Generalized FNO approximation on compact sets

If $C=\mathrm{\varnothing }$, choose the zero generalized FNO from [F4]; the uniform error condition is vacuous. Otherwise let $n=2d-2$ and let $J_0:L^2(\mathrm{\Omega };{\mathbb{R}}^b)\to L^2({\mathbb{T}}^d;{\mathbb{R}}^b)$ be zero extension. Let $I_{d,n}:L^2({\mathbb{T}}^d;{\mathbb{R}}^b)\to L^2({\mathbb{T}}^n;{\mathbb{R}}^b)$ send $g(x_1,\dots ,x_d)$ to the function constant in the last $n-d=d-2$ coordinates. The chart-pair components identify ${⨁}_{j,k}{L}^2({\mathbb{T}}^n;{\mathbb{R}}^a)$ with $L^2({\mathbb{T}}^n;{\mathbb{R}}^{J^{2}a})$. Define $S:=I_{d,n}{J}_{0}T{L}_a$ on this direct-sum space. This is continuous by [F1] and [F3], and $G_a(C)$ is compact because $G_a$ is continuous and $C$ is compact.

Apply [F2] to the continuous map $S:L^2({\mathbb{T}}^n;{\mathbb{R}}^{J^{2}a})\to L^2({\mathbb{T}}^n;{\mathbb{R}}^b)$ on compact $G_a(C)$ with tolerance $\epsilon$. There is a torus FNO $\mathrm{\Phi }$ such that ${\sup }_{u\in G_a(C)}\mathrm{\parallel }S(u)-\mathrm{\Phi }(u){\mathrm{\parallel }}_{L^2({\mathbb{T}}^n;{\mathbb{R}}^b)}<\epsilon$.

Set ${\mathbb{T}}^0=\{()\}$ with unit mass, so the following integral is the identity when $d=2$. For any finite channel dimension $c$, define $A_c:L^2({\mathbb{T}}^n;{\mathbb{R}}^c)\to L^2({\mathbb{T}}^d;{\mathbb{R}}^c)$ by $(A_{c}v)(x)={\int }_{{\mathbb{T}}^{d-2}}v(x,z)dz$, componentwise. Cauchy–Schwarz on the unit-measure torus and Fubini give $\mathrm{\parallel }{A}_c\mathrm{\parallel }\le 1$. Let $R$ be restriction from ${\mathbb{T}}^d$ to $\mathrm{\Omega }$, also of norm at most one. Since $A_b{I}_{d,n}=I$, $R{J}_0=I$, and $L_a{G}_a=I$, the generalized FNO $\mathrm{\Psi }:=R\circ A_b\circ \mathrm{\Phi }\circ G_a$ satisfies $\mathrm{\parallel }T(h)-\mathrm{\Psi }(h){\mathrm{\parallel }}_{L^2(\mathrm{\Omega };{\mathbb{R}}^b)}=\mathrm{\parallel }R{A}_b(S(G_{a}h)-\mathrm{\Phi }(G_{a}h)){\mathrm{\parallel }}_{L^2(\mathrm{\Omega };{\mathbb{R}}^b)}\le \mathrm{\parallel }S(G_{a}h)-\mathrm{\Phi }(G_{a}h){\mathrm{\parallel }}_{L^2({\mathbb{T}}^n;{\mathbb{R}}^b)}<\epsilon$ for every $h\in C$. Write the torus FNO as $\mathrm{\Phi }=Q_{\mathrm{\Phi }}\circ {\mathrm{\Phi }}_0$, where ${\mathrm{\Phi }}_0$ ends before its final constant affine channel map $Q_{\mathrm{\Phi }}(v)=Wv+c$ and has $r$ output channels. Its initial affine channel lift is composed into the first affine part of ${\mathrm{\Phi }}_0$ after $G_a$. Normalized averaging commutes with the final map, $A_b{Q}_{\mathrm{\Phi }}=Q_{\mathrm{\Phi }}{A}_r$, and restriction commutes with the same pointwise map. Hence $R{A}_b\mathrm{\Phi }{G}_a=Q_{\mathrm{\Phi }}R{A}_r{\mathrm{\Phi }}_0{G}_a$. This has the generalized architecture: identity pointwise lift, chart lift $G_a$, torus layers ${\mathrm{\Phi }}_0$, the constant-kernel dimension reduction $A_r$, restriction $R$, and final pointwise affine map $Q_{\mathrm{\Phi }}$.