Learning a generalized multiscale prolongation operator

TL;DR

The paper addresses solving Darcy flow in heterogeneous media with high-contrast random permeability by learning a generalized multiscale prolongation operator for a two-grid preconditioner. It introduces a subspace-distance loss and two data-augmentation strategies—symmetry transformations and Karhunen–Loève expansion—to train a four-level U-Net that outputs prolongation blocks directly from coarse-element coefficients. Results show the neural network-generated prolongation operator can be produced up to five times faster than solving local spectral problems, while preserving the efficiency of the two-grid preconditioner and generalizing to unseen permeability fields and different coarse-element sizes. This approach yields a universal operator applicable across boundary conditions and resolutions, with potential for further improvement by jointly learning the smoother along with the prolongation operator.

Abstract

In this research, we address Darcy flow problems with random permeability using iterative solvers, enhanced by a two-grid preconditioner based on a generalized multiscale prolongation operator, which has been demonstrated to be stable for high contrast profiles. To circumvent the need for repeatedly solving spectral problems with varying coefficients, we harness deep learning techniques to expedite the construction of the generalized multiscale prolongation operator. Considering linear transformations on multiscale basis have no impact on the performance of the preconditioner, we devise a loss function by the coefficient-based distance between subspaces instead of the plain $l^2$-norm of the difference of the corresponding multiscale bases. We discover that leveraging the inherent symmetry in the local spectral problem can effectively accelerate the neural network training process. In scenarios where training data are limited, we utilize the Karhunen-Loève expansion to augment the dataset. Extensive numerical experiments with various types of random coefficient models are exhibited, showing that the proposed method can significantly reduce the time required to generate the prolongation operator while maintaining the original efficiency of the two-grid preconditioner. Notably, the neural network demonstrates strong generalization capabilities, as evidenced by its satisfactory performance on unseen random permeability fields.

Figures (12)

• Figure 1: An illustration of the two-scale mesh: $\tau$ is a fine element and $K_{j}$ is a coarse element which contains $16$ fine elements.
• Figure 2: Two-level, three-level and four-level U-Net architectures.
• Figure 3: An intuitive diagram of distance defined in the loss function. $\overrightarrow{OA_{k}}$ is an orthonormal basis of subspace $X^{(2)}$, $\overrightarrow{OB_{k}}$ is the projection of $\overrightarrow{OA_k}$ onto the subspace $X^{(1)}$ and $\overrightarrow{A_{k}B_{k}}$ is the residual of $\overrightarrow{OA_{k}}$ relative to the subspace $X^{(1)}$.
• Figure 4: Examples of symmetry transformation of $4 \times 4$ grid.
• Figure 5: Test loss for neural networks of varying depths on the same dataset is displayed in two parts: left part shows results across all $30$ epochs, and right part presents an enlarged view of the results for the last $10$ epochs.

Theorems & Definitions (11)

• Definition 1
• Remark
• Theorem 3.1
• proof
• Definition 2
• Lemma 3.2
• Theorem 3.3
• proof
• Theorem 3.4
• proof