发表机构
University at Albany - State University of New York; Dartmouth College（纽约州立大学奥尔巴尼分校; 达特茅斯学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种改进的无导数损失方法，通过解析近似游走粒子局部运动以捕捉方向性拉伸并消除采样噪声，作为多尺度求解器更准确地再现二维湍流的宏观能量谱。
AI 中文摘要
无导数损失方法（DFLM）是一种无网格神经网络方法，通过模拟随机游走粒子而非直接计算导数来求解偏微分方程。尽管DFLM此前已被应用于Navier--Stokes方程，但将其扩展到湍流时，其最简单实现暴露出两个局限性。游走粒子未考虑流动的旋转、方向性拉伸运动，且随机采样用于评估目标的游走粒子会在学习到的解中引入噪声。我们通过游走粒子局部运动的解析近似来解决这两个局限性，该近似能够捕捉方向性拉伸并消除采样噪声，同时保持计算效率。我们将所得方法作为一种非侵入式多尺度求解器进行研究，该求解器能够在不于细网格上解析每个尺度、也不显式耦合粗细尺度模型的情况下学习宏观流动行为。对于两个二维湍流，所提方法再现完全解析参考模拟的宏观能量谱比标准方法更准确，尤其是在方向性拉伸最强的区域，从而证实了DFLM作为湍流流体系统多尺度求解器的有效性。
英文摘要
The derivative-free loss method (DFLM) is a mesh-free neural network approach for solving partial differential equations by simulating stochastic walkers rather than computing derivatives directly. Although DFLM has previously been applied to the Navier--Stokes equations, extending it to turbulent flows reveals two limitations of its simplest implementation. The walkers do not account for the swirling, directionally stretching motion of the flow, and randomly sampling the walkers used to evaluate the target introduces noise into the learned solution. We address both limitations with an analytic approximation of the walkers' local motion that captures directional stretching and eliminates this sampling noise, while remaining computationally efficient. We study the resulting method as a non-intrusive multiscale solver capable of learning macroscopic flow behavior without resolving every scale on a fine grid or explicitly coupling coarse- and fine-scale models. For two turbulent two-dimensional flows, the proposed method reproduces the macroscopic energy spectrum of fully resolved reference simulations more accurately than the standard method, particularly where directional stretching is strongest, confirming DFLM's efficacy as a multiscale solver for turbulent fluid systems.
Comments23 pages, 8 figures
