Multiscale Corrections by Continuous Super-Resolution

TL;DR

The paper tackles the challenge of simulating multiscale PDEs without prohibitive fine-mesh computations by introducing NH-CSR, a coefficient-guided continuous super-resolution framework. It maps a coarse finite element solution and the coefficient map A to a high-resolution solution using a three-part architecture: global feature extraction, Gabor wavelet-based local encoding (WIRE), and a multiscale implicit image function (MS-IIF). A novel loss combining L1 data fidelity with a stochastic cosine similarity term enforces both pixel accuracy and non-local structural alignment, enabling robust in-distribution and out-of-distribution upscaling. Empirical results show NH-CSR outperforms state-of-the-art continuous SR methods on synthetic multiscale FE data and real-world soil patterns, demonstrating strong generalization and practical impact for numerical homogenization and multiscale modeling.

Abstract

Finite element methods typically require a high resolution to satisfactorily approximate micro and even macro patterns of an underlying physical model. This issue can be circumvented by appropriate multiscale strategies that are able to obtain reasonable approximations on under-resolved scales. In this paper, we study the implicit neural representation and propose a continuous super-resolution network as a correction strategy for multiscale effects. It can take coarse finite element data to learn both in-distribution and out-of-distribution high-resolution finite element predictions. Our highlight is the design of a local implicit transformer, which is able to learn multiscale features. We also propose Gabor wavelet-based coordinate encodings, which can overcome the bias of neural networks learning low-frequency features. Finally, perception is often preferred over distortion, so scientists can recognize the visual pattern for further investigation. However, implicit neural representation is known for its lack of local pattern supervision. We propose to use stochastic cosine similarities to compare the local feature differences between prediction and ground truth. It shows better performance on structural alignments. Our experiments show that our proposed strategy achieves superior performance as an in-distribution and out-of-distribution super-resolution strategy.

Figures (11)

• Figure 1: Continuous super-resolution for finite element data. We use the proposed method to apply multiscale upscaling to the low-resolution image, both in-distribution (orange region, upscaling factor not larger than 16) and out-of-distribution (green region, upscaling factor larger than 16). We see from the comparison that ours achieves better visualization than the state-of-the-art LIIF LIIF approach. Note that the actual values are first normalized to [0, 255], then multiplied by a factor of 64, and taken modulo 255 to again be in the value range between 0 and 255. Finally, the result is converted to the RGB color space using the Matplotlib colormap to better visualize differences. This procedure aims at highlighting fine differences in simulation results and leads to the observed wavy patterns.
• Figure 2: The proposed super-resolution model for numerical homogenization. (a) shows the overall architecture and (b) shows the detailed operator of LIIF used in coarse and fine LIIF modules. Given a coarse fine element solution represented by $X$ and the corresponding coefficient $A$, we learn a conditional feature extractor using off-the-self neural networks. The extracted feature is then sampled based on the random coordinate map $c$, so that we obtain sampled feature points and grid points. Coarse and fine implicit image functions (IIF) take as input the paired feature and grid points learned by the proposed WIRE operation to form the regression model for super-resolution. The results of coarse- and fine-grained pixel regression are summed together with the sampled LR pixels to obtain the final super-resolved pixels.
• Figure 3: Examples of coefficient maps used for finite element data generation. We show six examples of different coefficients. The visualization is done by normalizing the coefficient value to [0, 255].
• Figure 4: Visual comparison of the coarse finite element solutions. We use the proposed method to apply multiscale upsampling to the coarse finite solution, both in-distribution (upsampling factor no larger than 16) and out-of-distribution (upsampling factor larger than 16) scenarios. Ours achieves the best visualization among others. As before, the pixel values are multiplied by a factor 8 and shifted to a value range between 0 and 255 to better visualize differences. More precisely, the multiplied values are taken modulo 255.
• Figure 5: Plots of sliced pixels from different super-resolution approaches. To better visualize the individual pixel differences, we use different methods to apply 16-times super-resolution on one coarse finite element data, then we slice the middle row and column of pixels (indicated by the red lines) and plot their values by coordinates. We can see that our approach better aligns with the HR data.