Finite-sample generalization of learned inverse maps

Fix $d\ge 2$, a bounded connected smooth domain $\mathrm{\Omega }\subset (0,1{)}^d$, a fixed collar ${\mathrm{\Omega }}^{\mathrm{\prime }}⋐\mathrm{\Omega }$, an admissible conductivity class $X_d(R)$ with fixed ellipticity and regularity parameters, a compact perturbation set $K\subset L^2(\mathrm{\partial }\mathrm{\Omega }\times \mathrm{\partial }\mathrm{\Omega };\mathbb{C})$, and a threshold $\delta \ge 0$. Let $K_{\delta }=\{\eta \in K:\mathrm{\parallel }\eta {\mathrm{\parallel }}_2\le \delta \}$ and $\mathcal{Z}=\{({\kappa }_{\gamma }+\eta ,\gamma ):\gamma \in X_d(R),\eta \in K_{\delta }\}$ in $\mathcal{H}\times \mathcal{Y}$, where $\mathcal{H}=L^2(\mathrm{\partial }\mathrm{\Omega }\times \mathrm{\partial }\mathrm{\Omega };\mathbb{C})$ is viewed as a real Hilbert space, $\mathcal{Y}=L^2(\mathrm{\Omega };\mathbb{R})$, and $\mathfrak{R}f=(\mathrm{Re}f,\mathrm{Im}f)\in L^2(\mathrm{\partial }\mathrm{\Omega }\times \mathrm{\partial }\mathrm{\Omega };{\mathbb{R}}^2)$ is the isometric realification. Assume the Axiom of Choice and Countable Choice. Fix an extension ${\tilde{\mathrm{\Psi }}}^{\star }:\mathcal{H}\to \mathcal{Y}$ of the exact-data inverse, so ${\tilde{\mathrm{\Psi }}}^{\star }({\kappa }_{\gamma })=\gamma$, and fix $C_{\star }>0$ and a continuous nondecreasing concave modulus ${\omega }_{\star }:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with ${\omega }_{\star }(0)=0$ such that $\mathrm{\parallel }{\tilde{\mathrm{\Psi }}}^{\star }(f)-{\tilde{\mathrm{\Psi }}}^{\star }(g){\mathrm{\parallel }}_{\mathcal{Y}}\le C_{\star }{\omega }_{\star }(\mathrm{\parallel }f-g{\mathrm{\parallel }}_{\mathcal{H}})(f,g\in \mathcal{H})\mathrm{.}$ These extension data are fixed independently of the architecture budgets, sample size and joint law. Assume $\mathcal{Z}$ is nonempty and compact, and set $X=\{x\in \mathcal{H}:(x,y)\in \mathcal{Z}\text{ for some }y\in \mathcal{Y}\}$. Fix a smooth globally Lipschitz nonpolynomial activation ${\sigma }_0$.

Choose once and for all a finite smooth boundary chart family and square partition of unity as in the boundary chart-lifting lemma. For every positive finite channel count $c$, let $G_c$ be the componentwise application of this same scalar chart construction, so $G_c:L^2(\mathrm{\partial }\mathrm{\Omega }\times \mathrm{\partial }\mathrm{\Omega };{\mathbb{R}}^c)\to L^2({\mathbb{T}}^n;{\mathbb{R}}^{J^{2}c})$ for its fixed number $J^2$ of chart pairs. Each pointwise lift $S_0$ receives the two realified input channels; if its output has $c$ channels, take $E=G_c$. The family $(G_c)$, the chart weights and the output restriction $R$ to $\mathrm{\Omega }$ are fixed and have no trainable parameters; $c$ is already part of the counted finite architecture index, not an additional scalar parameter. Write $n=2d-2$. The dimension-reduction layer $F$ has a matrix-valued kernel on ${\mathbb{T}}^{n-d}$; if $n=d$, take $F$ to be the identity. For each integer $s\ge 1$ and $B>0$, define ${\mathcal{F}}_{s,B}\subset C(X,\mathcal{Y})$ as the restrictions to $X$ of maps $x\mapsto \mathrm{\Phi }(\mathfrak{R}x)$, where $\mathrm{\Phi }$ is a generalized Fourier neural operator with the following finite parameterization:

1. The pointwise lift $S_0$ and output network $Q$ are fully connected finite-dimensional networks. At each point they receive the fixed ambient coordinate and the current real field vector; hidden affine layers are followed coordinatewise by ${\sigma }_0$, and the final layer is affine. Their depths, hidden widths and all variable output channel widths are at most $s$; the final output of $Q$ is the single real target channel.
2. The Fourier-operator stack has at most $s$ layers and at most $s$ channels in each layer. Each channel matrix is a trainable real matrix. Each Fourier multiplier has matrix coefficients $R_{\mathrm{\ell },k}$ supported on $\mathrm{\parallel }k{\mathrm{\parallel }}_{\mathrm{\infty }}\le s$, with $R_{\mathrm{\ell },-k}=\overline{R_{\mathrm{\ell },k}}$ so real fields remain real. Each coordinate bias $b_{\mathrm{\ell }}$ is a real trigonometric polynomial on ${\mathbb{T}}^n$ supported on the same finite mode range.
3. The dimension-reduction kernel is a real matrix-valued trigonometric polynomial $K(z)={\sum }_{\mathrm{\parallel }k{\mathrm{\parallel }}_{\mathrm{\infty }}\le s}{K}_k{e}^{2\pi ik\cdot z},K_{-k}=\overline{K_k},$ with coefficients of the dimensions required by the adjacent channel counts.
4. The number of independent trainable real scalar coefficients in all pointwise networks, channel matrices, Fourier multipliers, coordinate biases, and dimension-reduction kernels together is at most $s$. For complex Fourier coefficients, independent real and imaginary parts are counted separately, subject to conjugate symmetry. Every trainable real scalar coefficient has absolute value at most $B$. The fixed activation, chart maps, partition weights, realification, restriction, and Fourier basis are not trainable parameters.

