Learning homogenization for elliptic operators (Q6583661)

The authors consider the classical linear multiscale elliptic equation in a bounded domain \(\Omega \subset \mathbb{R}^{d}\): \(-\nabla _{x}\cdot (A^{\varepsilon }\nabla _{x}u^{\varepsilon })=f\), \(x\in \Omega \), with the homogeneous Dirichlet condition: \(u^{\varepsilon }=0\), \(x\in \partial \Omega \), where \(A^{\varepsilon }(x)=A(x/\varepsilon )\), \(A(\cdot )\) being 1-periodic and positive definite: \(A:\mathbb{T}^{d}\rightarrow \mathbb{R} _{\mathrm{sym},\succ 0}^{d\times d}\). The source term \(f\in L^{2}(\Omega ;\mathbb{R})\) and has no microscale variation with respect to \(x/\varepsilon \). The homogenized equation is: \(-\nabla _{x}\cdot (\overline{A}\nabla _{x}u)=f\), \(x\in \Omega \), \(u=0\), \(x\in \partial \Omega \), where \(\overline{A}\) is given by \( \overline{A}=\int_{\mathbb{T}^{d}}(A(y)+A(y)\nabla \chi (y)^{T})dy\), and \( \chi :\mathbb{T}^{d}\rightarrow \mathbb{R}^{d}\) solves the cell problem \( -\nabla \cdot (A\nabla \chi )=\nabla \cdot A\), with \(\chi \) 1-periodic. The authors define the solution map \(G:A\rightarrow \chi \) and the map \( F:A\rightarrow \overline{A}\). \N\NThe main results of the paper state and prove universal approximation theorems for the maps \(G\) and \(F\), using the notion of Fourier neural operator. Let \(K\subset PD_{\alpha, \beta }=\{A\in L^{\infty }(\mathbb{T}^{d};\mathbb{R}^{d\times d}):\forall (y,\xi )\in \mathbb{T}^{d}\times \mathbb{R}^{d}\), \(\alpha \left\vert \xi \right\vert ^{2}\leq \left\langle \xi, A(y)\xi \right\rangle \leq \beta \left\vert \xi \right\vert ^{2}\}\), such that \(K\) is compact in \(L^{2}(\mathbb{T}^{d}; \mathbb{R}^{d})\). Then, for any \(\varepsilon >0\), there exists a Fourier neural operator \(\Psi :K\rightarrow H^{1}(\mathbb{T}^{d};\mathbb{R}^{d})\) such that \(\sup_{A\in K}\left\Vert G(A)-\Psi (A)\right\Vert _{\overset{.}{H} ^{1}}<\varepsilon \), and a Fourier neural operator \(\Phi :K\rightarrow L^{\infty }(\mathbb{T}^{d};\mathbb{R}^{d\times d})\) such that \(\sup_{A\in K}\sup_{x\in \mathbb{T}^{d}}\left\Vert F(A)-\Phi (A)(x)\right\Vert _{F}<\varepsilon \). \N\NFor the proofs, the authors refer to the paper by \textit{N. Kovachki} et al. [J. Mach. Learn. Res. 22, Paper No. 290, 76 p. (2021; Zbl 07626805)]. In the last part of their paper, the authors present numerical simulations for different microstructures which satisfy the hypotheses of the main results: smooth microstructures, star-shaped or square inclusions, and Voronoi interfaces. They analyze relative \(H^{1}\) and relative \(W^{1,10}\) errors between the true solution \(\chi \) and its approximation through the Fourier neural operator. They also analyze a relative error for \(\overline{A}\).