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This research establishes a rigorous theoretical framework for physics-informed learning, proving that differentiating microscopic coefficients in nonlinear multiscale equations introduces finite-sample statistical ill-conditioning independent of the optimizer's choice. By isolating this penalty within variational formulations while maintaining stability constants across scales, the study clarifies fundamental limits on neural solvers for elliptic partial differential equations with oscillating parameters. The work advances mathematical machine learning by quantifying how unresolved multiscale features constrain approximation error rather than optimization dynamics alone.
A Statistical Formulation Gap for Nonlinear Multiscale Physics-Informed Learning
arXiv:2607.15702v1 Announce Type: cross Abstract: 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.
Abstract: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.
Numerical Analysis (math.NA); Machine Learning (cs.LG); Analysis of PDEs (math.AP)
arXiv:2607.15702 [math.NA]
(or arXiv:2607.15702v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2607.15702 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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