A Statistical Formulation Gap for Nonlinear Multiscale Physics-Informed Learning

We prove a finite-sample formulation gap for physics-informed learning of nonlinear multiscale elliptic equations. For a uniformly monotone divergence-form class with coefficients oscillating at scale $\epsilon$, we derive a finite-width, finite-sample, and finite-iteration error bound for a boundary-compatible variational neural solver. Its stability, sampling, and optimization constants are independent of $\epsilon$, while all unresolved scale dependence is isolated in the best-approximation error.
We then construct a minimal obstruction witness for a one-dimensional periodic diffusion equation with cubic reaction. Already on the one-parameter family $v_c(x)=cx(1-x)$, the empirical Rademacher complexity of the strong residual is bounded below by a constant multiple of $(\epsilon\sqrt{N})^{-1}$, while that of the squared strong-residual loss is bounded below by a constant multiple of $(\epsilon^2\sqrt{N})^{-1}$. These are optimizer-independent properties of the sampled residual and loss classes, rather than neural-tangent-kernel conditioning statements. The corresponding variational energy complexity is bounded above by a constant multiple of $N^{-1/2}$ uniformly in $\epsilon$.
A tensor-product construction shows that the same obstruction rates persist in every spatial dimension. Numerical evaluation gives fitted exponents $0.9971$, $1.9860$, and $-0.0028$ for the strong residual, squared strong loss, and variational energy, respectively. Thus, differentiating the microscopic coefficient creates finite-sample statistical ill-conditioning. The variational formulation removes this statistical penalty but does not remove the separate multiscale approximation problem.