Neural Network Approximation of Solutions to Fractional Parabolic Partial Differential Equations

We establish a dimension-efficient neural network approximation theory for solutions to fractional parabolic equations with lower-order drift and potential terms. By introducing anisotropic spectral Barron spaces, which measure temporal and spatial regularity separately in frequency space, we first develop a dimension-independent maximal regularity theory for these equations, using dimension-independent multiplication estimates and the method of continuity to incorporate the lower-order terms. A key technical novelty is the application of the Vandermonde matrix to the global-in-time extension of the finite-time fractional heat semigroup with sufficient regularity at the initial time, thereby enabling analysis of the forward-in-time evolution via the global space-time Fourier structure of anisotropic Barron norms. We also show that a corresponding uniform-in-time estimate of the spectral Barron regularity generally fails. Finally, we derive $n^{-1/2}$ two-layer approximation bounds in mixed Sobolev norms for non-constant periodic activations and, under additional anisotropic Barron regularity, for non-periodic activations satisfying a polynomial-decay condition.

Extreme Weather Nowcasting via Local Precipitation Pattern Prediction

Accurate forecasting of extreme weather events such as heavy rainfall or storms is critical for risk management and disaster mitigation. Although high-resolution radar observations have spurred extensive research on nowcasting models, precipitation nowcasting remains particularly challenging due to pronounced spatial locality, intricate fine-scale rainfall structures, and variability in forecasting horizons. While recent diffusion-based generative ensembles show promising results, they are computationally expensive and unsuitable for real-time applications. In contrast, deterministic models are computationally efficient but remain biased toward normal rainfall. Furthermore, the benchmark datasets commonly used in prior studies are themselves skewed--either dominated by ordinary rainfall events or restricted to extreme rainfall episodes--thereby hindering general applicability in real-world settings. In this paper, we propose exPreCast, an efficient deterministic framework for generating finely detailed radar forecasts, and introduce a newly constructed balanced radar dataset from the Korea Meteorological Administration (KMA), which encompasses both ordinary precipitation and extreme events. Our model integrates local spatiotemporal attention, a texture-preserving cubic dual upsampling decoder, and a temporal extractor to flexibly adjust forecasting horizons. Experiments on established benchmarks (SEVIR and MeteoNet) as well as on the balanced KMA dataset demonstrate that our approach achieves state-of-the-art performance, delivering accurate and reliable nowcasts across both normal and extreme rainfall regimes.

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.