neural network, a way to track parameters and gradients, and an example that trains on a pendulum style system. It should also include a classic RK4 solver so I can compare the neural network against a normal numerical solution. Please make it easy to run the pendulum training and the PiNN example from the command line, and keep the code readable since I want to learn from it.

If you need to check current C++ best practices or math details online, feel free to look them up. Also include a notebook or simple demo flow that shows the comparison clearly, ideally with plots or saved output so I can see whether the learned model matches the solver.