The Limits of Exact Symmetry as a Transfer Prior: Dimensionless Invariants Give No Durable Advantage over Log-Scaled Inputs in 1D Neural Operators

A central promise of physics-aware representations is transfer: an embedding respecting a system's governing structure should extrapolate into regimes never seen in training. Fluid equations offer a clean test, admitting an exactly correct structural representation—dynamical similarity makes the solution operator depend on the dimensional parameters only through the dimensionless groups Re and Ma. In smooth, shock-free 1D Burgers and compressible flow we compare two operators with identical architectures and budgets: a naive operator conditioned on log-scaled raw parameters (log μ, log L, log c₀) and a similarity operator conditioned on (log Re, log Ma), evaluated on held-out conditions—length scales outside the training band and (Re, Ma) configurations disjoint from training. The invariant encoding improves out-of-band transfer only when training covers a single length scale (4.2× with an MLP); once training contains modest parameter diversity the advantage collapses, consistent with the naive operator reconstructing the invariants from its inputs. Gaps are geometric means over five seeds; the collapse appears in every MLP seed and is reproduced with a Fourier Neural Operator backbone. A positive control varies how hard the invariant is to obtain from the inputs: re-parameterizing so that it is no longer a linear function of the encoded parameters—while holding feature ranges or the extrapolation axis fixed—restores a durable 2.3–2.6× advantage. We read this as evidence that, in this setting, the value of an exact symmetry prior depends less on its correctness than on its recoverability: whether the invariant is a simple function of the inputs the network already receives.