用神经网络直接解流体方程，实现无网格三维湍流模拟。
Simulating Three-dimensional Turbulence with Physics-informed Neural Networks
- 基于物理方程训练神经网络，无需传统网格和数据
- 准确复现能量谱、动能、涡量等关键流动统计量
- 适合需要高效高精度湍流建模的研究者
湍流模拟是科学计算中最耗算力的问题之一，在高速流动下传统方法计算成本极高。物理信息神经网络（PINNs）通过直接从物理方程训练神经网络，提供了一种无需网格的连续求解新路径。本文展示，经过合理设计的PINNs可成功模拟二维与三维完全湍流，直接学习基本流体方程的解，无需传统计算网格或训练数据。该方法结合自适应网络结构、因果训练及先进优化策略，有效应对混沌动力学的学习挑战。在多个复杂湍流问题上的严格验证表明，PINNs能准确再现能量谱、动能、涡量和雷诺应力等关键流动统计特性。结果证明，神经方程求解器可处理复杂混沌系统，为超越传统计算限制的连续湍流建模开辟了新途径。
原文摘要 · Abstract (English)
Turbulent fluid flows are among the most computationally demanding problems in science, requiring enormous computational resources that become prohibitive at high flow speeds. Physics-informed neural networks (PINNs) represent a radically different approach that trains neural networks directly from physical equations rather than data, offering the potential for continuous, mesh-free solutions. Here we show that appropriately designed PINNs can successfully simulate fully turbulent flows in both two and three dimensions, directly learning solutions to the fundamental fluid equations without traditional computational grids or training data. Our approach combines several algorithmic innovations including adaptive network architectures, causal training, and advanced optimization methods to overcome the inherent challenges of learning chaotic dynamics. Through rigorous validation on challenging turbulence problems, we demonstrate that PINNs accurately reproduce key flow statistics including energy spectra, kinetic energy, enstrophy, and Reynolds stresses. Our results demonstrate that neural equation solvers can handle complex chaotic systems, opening new possibilities for continuous turbulence modeling that transcends traditional computational limitations.
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