Localized Physics-informed Gaussian Processes with Curriculum Training for Topology Optimization

2025-03-18

AI Summary

This work addresses key challenges in topology optimization—namely, the difficulty of simultaneously achieving mesh-free formulation and physics-informed embedding, ambiguous material interfaces, and susceptibility to local optima. We propose a mesh-free joint optimization framework grounded in Physics-Informed Gaussian Processes (PI-GPs). Our method employs a customized local neural network as the GP mean function, enabling concurrent optimization of design variables and state fields. To enhance accuracy in high-gradient regions, we introduce a weighted local residual loss, integrated with curriculum learning and continuous optimization strategies to improve global convergence. Evaluated on three benchmark Stokes flow problems targeting dissipation power minimization, the approach achieves accuracy comparable to COMSOL, yields sharp material interfaces, and significantly reduces PDE residual errors in high-gradient zones. To our knowledge, this is the first mesh-free, high-fidelity, physics-driven topology optimization framework.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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.

Problem

• Develops a meshfree topology optimization framework using physics-informed Gaussian processes.
• Uses a customized deep neural network for localized learning and capturing intricate topologies.
• Incorporates curriculum training to avoid local optimality and promote global solutions.

Innovation

• Physics-informed Gaussian Processes for topology optimization
• Customized DNN with localized learning capacity
• Curriculum training to avoid local optimality