[2603.24002] Stochastic Dimension-Free Zeroth-Order Estimator for High-Dimensional and High-Order PINNs

Computer Science > Machine Learning arXiv:2603.24002 (cs) [Submitted on 25 Mar 2026] Title:Stochastic Dimension-Free Zeroth-Order Estimator for High-Dimensional and High-Order PINNs Authors:Zhangyong Liang, Ji Zhang View a PDF of the paper titled Stochastic Dimension-Free Zeroth-Order Estimator for High-Dimensional and High-Order PINNs, by Zhangyong Liang and 1 other authors View PDF HTML (experimental) Abstract:Physics-Informed Neural Networks (PINNs) for high-dimensional and high-order partial differential equations (PDEs) are primarily constrained by the $\mathcal{O}(d^k)$ spatial derivative complexity and the $\mathcal{O}(P)$ memory overhead of backpropagation (BP). While randomized spatial estimators successfully reduce the spatial complexity to $\mathcal{O}(1)$, their reliance on first-order optimization still leads to prohibitive memory consumption at scale. Zeroth-order (ZO) optimization offers a BP-free alternative; however, naively combining randomized spatial operators with ZO perturbations triggers a variance explosion of $\mathcal{O}(1/\varepsilon^2)$, leading to numerical divergence. To address these challenges, we propose the \textbf{S}tochastic \textbf{D}imension-free \textbf{Z}eroth-order \textbf{E}stimator (\textbf{SDZE}), a unified framework that achieves dimension-independent complexity in both space and memory. Specifically, SDZE leverages \emph{Common Random Numbers Synchronization (CRNS)} to algebraically cancel the $\mathcal{O}(1/\varepsilon^2)$ var...