Sobolev Approximation of Deep ReLU Network in Log-weighted Barron Space

Universal approximation theorems show that neural networks can approximate any continuous function; however, the number of parameters may grow exponentially with the ambient dimension, so these results do not fully explain the practical success of deep models on high-dimensional data. Barron space theory addresses this: if a target function belongs to a Barron space, a two-layer network with $n$ parameters achieves an $O(n^{-1/2})$ approximation error in $L^2$. Yet classical Barron spaces $\mathscr{B}^{s+1}$ still require stronger regularity than Sobolev spaces $H^s$, and existing depth-sensitive results often assume constraints such as $sL \le 1/2$. In this paper, we introduce a log-weighted Barron space $\mathscr{B}^{\log}$, which requires a strictly weaker assumption than $\mathscr{B}^s$ for any $s>0$. For this new function space, we first study embedding properties and carry out a statistical analysis via the Rademacher complexity. Then we prove that functions in $\mathscr{B}^{\log}$ can be approximated by deep ReLU networks with explicit depth dependence. We then define a family $\mathscr{B}^{s,\log}$, establish approximation bounds in the $H^1$ norm, and identify maximal depth scales under which these rates are preserved. Our results clarify how depth reduces regularity requirements for efficient representation, offering a more precise explanation for the performance of deep architectures beyond the classical Barron setting, and for their stable use in high-dimensional problems used today.

Beyond Derivative Pathology of PINNs: Variable Splitting Strategy with Convergence Analysis

Physics-informed neural networks (PINNs) have recently emerged as effective methods for solving partial differential equations (PDEs) in various problems. Substantial research focuses on the failure modes of PINNs due to their frequent inaccuracies in predictions. However, most are based on the premise that minimizing the loss function to zero causes the network to converge to a solution of the governing PDE. In this study, we prove that PINNs encounter a fundamental issue that the premise is invalid. We also reveal that this issue stems from the inability to regulate the behavior of the derivatives of the predicted solution. Inspired by the \textit{derivative pathology} of PINNs, we propose a \textit{variable splitting} strategy that addresses this issue by parameterizing the gradient of the solution as an auxiliary variable. We demonstrate that using the auxiliary variable eludes derivative pathology by enabling direct monitoring and regulation of the gradient of the predicted solution. Moreover, we prove that the proposed method guarantees convergence to a generalized solution for second-order linear PDEs, indicating its applicability to various problems.