Yahong Yang

Yahong Yang, Qipin Chen, Wenrui Hao

In this paper, we present a novel training approach called the Homotopy Relaxation Training Algorithm (HRTA), aimed at accelerating the training process in contrast to traditional methods. Our algorithm incorporates two key mechanisms: one involves building a homotopy activation function that seamlessly connects the linear activation function with the ReLU activation function; the other technique entails relaxing the homotopy parameter to enhance the training refinement process. We have conducted an in-depth analysis of this novel method within the context of the neural tangent kernel (NTK), revealing significantly improved convergence rates. Our experimental results, especially when considering networks with larger widths, validate the theoretical conclusions. This proposed HRTA exhibits the potential for other activation functions and deep neural networks.

Yahong Yang, Yang Xiang

In this paper, we establish a neural network to approximate functionals, which are maps from infinite dimensional spaces to finite dimensional spaces. The approximation error of the neural network is $O(1/\sqrt{m})$ where $m$ is the size of networks, which overcomes the curse of dimensionality. The key idea of the approximation is to define a Barron spectral space of functionals.

Yahong Yang, Yulong Lu

This paper establishes the nearly optimal rate of approximation for deep neural networks (DNNs) when applied to Korobov functions, effectively overcoming the curse of dimensionality. The approximation results presented in this paper are measured with respect to $L_p$ norms and $H^1$ norms. Our achieved approximation rate demonstrates a remarkable "super-convergence" rate, outperforming traditional methods and any continuous function approximator. These results are non-asymptotic, providing error bounds that consider both the width and depth of the networks simultaneously.

Chuqi Chen, Yahong Yang, Yang Xiang et al.

Neural network-based approaches have recently shown significant promise in solving partial differential equations (PDEs) in science and engineering, especially in scenarios featuring complex domains or incorporation of empirical data. One advantage of the neural network methods for PDEs lies in its automatic differentiation (AD), which necessitates only the sample points themselves, unlike traditional finite difference (FD) approximations that require nearby local points to compute derivatives. In this paper, we quantitatively demonstrate the advantage of AD in training neural networks. The concept of truncated entropy is introduced to characterize the training property. Specifically, through comprehensive experimental and theoretical analyses conducted on random feature models and two-layer neural networks, we discover that the defined truncated entropy serves as a reliable metric for quantifying the residual loss of random feature models and the training speed of neural networks for both AD and FD methods. Our experimental and theoretical analyses demonstrate that, from a training perspective, AD outperforms FD in solving PDEs.

Yahong Yang, Juncai He

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.

Jiaqi Luo, Yahong Yang, Yuan Yuan et al.

This paper introduces Residual-based Smote (RSmote), an innovative local adaptive sampling technique tailored to improve the performance of Physics-Informed Neural Networks (PINNs) through imbalanced learning strategies. Traditional residual-based adaptive sampling methods, while effective in enhancing PINN accuracy, often struggle with efficiency and high memory consumption, particularly in high-dimensional problems. RSmote addresses these challenges by targeting regions with high residuals and employing oversampling techniques from imbalanced learning to refine the sampling process. Our approach is underpinned by a rigorous theoretical analysis that supports the effectiveness of RSmote in managing computational resources more efficiently. Through extensive evaluations, we benchmark RSmote against the state-of-the-art Residual-based Adaptive Distribution (RAD) method across a variety of dimensions and differential equations. The results demonstrate that RSmote not only achieves or exceeds the accuracy of RAD but also significantly reduces memory usage, making it particularly advantageous in high-dimensional scenarios. These contributions position RSmote as a robust and resource-efficient solution for solving complex partial differential equations, especially when computational constraints are a critical consideration.

Wenrui Hao, Xinliang Liu, Yahong Yang

Solving nonlinear partial differential equations (PDEs) with multiple solutions using neural networks has found widespread applications in various fields such as physics, biology, and engineering. However, classical neural network methods for solving nonlinear PDEs, such as Physics-Informed Neural Networks (PINN), Deep Ritz methods, and DeepONet, often encounter challenges when confronted with the presence of multiple solutions inherent in the nonlinear problem. These methods may encounter ill-posedness issues. In this paper, we propose a novel approach called the Newton Informed Neural Operator, which builds upon existing neural network techniques to tackle nonlinearities. Our method combines classical Newton methods, addressing well-posed problems, and efficiently learns multiple solutions in a single learning process while requiring fewer supervised data points compared to existing neural network methods.

