When Do PINNs Beat Classical Numerical Methods? A 1D vs 5D Experiment

When solving the Schrödinger equation, the best numerical method dramatically changes based on the problem's dimensionality. For a simple one-dimensional problem, a classical finite-difference solver vastly outperforms a physics-informed neural network (PINN), achieving better accuracy thousands of times faster. However, the tables turn in a five-dimensional scenario, where the classical method becomes computationally impossible due to the "curse of dimensionality," while the PINN efficiently finds an accurate solution. This success hinged on a crucial training strategy, where switching from the Adam optimizer to L-BFGS for refinement slashed the PINN's final error by a factor of 50.