A Pedagogical Framework for Physics-Informed Machine Learning: From Classical Pendulum to Quantum Anharmonic Oscillator Using PyTorch on Modern GPU Hardware

Abstract

We present a five-module pedagogical framework for teaching physics-informed machine learning (ML) through two progressively complex physical systems: a driven, damped nonlinear pendulum and a one-dimensional quantum anharmonic oscillator. Five model architectures are implemented and compared: a standard artificial neural network (ANN), a one-dimensional convolutional neural network (CNN), a long short-term memory (LSTM) network, and two physics-informed neural networks (PINNs) -- one per physical system. All models are implemented in PyTorch~2.9 and executed on an NVIDIA RTX~5090 GPU, making the framework directly applicable to modern deep learning laboratory courses. Quantitative benchmarks show that data-driven models achieve mean absolute errors of $1.3\times10^{-2}$~rad (pendulum ANN) and $4.4\times10^{-5}$~a.u.\ (quantum CNN), while the curriculum-trained pendulum PINN reaches an MAE of $3.1\times10^{-2}$~rad using only collocation points. A systematic CPU-vs-GPU benchmark reveals speedups ranging from $1.2\times$ (small ANN) to $24.6\times$ (LSTM), providing a concrete pedagogical demonstration of when GPU acceleration is -- and is not -- warranted. The framework is packaged as self-contained Jupyter notebooks designed for a graduate-level \emph{Deep Neural Networks for Physical Systems} course, with embedded reflection questions that guide students from data-driven thinking toward physics-constrained formulations.