用物理信息高斯过程实现无网格拓扑优化，自动捕捉复杂结构边界。
Localized Physics-informed Gaussian Processes with Curriculum Training for Topology Optimization
- 基于自定义深度网络的多输出高斯过程，共享均值函数建模设计与状态变量。
- 在斯托克斯流中最小化耗散功率，比商用软件COMSOL更精确且结构清晰。
- 局部加权损失与课程训练结合，有效避免局部最优，适合复杂拓扑设计。
我们提出一种基于物理信息高斯过程（GPs）的同步无网格拓扑优化（TO）框架。该框架通过高斯过程先验对所有设计变量和状态变量进行建模，其共享的多输出均值函数由定制化的深度神经网络（DNN）参数化。该均值函数的参数通过最小化包含性能指标、设计约束及状态方程残差的多组件损失函数来估计。该方法生成清晰的材料界面，并具备内在的延续性，有助于全局最优。独特之处包括：(1) 自定义的局部学习型DNN，可捕捉复杂拓扑并降低高梯度区域残差；(2) 局部加权损失函数，提升界面附近解的精度；(3) 采用课程训练策略，防止陷入局部最优。为验证框架有效性，我们在三个涉及斯托克斯流耗散功率最小化的实例上与商业软件COMSOL进行了对比。
原文摘要 · Abstract (English)
We introduce a simultaneous and meshfree topology optimization (TO) framework based on physics-informed Gaussian processes (GPs). Our framework endows all design and state variables via GP priors which have a shared, multi-output mean function that is parametrized via a customized deep neural network (DNN). The parameters of this mean function are estimated by minimizing a multi-component loss function that depends on the performance metric, design constraints, and the residuals on the state equations. Our TO approach yields well-defined material interfaces and has a built-in continuation nature that promotes global optimality. Other unique features of our approach include (1) its customized DNN which, unlike fully connected feed-forward DNNs, has a localized learning capacity that enables capturing intricate topologies and reducing residuals in high gradient fields, (2) its loss function that leverages localized weights to promote solution accuracy around interfaces, and (3) its use of curriculum training to avoid local optimality.To demonstrate the power of our framework, we validate it against commercial TO package COMSOL on three problems involving dissipated power minimization in Stokes flow.
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