Compliance Minimization via Physics-Informed Gaussian Processes

2025-07-14

AI Summary

To address limitations of existing machine learning approaches for compliance minimization (CM)—including ambiguous feature boundaries, high computational cost, and uncontrollable design complexity—this paper proposes a physics-informed, mesh-free concurrent solution framework. The method constructs a joint design-state surrogate model by integrating Gaussian processes (GPs) with a shared multi-output neural network; incorporates an interpretable complexity regularization term to suppress spectral bias; and enforces physical consistency via a residual-free physics-constrained loss function. Furthermore, parametric grid convolutional attention networks (PGCANs) coupled with curriculum learning enable super-resolution topology generation and mesoscale feature control. Experiments demonstrate that the proposed approach achieves significantly lower compliance values and reduced grayscale region ratios, exhibits faster convergence and sharper topologies, and outperforms state-of-the-art ML-based topology optimization methods in both design controllability and computational efficiency.

Technology Category

• Machine Learning: Hardware-aware ML
• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Machine learning (ML) techniques have recently gained significant attention for solving compliance minimization (CM) problems. However, these methods typically provide poor feature boundaries, are very expensive, and lack a systematic mechanism to control the design complexity. Herein, we address these limitations by proposing a mesh-free and simultaneous framework based on physics-informed Gaussian processes (GPs). In our approach, we parameterize the design and state variables with GP priors which have independent kernels but share a multi-output neural network (NN) as their mean function. The architecture of this NN is based on Parametric Grid Convolutional Attention Networks (PGCANs) which not only mitigate spectral bias issues, but also provide an interpretable mechanism to control design complexity. We estimate all the parameters of our GP-based representations by simultaneously minimizing the compliance, total potential energy, and residual of volume fraction constraint. Importantly, our loss function exclude all data-based residuals as GPs automatically satisfy them. We also develop computational schemes based on curriculum training and numerical integration to increase the efficiency and robustness of our approach which is shown to (1) produce super-resolution topologies with fast convergence, (2) achieve smaller compliance and less gray area fraction compared to traditional numerical methods, (3) provide control over fine-scale features, and (4) outperform competing ML-based methods.

Problem

• Improve feature boundaries in compliance minimization
• Reduce computational cost of ML-based CM methods
• Control design complexity systematically

Innovation

• Mesh-free physics-informed Gaussian processes framework
• Multi-output neural network with PGCANs architecture
• Curriculum training and numerical integration schemes