Fourier neural operators approximate continuous maps uniformly on compact torus sets

If $C=\mathrm{\varnothing }$, the zero operator from [F3] suffices. Otherwise let $P_N$ be the orthogonal projection onto real Fourier modes of frequency at most $N$. Strong convergence in [F1] is uniform on compact $C$: for $\eta >0$, choose a finite $\eta \mathrm{/}3$-net $f_1,\dots ,f_s$ for $C$, then take $N$ so that $\mathrm{\parallel }{P}_N{f}_i-f_i{\mathrm{\parallel }}_2<\eta \mathrm{/}3$ for every $i$. For $f$ within $\eta \mathrm{/}3$ of $f_i$, contraction of $I-P_N$ gives $\mathrm{\parallel }(I-P_N)f{\mathrm{\parallel }}_2<2\eta \mathrm{/}3$. Hence ${\sup }_{f\in C}\mathrm{\parallel }{P}_{N}f-f{\mathrm{\parallel }}_2\to 0$.

For each $f\in C$, let $m(f)$ be the least nonnegative integer such that $T$ has oscillation less than $\epsilon \mathrm{/}8$ on $B(f,2^{-m(f)})$, and set $r_f=2^{-m(f)}$. Continuity ensures that such an integer exists, so this defines all radii canonically without a simultaneous choice. Choose a finite subcover $B(f_i,r_i\mathrm{/}2)$ of $C$ by the HALF-radius balls, and take a positive $\delta <{\min }_i{r}_i\mathrm{/}4$. By step 1.1, increase $N$ until ${\sup }_{f\in C}\mathrm{\parallel }{P}_{N}f-f{\mathrm{\parallel }}_2<\delta$. For every $f\in C$ choose $i$ with $\mathrm{\parallel }f-f_i{\mathrm{\parallel }}_2<r_i\mathrm{/}2$; then $\mathrm{\parallel }{P}_{N}f-f_i{\mathrm{\parallel }}_2<3r_i\mathrm{/}4$, so both points lie in the full continuity ball $B(f_i,r_i)$ and $\mathrm{\parallel }T(P_{N}f)-T(f){\mathrm{\parallel }}_2<\epsilon \mathrm{/}8<\epsilon \mathrm{/}4$. The set $P_{N}C$ and its continuous image $T(P_{N}C)$ are compact. Fourier projections converge uniformly on the latter by the finite-net argument of step 1.1, so choose $M$ with ${\sup }_{z\in P_{N}C}\mathrm{\parallel }{P}_{M}T(z)-T(z){\mathrm{\parallel }}_2<\epsilon \mathrm{/}4$.

The range of $P_N$ is finite dimensional. In a real orthonormal trigonometric basis, define $g:P_{N}C\to {\mathbb{R}}^q$ by listing the coefficients of $P_{M}T(z)$. By [F2], approximate $g$ uniformly on compact $P_{N}C$ by a finite $\sigma$-network with Euclidean coefficient error less than $\epsilon \mathrm{/}8$; orthonormality makes the corresponding output-polynomial error less than $\epsilon \mathrm{/}8$ in $L^2$.

First apply the finite Fourier multiplier $P_N$ from step 2.1, retaining the WHOLE real Fourier polynomial, including both sine and cosine components of each nonzero frequency. Its coefficient vectors lie in a compact set, so all its pointwise values are uniformly bounded. Inject the finitely many retained real basis functions ${\phi }_j(x)$ as periodic bias channels. The multiplication map $(z,t)\mapsto zt$ is continuous on the compact ranges of $(P_{N}f)(x)$ and ${\phi }_j(x)$; approximate it by the pointwise networks of [F2], then use a zero-frequency multiplier to average each product. Since $\int (P_{N}f)(x){\phi }_j(x)dx$ is exactly the real coefficient of ${\phi }_j$, these constant channels approximate ALL desired coefficients without using an impossible translation-invariant cosine-only projection. Mandatory activations can be passed with arbitrarily small error on these bounded channels: choose $c$ with ${\sigma }^{\mathrm{\prime }}(c)\mathrm{\ne }0$, encode $z$ as $c+\tau z$, and decode affinely by $(\sigma (c+\tau z)-\sigma (c))\mathrm{/}(\tau {\sigma }^{\mathrm{\prime }}(c))$, which converges uniformly to $z$ on every bounded interval as $\tau \to 0$. Approximate the finite products and implement the coefficient network from step 3.1 using finitely many such layers. At each stage uniform continuity on the compact finite coefficient ranges propagates arbitrarily small errors through the finite composition; zero-mode averaging has norm one.

Inject the finitely many output basis functions ${\phi }_k(x)$ as periodic bias channels. The output coefficient vector from step 4.1 ranges over a compact set; the continuous map from that vector and $({\phi }_k(x){)}_k$ to ${\sum }_k{c}_k{\phi }_k(x)$ is uniformly approximated on their compact product by [F2]. The final affine output approximates this output Fourier polynomial with arbitrarily small uniform, hence $L^2$, error because normalized torus measure is one.

Choose the coefficient-extraction and Fourier-synthesis errors in steps 4.1–5.1 so their sum is less than $\epsilon \mathrm{/}4$. Combining this with the input truncation, output truncation and finite coefficient-map errors from steps 2.1 and 3.1 yields ${\sup }_{f\in C}\mathrm{\parallel }T(f)-\mathrm{\Phi }(f){\mathrm{\parallel }}_2<\epsilon \mathrm{/}4+\epsilon \mathrm{/}4+\epsilon \mathrm{/}8+\epsilon \mathrm{/}4<\epsilon$. All cutoffs and channel counts are finite.