用多专家与多尺度机制提升偏微分方程模拟精度
M$^{2}$M: Learning controllable Multi of experts and multi-scale operators are the Partial Differential Equations need
- 设计多专家门控网络，动态分配不同区域的求解专家
- 在2D纳维-斯托克斯方程上实现更高精度模拟
- 适合需要高精度与可解释性的物理仿真研究者
学习偏微分方程（PDE）的演化动态对理解动态系统至关重要，但现有方法难以有效捕捉其多尺度特征——部分区域快速振荡，其他区域变化缓慢。本文提出一种多尺度、多专家（M²M）神经算子框架，采用分而治之策略训练多专家门控网络以实现动态路由。引入可控制的先验门控机制，决定专家选择权，提升模型效率。通过比例-积分（PI）控制策略精确调整分配规则，实现通用可控优化。在基准2D纳维-斯托克斯方程及自定义多尺度数据集上测试，M²M相比基线方法显著提升模拟精度并增强可解释性。
原文摘要 · Abstract (English)
Learning the evolutionary dynamics of Partial Differential Equations (PDEs) is critical in understanding dynamic systems, yet current methods insufficiently learn their representations. This is largely due to the multi-scale nature of the solution, where certain regions exhibit rapid oscillations while others evolve more slowly. This paper introduces a framework of multi-scale and multi-expert (M$^2$M) neural operators designed to simulate and learn PDEs efficiently. We employ a divide-and-conquer strategy to train a multi-expert gated network for the dynamic router policy. Our method incorporates a controllable prior gating mechanism that determines the selection rights of experts, enhancing the model's efficiency. To optimize the learning process, we have implemented a PI (Proportional, Integral) control strategy to adjust the allocation rules precisely. This universal controllable approach allows the model to achieve greater accuracy. We test our approach on benchmark 2D Navier-Stokes equations and provide a custom multi-scale dataset. M$^2$M can achieve higher simulation accuracy and offer improved interpretability compared to baseline methods.
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