chiikaawaa/araneae ? reverse-engineered prompt

Build me a small C++ project that can do autograd and train a physics informed neural network from scratch, without using any ready made machine learning libraries.

I want it to be able to learn a simple system like a pendulum, compare the model against an RK4 solver, and show that the network can fit the dynamics over time. Please make the code easy to run from the command line, with a clear training program and a separate solver or demo program.

If there’s a good way to organize the math pieces like parameters, the MLP, and the autograd logic so the project is cleaner and easier to understand, please do that. Keep the implementation self contained and beginner friendly, and make sure the sample run actually produces useful output so I can see the training working. If you need to look up anything current about C++ compiler support or plotting or notebook integration, go ahead and check online.