Data-induced multiscale losses and efficient multirate gradient descent schemes

This paper investigates the impact of multiscale data on machine learning algorithms, particularly in the context of deep learning. A dataset is multiscale if its distribution shows large variations in scale across different directions. This paper reveals multiscale structures in the loss landscape, including its gradients and Hessians inherited from the data. Correspondingly, it introduces a novel gradient descent approach, drawing inspiration from multiscale algorithms used in scientific computing. This approach seeks to transcend empirical learning rate selection, offering a more systematic, data-informed strategy to enhance training efficiency, especially in the later stages.

Deeper or Wider: A Perspective from Optimal Generalization Error with Sobolev Loss

Constructing the architecture of a neural network is a challenging pursuit for the machine learning community, and the dilemma of whether to go deeper or wider remains a persistent question. This paper explores a comparison between deeper neural networks (DeNNs) with a flexible number of layers and wider neural networks (WeNNs) with limited hidden layers, focusing on their optimal generalization error in Sobolev losses. Analytical investigations reveal that the architecture of a neural network can be significantly influenced by various factors, including the number of sample points, parameters within the neural networks, and the regularity of the loss function. Specifically, a higher number of parameters tends to favor WeNNs, while an increased number of sample points and greater regularity in the loss function lean towards the adoption of DeNNs. We ultimately apply this theory to address partial differential equations using deep Ritz and physics-informed neural network (PINN) methods, guiding the design of neural networks.

Deep Neural Networks and Finite Elements of Any Order on Arbitrary Dimensions

In this study, we establish that deep neural networks employing ReLU and ReLU$^2$ activation functions can effectively represent Lagrange finite element functions of any order on various simplicial meshes in arbitrary dimensions. We introduce two novel formulations for globally expressing the basis functions of Lagrange elements, tailored for both specific and arbitrary meshes. These formulations are based on a geometric decomposition of the elements, incorporating several insightful and essential properties of high-dimensional simplicial meshes, barycentric coordinate functions, and global basis functions of linear elements. This representation theory facilitates a natural approximation result for such deep neural networks. Our findings present the first demonstration of how deep neural networks can systematically generate general continuous piecewise polynomial functions on both specific or arbitrary simplicial meshes.

Expressivity and Approximation Properties of Deep Neural Networks with ReLU$^k$ Activation

In this paper, we investigate the expressivity and approximation properties of deep neural networks employing the ReLU$^k$ activation function for $k \geq 2$. Although deep ReLU networks can approximate polynomials effectively, deep ReLU$^k$ networks have the capability to represent higher-degree polynomials precisely. Our initial contribution is a comprehensive, constructive proof for polynomial representation using deep ReLU$^k$ networks. This allows us to establish an upper bound on both the size and count of network parameters. Consequently, we are able to demonstrate a suboptimal approximation rate for functions from Sobolev spaces as well as for analytic functions. Additionally, through an exploration of the representation power of deep ReLU$^k$ networks for shallow networks, we reveal that deep ReLU$^k$ networks can approximate functions from a range of variation spaces, extending beyond those generated solely by the ReLU$^k$ activation function. This finding demonstrates the adaptability of deep ReLU$^k$ networks in approximating functions within various variation spaces.

MgNO: Efficient Parameterization of Linear Operators via Multigrid

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the $i$-th neuron in a nonlinear operator layer is defined by $\mathcal O_i(u) = \sigma\left( \sum_j \mathcal W_{ij} u + \mathcal B_{ij}\right)$. Here, $\mathcal W_{ij}$ denotes the bounded linear operator connecting $j$-th input neuron to $i$-th output neuron, and the bias $\mathcal B_{ij}$ takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to other CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs).

On the Optimal Expressive Power of ReLU DNNs and Its Application in Approximation with Kolmogorov Superposition Theorem

This paper is devoted to studying the optimal expressive power of ReLU deep neural networks (DNNs) and its application in approximation via the Kolmogorov Superposition Theorem. We first constructively prove that any continuous piecewise linear functions on $[0,1]$, comprising $O(N^2L)$ segments, can be represented by ReLU DNNs with $L$ hidden layers and $N$ neurons per layer. Subsequently, we demonstrate that this construction is optimal regarding the parameter count of the DNNs, achieved through investigating the shattering capacity of ReLU DNNs. Moreover, by invoking the Kolmogorov Superposition Theorem, we achieve an enhanced approximation rate for ReLU DNNs of arbitrary width and depth when dealing with continuous functions in high-dimensional spaces.

Linear Regression on Manifold Structured Data: the Impact of Extrinsic Geometry on Solutions

In this paper, we study linear regression applied to data structured on a manifold. We assume that the data manifold is smooth and is embedded in a Euclidean space, and our objective is to reveal the impact of the data manifold's extrinsic geometry on the regression. Specifically, we analyze the impact of the manifold's curvatures (or higher order nonlinearity in the parameterization when the curvatures are locally zero) on the uniqueness of the regression solution. Our findings suggest that the corresponding linear regression does not have a unique solution when the embedded submanifold is flat in some dimensions. Otherwise, the manifold's curvature (or higher order nonlinearity in the embedding) may contribute significantly, particularly in the solution associated with the normal directions of the manifold. Our findings thus reveal the role of data manifold geometry in ensuring the stability of regression models for out-of-distribution inferences.

An Enhanced V-cycle MgNet Model for Operator Learning in Numerical Partial Differential Equations

This study used a multigrid-based convolutional neural network architecture known as MgNet in operator learning to solve numerical partial differential equations (PDEs). Given the property of smoothing iterations in multigrid methods where low-frequency errors decay slowly, we introduced a low-frequency correction structure for residuals to enhance the standard V-cycle MgNet. The enhanced MgNet model can capture the low-frequency features of solutions considerably better than the standard V-cycle MgNet. The numerical results obtained using some standard operator learning tasks are better than those obtained using many state-of-the-art methods, demonstrating the efficiency of our model.Moreover, numerically, our new model is more robust in case of low- and high-resolution data during training and testing, respectively.

FV-MgNet: Fully Connected V-cycle MgNet for Interpretable Time Series Forecasting

By investigating iterative methods for a constrained linear model, we propose a new class of fully connected V-cycle MgNet for long-term time series forecasting, which is one of the most difficult tasks in forecasting. MgNet is a CNN model that was proposed for image classification based on the multigrid (MG) methods for solving discretized partial differential equations (PDEs). We replace the convolutional operations with fully connected operations in the existing MgNet and then apply them to forecasting problems. Motivated by the V-cycle structure in MG, we further propose the FV-MgNet, a V-cycle version of the fully connected MgNet, to extract features hierarchically. By evaluating the performance of FV-MgNet on popular data sets and comparing it with state-of-the-art models, we show that the FV-MgNet achieves better results with less memory usage and faster inference speed. In addition, we develop ablation experiments to demonstrate that the structure of FV-MgNet is the best choice among the many variants.

Side-effects of Learning from Low Dimensional Data Embedded in an Euclidean Space

The low dimensional manifold hypothesis posits that the data found in many applications, such as those involving natural images, lie (approximately) on low dimensional manifolds embedded in a high dimensional Euclidean space. In this setting, a typical neural network defines a function that takes a finite number of vectors in the embedding space as input. However, one often needs to consider evaluating the optimized network at points outside the training distribution. This paper considers the case in which the training data is distributed in a linear subspace of $\mathbb R^d$. We derive estimates on the variation of the learning function, defined by a neural network, in the direction transversal to the subspace. We study the potential regularization effects associated with the network's depth and noise in the codimension of the data manifold. We also present additional side effects in training due to the presence of noise.