Neural numerical homogenization based on Deep Ritz corrections

TL;DR

The paper tackles multiscale parabolic PDEs with highly oscillatory coefficients $a(t,x)$, common in battery-like materials, by combining Localized Orthogonal Decomposition (LOD) with a Deep Ritz-based neural correction to learn coefficient-dependent local corrections. A neural surrogate is trained to reproduce energy-minimizing corrections on small patches, enabling online updates without recomputing the entire multiscale space when $a$ varies in time or is uncertain. Key contributions include a self-adaptive loss balancing scheme within a Deep Ritz framework, a localization-aware training procedure on patches, and demonstration on two battery-inspired examples showing accurate coarse solves with substantial online efficiency gains. This approach offers a practical route to robust, efficient space-time homogenization for complex multiscale coefficients while preserving the accuracy of classical LOD methods.

Abstract

Numerical homogenization methods aim at providing appropriate coarse-scale approximations of solutions to (elliptic) partial differential equations that involve highly oscillatory coefficients. The localized orthogonal decomposition (LOD) method is an effective way of dealing with such coefficients, especially if they are non-periodic and non-smooth. It modifies classical finite element basis functions by suitable fine-scale corrections. In this paper, we make use of the structure of the LOD method, but we propose to calculate the corrections based on a Deep Ritz approach involving a parametrization of the coefficients to tackle temporal variations or uncertainties. Numerical examples for a parabolic model problem are presented to assess the performance of the approach.

Figures (11)

• Figure 1: Examples of possible patch configurations
• Figure 2: Illustration of a fully-connected feed-forward artificial neural network. The neurons are depicted as circles, and the connections represented by black arrows.
• Figure 3: The respective parts of the total loss function for the correction of the basis function associated to node $(0.5,0.5)$ in Example 1. On the left the energy functional loss $\widehat{\mathcal{L}}_\mathrm{energy}$ and on the right the interpolation loss $\widehat{\mathcal{L}}_\mathrm{interp}$ in a log-log-plot is shown.
• Figure 4: Illustrations of different corrected basis functions $\widetilde{\Lambda}_j^\ell$ for each possible patch configurations for $\ell=1$ in Example 1. The top row shows the classical LOD basis functions, while the bottom row depicts their learned analogs.
• Figure 5: Examples of the cross sections with fixed $y$-coordinate through the domain of three different basis functions and the learned counterparts.

Theorems & Definitions (12)

• Definition 2.1: Patch
• Definition 2.2: Local projective quasi-interpolation operator
• Definition 2.3: Localized Corrections
• Definition 3.1: Realization and network function
• Remark 3.2
• Definition 3.3: Localization of the network function
• Remark 3.4
• Remark 3.5
• Definition 3.6: Loss functions
• Remark 3.7