Yahong Yang, Juncai He

This paper establishes a comprehensive approximation result for deep fully-connected neural networks with commonly-used and general activation functions in Sobolev spaces $W^{n,\infty}$, with errors measured in the $W^{m,p}$-norm for $m < n$ and $1\le p \le \infty$. The derived rates surpass those of classical numerical approximation techniques, such as finite element and spectral methods, exhibiting a phenomenon we refer to as \emph{super-convergence}. Our analysis shows that deep networks with general activations can approximate weak solutions of partial differential equations (PDEs) with superior accuracy compared to traditional numerical methods at the approximation level. Furthermore, this work closes a significant gap in the error-estimation theory for neural-network-based approaches to PDEs, offering a unified theoretical foundation for their use in scientific computing.

Yahong Yang, Wei Zhu

We investigate the generalization error of group-invariant neural networks within the Barron framework. Our analysis shows that incorporating group-invariant structures introduces a group-dependent factor $δ_{G,Γ,σ} \le 1$ into the approximation rate. When this factor is small, group invariance yields substantial improvements in approximation accuracy. On the estimation side, we establish that the Rademacher complexity of the group-invariant class is no larger than that of the non-invariant counterpart, implying that the estimation error remains unaffected by the incorporation of symmetry. Consequently, the generalization error can improve significantly when learning functions with inherent group symmetries. We further provide illustrative examples demonstrating both favorable cases, where $δ_{G,Γ,σ}\approx |G|^{-1}$, and unfavorable ones, where $δ_{G,Γ,σ}\approx 1$. Overall, our results offer a rigorous theoretical foundation showing that encoding group-invariant structures in neural networks leads to clear statistical advantages for symmetric target functions.

Chuqi Chen, Yahong Yang, Yang Xiang et al.

Solving partial differential equations (PDEs) using neural networks has become a central focus in scientific machine learning. Training neural networks for singularly perturbed problems is particularly challenging due to certain parameters in the PDEs that introduce near-singularities in the loss function. In this study, we overcome this challenge by introducing a novel method based on homotopy dynamics to effectively manipulate these parameters. From a theoretical perspective, we analyze the effects of these parameters on training difficulty in these singularly perturbed problems and establish the convergence of the proposed homotopy dynamics method. Experimentally, we demonstrate that our approach significantly accelerates convergence and improves the accuracy of these singularly perturbed problems. These findings present an efficient optimization strategy leveraging homotopy dynamics, offering a robust framework to extend the applicability of neural networks for solving singularly perturbed differential equations.

Yulong Lu, Tong Mao, Jinchao Xu et al.

Deep neural networks have been widely used as universal approximators for functions with inherent physical structures, including permutation symmetry. In this paper, we construct symmetric deep neural networks to approximate symmetric Korobov functions and prove that both the convergence rate and the constant prefactor scale at most polynomially with respect to the ambient dimension. This represents a substantial improvement over prior approximation guarantees that suffer from the curse of dimensionality. Building on these approximation bounds, we further derive a generalization-error rate for learning symmetric Korobov functions whose leading factors likewise avoid the curse of dimensionality.

Yahong Yang, Sun Lee, Jeff Calder et al.

We derive an energy-based continuum limit for $\varepsilon$-graphs endowed with a general connectivity functional. We prove that the discrete energy and its continuum counterpart differ by at most $O(\varepsilon)$; the prefactor involves only the $W^{1,1}$-norm of the connectivity density as $\varepsilon\to0$, so the error bound remains valid even when that density has strong local fluctuations. As an application, we introduce a neural-network procedure that reconstructs the connectivity density from edge-weight data and then embeds the resulting continuum model into a brain-dynamics framework. In this setting, the usual constant diffusion coefficient is replaced by the spatially varying coefficient produced by the learned density, yielding dynamics that differ significantly from those obtained with conventional constant-diffusion models.

Yahong Yang, Haizhao Yang, Yang Xiang

This paper addresses the problem of nearly optimal Vapnik--Chervonenkis dimension (VC-dimension) and pseudo-dimension estimations of the derivative functions of deep neural networks (DNNs). Two important applications of these estimations include: 1) Establishing a nearly tight approximation result of DNNs in the Sobolev space; 2) Characterizing the generalization error of machine learning methods with loss functions involving function derivatives. This theoretical investigation fills the gap of learning error estimations for a wide range of physics-informed machine learning models and applications including generative models, solving partial differential equations, operator learning, network compression, distillation, regularization, etc.