For each fixed $s$, only finitely many architecture index tuples occur; equip their disjoint union of bounded real parameter boxes with its Borel structure, and give ${\mathcal{F}}_{s,B}$ the Borel sigma-algebra induced by the supremum norm on $C(X,\mathcal{Y})$. Consider only $s,B$ for which this class is nonempty and ${\sup }_{\mathrm{\Psi }\in {\mathcal{F}}_{s,B},x\in X}\mathrm{\parallel }\mathrm{\Psi }(x){\mathrm{\parallel }}_{\mathcal{Y}}<\mathrm{\infty }$. For $z=(x,y)\in \mathcal{Z}$ and $\mathrm{\Psi }\in {\mathcal{F}}_{s,B}$, define $\mathrm{\ell }(z,\mathrm{\Psi })=\mathrm{\parallel }\mathrm{\Psi }(x)-y{\mathrm{\parallel }}_{\mathcal{Y}}^2$. For a Borel probability law $P$ on $\mathcal{Z}$, put $R_P(\mathrm{\Psi })={\mathbb{E}}_{Z\sim P}\mathrm{\ell }(Z,\mathrm{\Psi })$ and ${\widehat{R}}_N(\mathrm{\Psi })=N^{-1}{\sum }_{i=1}^N\mathrm{\ell }(Z_i,\mathrm{\Psi })$ for iid $Z_i\sim P$. Determine the minimax excess-risk behavior ${\mathfrak{R}}_N(s,B):={\inf }_{{\widehat{\mathrm{\Psi }}}_N}{\sup }_{P\in \mathcal{P}(\mathcal{Z})}{\mathbb{E}}_{P^N}[R_P({\widehat{\mathrm{\Psi }}}_N)-{\inf }_{\mathrm{\Psi }\in {\mathcal{F}}_{s,B}}{R}_P(\mathrm{\Psi })],$ where $\mathcal{P}(\mathcal{Z})$ is the set of Borel probability laws on $\mathcal{Z}$ and the infimum is over Borel learners ${\widehat{\mathrm{\Psi }}}_N:{\mathcal{Z}}^N\to {\mathcal{F}}_{s,B}$. Characterize growth regimes $s=s_N$, $B=B_N$ in which estimation and continuum approximation errors can be balanced while retaining the deterministic noise term $C_{\star }{\omega }_{\star }(\delta )$.

The continuum noise-robust existence statement.

The convention for expected loss under Borel joint input-target laws.

The dimension-dependent admissible conductivity class X_d(R).

The mean-zero smoothed NtD kernel kappa_gamma and noise convention.

The generalized-FNO architecture on the boundary manifold.

The fixed activation hypotheses used by the FNO approximation route.

de Hoop et al., Theorem 4.8 and §5 numerical experiments

Reinhardt, Wang, and Zech (2026), Statistical Learning Theory for Neural Operators

The question asks for a quantitative minimax characterization for this particular noisy EIT joint data set and bounded architecture. It does not assert that finite-sample learning theory for neural operators is absent. Kovachki–Lanthaler–Mhaskar already study continuum-FNO metric entropy and empirical-risk learning for operator approximation classes with a specified algebraic relation between model size and approximation error. Grohs–Lanthaler–Trautner prove sampling limitations for specified ReLU approximation classes and input measures. Their target families, activation hypotheses, sampling models and risks differ from the all-joint-laws excess-risk expression above; their bounds are not silently transferred to this EIT class.

Reinhardt–Wang–Zech prove empirical-risk concentration bounds for operator regression under i.i.d. random design and additive white or sub-Gaussian output noise (Theorems 5–6), and algebraic ERM rates for ReLU/RePU FrameNet classes on holomorphic operators, including a parametric Darcy-flow problem (Theorems 23 and 27). Their results are close in subject but do not give the minimax excess-risk behavior over arbitrary Borel joint laws for the bounded generalized-FNO EIT class specified here.

There is also an established elementary estimation baseline. If a uniformly bounded model class has a finite uniform $ϵ$-net of size $m$, empirical square-loss minimization over that finite list is a Borel learner, and its expected excess risk is at most $2A\sqrt{\log (2m)\mathrm{/}(2N)}+2(C+D)ϵ$, uniformly over all joint laws, where $C$ bounds model outputs, $D$ bounds targets and $A=(C+D{)}^2$. Thus deriving some finite-sample upper bound is not the unresolved claim. With fixed finite scalar parametrizations, sufficiently slow budget growth can also be obtained qualitatively by a diagonal choice of compact model classes and finite nets; this gives no efficient quantitative rate. The remaining requested analysis is matching upper and lower behavior for the specified EIT geometry and quantitative approximation versus the budgets $(s,B)$, which the continuum existence theorem alone does not provide. The deterministic reconstruction term remains $C_{\star }{\omega }_{\star }(\delta )$, with the corrected kernel modulus involving $\sqrt{\delta }$; its contribution to square risk is handled by squaring the reconstruction bound.

The nonempty compact support, Borel model-class convention, nonempty learner range and uniform output bound make the displayed risks finite and the infimum over Borel learners nonempty. The chart lift and restriction are fixed, and every learned layer, multiplier, bias and dimension-reduction kernel has the finite Fourier or affine parameterization counted in the statement. This source comparison, updated 21 October 2026, delimits the registered research question. It is not a certification that no matching result exists anywhere in the literature, and no generalization theorem is inferred from the EIT paper's finite numerical experiments.