Learning local Dirichlet-to-Neumann maps of nonlinear elliptic PDEs with rough coefficients

TL;DR

This work tackles nonlinear elliptic PDEs with rough, high-contrast coefficients by extending the Multiscale Finite Element Method (MsFEM) with learned local Dirichlet-to-Neumann (DtN) maps. The authors develop a discrete, substructured framework where subdomain coupling is mediated by DtN maps, and they replace coarse DtN maps with surrogate operators learned by neural networks, trained with losses on values, derivatives, and monotonicity. The key contributions include a detailed discrete substructuring formulation, explicit expressions for discrete DtN maps and their derivatives, and a practical workflow for training surrogate DtN operators that accelerate Newton iterations while preserving accuracy. Numerical experiments in 1D and 2D demonstrate that learned DtN surrogates achieve percent-level accuracy with modest training and substantially reduce online computation, offering a scalable approach for multiscale nonlinear PDEs, with planned extensions to nonperiodic media and enhanced training strategies.

Abstract

Partial differential equations (PDEs) involving high contrast and oscillating coefficients are common in scientific and industrial applications. Numerical approximation of these PDEs is a challenging task that can be addressed, for example, by multi-scale finite element analysis. For linear problems, multi-scale finite element method (MsFEM) is well established and some viable extensions to non-linear PDEs are known. However, some features of the method seem to be intrinsically based on linearity-based. In particular, traditional MsFEM rely on the reuse of computations. For example, the stiffness matrix can be calculated just once, while being used for several right-hand sides, or as part of a multi-level iterative algorithm. Roughly speaking, the offline phase of the method amounts to pre-assembling the local linear Dirichlet-to-Neumann (DtN) operators. We present some preliminary results concerning the combination of MsFEM with machine learning tools. The extension of MsFEM to nonlinear problems is achieved by means of learning local nonlinear DtN maps. The resulting learning-based multi-scale method is tested on a set of model nonlinear PDEs involving the $p-$Laplacian and degenerate nonlinear diffusion.

Figures (14)

• Figure 1: Heterogeneous domain $K_\epsilon(\mathbf{x})$ with $N=5 \times 5$ coarse cells. Nonoverlapping skeleton $\Gamma$ is denoted by thick dark blue lines.
• Figure 2: Representation of the learning workflow for the operator. Each neural network is depicted as a graph and will be responsible for learning each component of the $\widetilde{\text{DtN}}_{H,i}$ operator. The first column of neurons (from left to right in NN) is the input layer, taking inputs $(u_1, u_2, \cdots, u_s)$ , and the last column is the output layer with output $f_{N_{H i}} \approx \widetilde{\text{DtN}}_{H,i}$. The intermediate columns represent the hidden layers.
• Figure 3: Partitioning of the model domain and the plot of $K_\epsilon(x)$
• Figure 4: Qualitative comparison of the surfaces $z = {\rm DtN}_{H,i}((u_{\rm left}, u_{\rm right}))^{1}$ (solid) and $z = \widetilde{\rm DtN} _{H,i}((u_{\rm left}, u_{\rm right}))^{1}$ (wireframe). Left column (blue): only DtN values included. Middle column (green): DtN and DtN derivatives included. Right column (purple): DtN values, derivatives, and monotonicity assertion included. Top to bottom: $n_s = 2^2$ training points, $n_s=2^2+1$ training points, $n_s=3^2$ training points.
• Figure 5: Left: Interpolation relative $L^2$ error. Right: Solution relative $L^2$ error.