Ermakov--Based embeddings as a parameter-free layer for physics-informed neural networks
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 $ρ$, 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 ($c=-1$, $ω\equiv0$), diffusion in a time-dependent confining trap ($c=-1$, $ω\neq0$), and the quantum oscillator with time-dependent frequency ($c=+1$). An exact moment identity gives $ρ$ in closed form from the initial data alone, so no auxiliary network or knowledge of the solution is needed. Our central finding is comparative: each family requires a different component of the construction and has a different honest control. In family I, the diffusive-time reparametrisation alone reproduces most of the gain, so that family is a poor test of the idea and shows why an ablation is mandatory. In family II no time reparametrisation exists, the control degrades accuracy, and the amplitude prefactor becomes essential (40.8x). In family III the adiabatic approximation also fails, and the quadratic phase factor is what matters (5.7x). The method does not help when the reduced solution does not converge to a fixed point: its extrapolation advantage then disappears. The reduction itself is the classical scaling-variable change of the self-similar-asymptotics literature; what is new is the moment identity that makes it architectural for time-dependent coefficients, and the comparative anatomy above.