LSR-Net：学习非线性流体动力学的前向演化算子
LSR-Net: Learning the Forward Evolution Operator for Nonlinear Fluid Dynamics
- The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））
- Universitas Diponegoro（迪波尼戈罗大学）
- East China Normal University（华东师范大学）
- The Hong Kong University of Science and Technology（香港科技大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出LSR-Net神经算子架构，通过分解长程与短程核高效学习非线性流体动力学演化算子，在多个基准上显著提升预测精度。
AI中文摘要：
我们提出了长-短程神经网络（LSR-Net），这是一种新颖的神经算子架构，专为数据驱动的前向演化建模而设计，并将其扩展到非线性流体动力学的预测。LSR-Net仅从初始状态和未来状态快照对中学习动力系统的演化算子，在堆叠的网络块内将可学习的积分核拆分为长程（LR）和短程（SR）分量。SR分量使用标准卷积来捕获局部动力学，而LR分量采用指数和（SOE）表示。这使得全局交互可以高效地计算为可训练的傅里叶乘子，将计算复杂度降低到O(n log n)，其中n是输入快照中的像素数，并且每个通道仅需少量参数。LSR-Net在三个具有挑战性的二维基准上进行了评估：耦合Burgers方程、具有空间变化系数的波动方程以及非线性浅水方程（SWE）。结果表明，LSR-Net在预测精度上显著优于基线短程网络（SR-Net）以及FNO和DeepONets，通过有效捕获局部细尺度结构和关键的全局模式交互，实现了大幅更低的相对误差。
英文摘要：
We introduce the Long-Short-Range Neural Network (LSR-Net), a novel neural operator architecture designed for data-driven forward evolution modeling, and extends it to the prediction of nonlinear fluid dynamics. LSR-Net learns the evolution operator of a dynamical system solely from pairs of initial and future state snapshots, which splits the learnable integral kernel into long-range (LR) and short-range (SR) components within stacked network blocks. While the SR component uses standard convolutions to capture local dynamics, the LR component employs a sum-of-exponentials (SOE) representation. This allows for the efficient computation of global interactions as a trainable Fourier multiplier, reducing computational complexity to $O(n \log n)$ where $n$ is the number of pixels in an input snapshot and requiring only a few parameters per channel. LSR-Net is evaluated on three challenging 2D benchmarks: the coupled Burgers equation, the wave equation with a spatially varying coefficient, and the nonlinear shallow water equation {(SWE)}. Results demonstrate that LSR-Net significantly outperforms the baseline short-range network (SR-Net) as well as FNO and DeepONets in predictive accuracy, achieving substantially lower relative errors by effectively capturing both local fine-scale structures and crucial global pattern interactions.
