arXiv subjects

Yahong Yang

Publications and source records attributed to Yahong Yang.

At least 19 recordsLinked to original sources

arXiv2026

Generalization Guarantees for Multi-Input Neural Operator Learning in Sobolev Spaces

We develop approximation and generalization error estimates for multi-input neural operators, with the output error measured in Sobolev norms. In contrast to standard operator-learning settings with a single input function, our framework allows multiple input functions defined on possibly different domains, with different dimensions and Sobolev regularities. The derived rates explicitly quantify the contribution of each input space to the final error bound. In particular, in the balanced regime, the approximation and generalization rates are governed by the interaction between the input dimensions, regularities, and Sobolev orders, while the dependence on the model complexity retains a $\log\log/\log$-type structure. Our analysis provides a general theoretical framework for multi-input operator learning, including Sobolev training, and is applicable to operator learning problems arising from partial differential equations and scientific computing.

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arXiv2026

Deep Neural Networks with General Activations: Super-Convergence in Sobolev Norms

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.

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arXiv2025

Multiscale Neural Networks for Approximating Green's Functions

Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) in the applications of physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is learning Green's functions. However, Green's functions are notoriously difficult to learn due to their poor regularity, which typically requires larger NNs and longer training times. In this paper, we address these challenges by leveraging multiscale NNs to learn Green's functions. Through theoretical analysis using multiscale Barron space methods and experimental validation, we show that the multiscale approach significantly reduces the necessary NN size and accelerates training.

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arXiv2025

On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions

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.

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arXiv2025

Energy Approach from $\varepsilon$-Graph to Continuum Diffusion Model with Connectivity Functional

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.

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arXiv2025

DeepONet for Solving Nonlinear Partial Differential Equations with Physics-Informed Training

In this paper, we investigate the applications of operator learning, specifically DeepONet, for solving nonlinear partial differential equations (PDEs). Unlike conventional function learning methods that require training separate neural networks for each PDE, operator learning enables generalization across different PDEs without retraining. This study examines the performance of DeepONet in physics-informed training, focusing on two key aspects: (1) the approximation capabilities of deep branch and trunk networks, and (2) the generalization error in Sobolev norms. Our results show that complex branch networks provide substantial performance gains, while trunk networks are most effective when kept relatively simple. Furthermore, we derive a bound on the generalization error of DeepONet for solving nonlinear PDEs by analyzing the Rademacher complexity of its derivatives in terms of pseudo-dimension. This work bridges a critical theoretical gap by delivering rigorous error estimates. This paper fills a theoretical gap by providing error estimates for a wide range of physics-informed machine learning models and applications.

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arXiv2025

Statistical Learning Guarantees for Group-Invariant Barron Functions

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.

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arXiv2025

Nearly Optimal Approximation Rates for Deep Super ReLU Networks on Sobolev Spaces

This paper introduces deep super ReLU networks (DSRNs) as a method for approximating functions in Sobolev spaces measured by Sobolev norms $W^{m,p}$ for $m\in\mathbb{N}$ with $m\ge 2$ and $1\le p\le +\infty$. Standard ReLU deep neural networks (ReLU DNNs) cannot achieve this goal. DSRNs consist primarily of ReLU DNNs, and several layers of the square of ReLU added at the end to smooth the networks output. This approach retains the advantages of ReLU DNNs, leading to the straightforward training. The paper also proves the optimality of DSRNs by estimating the VC-dimension of higher-order derivatives of DNNs, and obtains the generalization error in Sobolev spaces via an estimate of the pseudo-dimension of higher-order derivatives of DNNs.

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arXiv2025

Learn Singularly Perturbed Solutions via Homotopy Dynamics

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.

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arXiv2025

On pattern formation in the thermodynamically-consistent variational Gray-Scott model

In this paper, we explore pattern formation in a four-species variational Gary-Scott model, which includes all reverse reactions and introduces a virtual species to describe the birth-death process in the classical Gray-Scott model. This modification transforms the classical Gray-Scott model into a thermodynamically consistent closed system. The classical two-species Gray-Scott model can be viewed as a subsystem of the variational model in the limiting case when the small parameter $ε$, related to the reaction rate of the reverse reactions, approaches zero. We numerically explore pattern formation in this physically more complete Gray-Scott model in one spatial dimension, using non-uniform steady states of the classical model as initial conditions. By decreasing $ε$, we observed that the stationary pattern in the classical Gray-Scott model can be stabilized as the transient state in the variational model for a significantly small $ε$. Additionally, the variational model admits oscillating and traveling-wave-like patterns for small $ε$. The persistent time of these patterns is on the order of $O(ε^{-1})$. We also analyze the energy stability of two uniform steady states in the variational Gary-Scott model for fixed $ε$. Although both states are stable in a certain sense, the gradient flow type dynamics of the variational model exhibit a selection effect based on the initial conditions, with pattern formation occurring only if the initial condition does not converge to the boundary steady state, which corresponds to the trivial uniform steady state in the classical Gray-Scott model.

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arXiv2025

Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equations

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.

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arXiv2025

An Imbalanced Learning-based Sampling Method for Physics-informed Neural Networks

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.

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arXiv2024

Quantifying Training Difficulty and Accelerating Convergence in Neural Network-Based PDE Solvers

Neural network-based methods have emerged as powerful tools for solving partial differential equations (PDEs) in scientific and engineering applications, particularly when handling complex domains or incorporating empirical data. These methods leverage neural networks as basis functions to approximate PDE solutions. However, training such networks can be challenging, often resulting in limited accuracy. In this paper, we investigate the training dynamics of neural network-based PDE solvers with a focus on the impact of initialization techniques. We assess training difficulty by analyzing the eigenvalue distribution of the kernel and apply the concept of effective rank to quantify this difficulty, where a larger effective rank correlates with faster convergence of the training error. Building upon this, we discover through theoretical analysis and numerical experiments that two initialization techniques, partition of unity (PoU) and variance scaling (VS), enhance the effective rank, thereby accelerating the convergence of training error. Furthermore, comprehensive experiments using popular PDE-solving frameworks, such as PINN, Deep Ritz, and the operator learning framework DeepOnet, confirm that these initialization techniques consistently speed up convergence, in line with our theoretical findings.

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arXiv2024

Homotopy Relaxation Training Algorithms for Infinite-Width Two-Layer ReLU Neural Networks

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.

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arXiv2024

Newton Informed Neural Operator for Computing Multiple Solutions of Nonlinear Partials Differential Equations

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.

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arXiv2024

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.

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arXiv2023

Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Norms

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.

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arXiv2023

Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivatives

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.

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