Fourier neural-operator layer

Let ${\mathbb{T}}^d=(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^d$ with normalized Lebesgue measure, represented by the unit cube $[0,1{]}^d$, and let $v\in C({\mathbb{T}}^d;{\mathbb{R}}^q)$ be a periodic vector field. Let $K\ge 0$ be a frequency cutoff, and let $R_k\in {\mathbb{C}}^{q^{\mathrm{\prime }}\times q}$ be trainable matrices for integer multi-indices $k$ with $\mathrm{\parallel }k{\mathrm{\parallel }}_{\mathrm{\infty }}\le K$. Define the truncated Fourier integral layer by ${\mathcal{K}}_{R}v(x)={\sum }_{\mathrm{\parallel }k{\mathrm{\parallel }}_{\mathrm{\infty }}\le K}{e}^{2\pi ik\cdot x}{R}_k\widehat{v}(k),\widehat{v}(k)={\int }_{{\mathbb{T}}^d}{e}^{-2\pi ik\cdot y}v(y)dy\mathrm{.}$ For real-valued outputs require $R_{-k}=\overline{R_k}$ and $R_0$ real. A Fourier neural-operator layer is $v_{t+1}(x)=\sigma (W_t{v}_t(x)+{\mathcal{K}}_R{v}_t(x)+b_t(x)),$ where $W_t$ is a real pointwise channel map, $b_t\in C({\mathbb{T}}^d;{\mathbb{R}}^{q^{\mathrm{\prime }}})$ is a continuous periodic real bias, and $\sigma :\mathbb{R}\to \mathbb{R}$ is continuous and acts coordinatewise. The finite Fourier sum and these conditions give $v_{t+1}\in C({\mathbb{T}}^d;{\mathbb{R}}^{q^{\mathrm{\prime }}})$. This definition gives a Fourier-parameterized layer only. It does not assert that the architecture is universal in any specified function-space topology.

neural operator

measure space

lebesgue measure and the lebesgue sigma algebra

integrable real and complex functions and their integrals

integral over a measurable set

Kovachki et al., §4.4, equation (26): Fourier neural-operator multiplier