Deep Learning-based Surrogate Modelling of the LOD Method for Multiscale Problems

Multiscale problems are notoriously difficult to tackle using traditional numerical methods, as accurately resolving fine-scale features often requires prohibitively fine discretizations. This challenge is particularly pronounced in applications such as materials science, fluid dynamics, climate systems, chemical processes, and complex networks. Recent neural operator models provide a promising data-driven alternative, but frequently struggle to achieve sufficient accuracy in the presence of strongly heterogeneous or oscillatory coefficients. In this work, we focus on the solution of elliptic PDEs with rough and high-contrast inputs. The Localized Orthogonal Decomposition (LOD) method is a well-established numerical approach for such problems, but it comes, however, at a substantial computational cost. We investigate the performance of popular neural operator architectures on these challenging multiscale problems and identify key limitations in their ability to resolve fine-scale structure. To overcome these challenges, we introduce LOD-MSNO (LOD-Multiscale Neural Operator), a hybrid approach that leverages the LOD method as a strong multiscale prior by building on its representation of the solution as a linear combination of problem-adapted basis functions, while addressing its main computational bottlenecks through data-driven operator learning. We further provide theoretical error estimates for the proposed coefficient-learning framework. Lastly, we demonstrate the potential of our proposed method to outperform current neural operator baselines in terms of accuracy for challenging multiscale inputs, while mainly retaining the computational efficiency of neural operator models.

RepNet: Tackling spectral bias in deep neural networks via parameter reparameterization

Deep neural networks (DNNs) have achieved remarkable success in scientific computing, yet they often suffer from spectral bias in capturing oscillatory and multiscale behaviors. In this study, we investigate this limitation by examining the failure of shallow ReLU neural networks in fitting high-frequency functions. This observation identifies two important factors in resolving rapid oscillations: the initial slope scale and the distribution of partition points induced by the networks. Motivated by this analysis, we propose RepNet, a reparameterized DNN model for ReLU and tanh networks designed for high-frequency and multiscale problems. The key idea is to reparameterize the weights and biases in the first hidden layer, which enables effective control of the initial slope scale and provides an appropriate distribution of the initial partition points. Furthermore, treating the reparameterized weights and biases as trainable parameters allows the DNN to achieve adaptive frequency scaling during training. In addition, we derive quantitative estimates for the output and slope magnitudes of the reparameterized DNN to guide the initialization of the proposed method. Numerical experiments, including multiscale one- and four-dimensional function approximation, forward and inverse PDE problems in combination with physics-informed neural networks (PINNs), and operator learning, demonstrate that RepNet improves the predicted accuracy of vanilla DNNs in capturing highly oscillatory features with slightly additional computational cost. These results indicate that RepNet provides an effective and flexible approach for overcoming spectral bias and applying DNNs to multiscale problems.