Ermakov--Based embeddings as a parameter-free layer for physics-informed neural networks

arXiv:2611.03964v1 Announce Type: new Abstract: Evolution equations with time-dependent coefficients force a neural solver to spend capacity on a scale that the equation already determines. We show that a single classical object, the Ermakov-Pinney scale function $\rho$, can be written into the input layer of a physics-informed neural network as a parameter-free embedding. One construction covers three problem families through the two signs of the Pinney constant $c$: linear diffusion ($c=-1$, $\omega\equiv0$), diffusion in a time-dependent confining trap ($c=-1$, $\omega\neq0$), and the quantum oscillator with time-dependent frequency ($c=+1$). An exact moment identity gives $\rho$ in closed form from the initial data alone, so no auxiliary network or knowledge of the solution is needed.