Multi-material Multi-physics Topology Optimization with Physics-informed Gaussian Process Priors

2026-02-19

AI Summary

This work proposes a novel physics-informed Gaussian process (PIGP) framework to address the challenges of high computational cost, spectral bias, and the non-self-adjoint nature of objectives or constraints that hinder the scalability of existing machine learning methods in multi-material, multi-physics topology optimization. By employing a neural network–parameterized Gaussian process prior, the approach jointly models primal, adjoint, and design variables while simultaneously optimizing the objective function, multi-physics energy functionals, and design constraints. This study presents the first application of PIGP to multi-material, multi-physics topology optimization, achieving super-resolution topology generation, sharp interface representation, and strong physical interpretability. Efficient differentiation and integration strategies significantly accelerate training. The method’s effectiveness is validated on benchmark problems—including compliance minimization, thermal conduction, compliant mechanisms, and thermo-mechanical coupling—with results verified against both COMSOL simulations and open-source implementations.

Technology Category

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

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Machine learning (ML) has been increasingly used for topology optimization (TO). However, most existing ML-based approaches focus on simplified benchmark problems due to their high computational cost, spectral bias, and difficulty in handling complex physics. These limitations become more pronounced in multi-material, multi-physics problems whose objective or constraint functions are not self-adjoint. To address these challenges, we propose a framework based on physics-informed Gaussian processes (PIGPs). In our approach, the primary, adjoint, and design variables are represented by independent GP priors whose mean functions are parametrized via neural networks whose architectures are particularly beneficial for surrogate modeling of PDE solutions. We estimate all parameters of our model simultaneously by minimizing a loss that is based on the objective function, multi-physics potential energy functionals, and design-constraints. We demonstrate the capability of the proposed framework on benchmark TO problems such as compliance minimization, heat conduction optimization, and compliant mechanism design under single- and multi-material settings. Additionally, we leverage thermo-mechanical TO with single- and multi-material options as a representative multi-physics problem. We also introduce differentiation and integration schemes that dramatically accelerate the training process. Our results demonstrate that the proposed PIGP framework can effectively solve coupled multi-physics and design problems simultaneously -- generating super-resolution topologies with sharp interfaces and physically interpretable material distributions. We validate these results using open-source codes and the commercial software package COMSOL.

Problem

• multi-material
• multi-physics
• topology optimization
• non-self-adjoint
• complex physics

Innovation

• Physics-informed Gaussian Process
• Multi-material Topology Optimization
• Multi-physics Coupling
• Neural Network Surrogate Modeling
• Adjoint Variable Integration