用多分辨率卷积网络预测任意网格上的物理场，比传统方法快且准。
Scalar Field Prediction on Meshes Using Interpolated Multi-Resolution Convolutional Neural Networks
- 通过插值多尺度特征图到网格节点，实现对任意网格的场预测
- 在应力场预测上中位R²达0.91，在温度场达0.99
- 适合需要快速模拟的工程设计场景，尤其拓扑变化大的问题
标量场（如应力或温度）常用于工程中的形状优化与设计问题。对于拓扑复杂且无法参数化的形状，基于数据驱动的标量场预测比传统有限元方法更快。然而，现有数据驱动方法仅适用于固定网格域，无法处理任意网格结构。本文提出一种在任意网格上预测标量场的方法：使用卷积神经网络，将多分辨率特征图插值至节点位置，再通过多层感知机预测偏微分方程在网格节点的解。模型在有限元计算的von Mises应力场数据上训练，训练后可对任意输入网格的节点进行应力值估计。在两个形状数据集上测试均表现优异，中位R²达0.91。此外，还应用于热传导问题的温度场预测，中位R²达0.99。该方法为工程设计中的有限元分析提供了灵活替代方案。代码与数据集已公开。
原文摘要 · Abstract (English)
Scalar fields, such as stress or temperature fields, are often calculated in shape optimization and design problems in engineering. For complex problems where shapes have varying topology and cannot be parametrized, data-driven scalar field prediction can be faster than traditional finite element methods. However, current data-driven techniques to predict scalar fields are limited to a fixed grid domain, instead of arbitrary mesh structures. In this work, we propose a method to predict scalar fields on arbitrary meshes. It uses a convolutional neural network whose feature maps at multiple resolutions are interpolated to node positions before being fed into a multilayer perceptron to predict solutions to partial differential equations at mesh nodes. The model is trained on finite element von Mises stress fields, and once trained it can estimate stress values at each node on any input mesh. Two shape datasets are investigated, and the model has strong performance on both, with a median R-squared value of 0.91. We also demonstrate the model on a temperature field in a heat conduction problem, where its predictions have a median R-squared value of 0.99. Our method provides a potential flexible alternative to finite element analysis in engineering design contexts. Code and datasets are available online.
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