Compliance Minimization via Physics-Informed Gaussian Processes

TL;DR

This work introduces a mesh-free, physics-informed GP framework (PIGP) for compliance minimization in topology optimization, where $u_1$, $u_2$, and $\rho$ are modeled by GP priors with independent kernels and a shared mean function $m(\boldsymbol{x};\boldsymbol{\theta})$ realized by a Parametric Grid Convolution Attention Network (PGCAN). By reformulating CM within the Deep Energy Method, the authors minimize the total potential energy and compliance simultaneously, while enforcing BCs and volume constraints implicitly through GP conditioning, and they employ adjoint-based gradients, adaptive FD-based gradient evaluations, and curriculum training to stabilize learning. Key contributions include the PGCAN-based shared mean, a simultaneous mesh-free optimization that avoids data residuals, and numerical strategies that yield super-resolution topologies with competitive compliance and reduced gray areas compared to SIMP, along with explicit control over design complexity via the resolution parameter $\text{Res}$. The framework demonstrates strong potential for high-resolution, feature-rich topology design and offers a scalable path toward 3D CM with improved gradient fidelity and efficiency, though future work aims to address FD-approximation accuracy and CP scalability.

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 comparable 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.

Figures (23)

• Figure 1: Deep energy method: The displacement field is obtained by minimizing the total potential functional in \ref{['eq pi']}. Kinematic admissibility in this architecture is ensured via \ref{['eq ab']}.
• Figure 2: PIGP framework for CM: the mean functions are parameterized with the multi-outputs from the PGCAN having three modules: convolutional neural network-based feature encoding, feature interpolation, and decoding via a shallow MLP. GPs are employed on those mean functions to impose displacement BCs and density constraints.
• Figure 3: Benchmark examples: We consider five cases for compliance minimization including (a) Messerschmitt-Bölkow-Blohm (MBB) beam, (b) Cantilever beam, (c) Uniformly loaded beam, (d) L-shape beam, and (e) Hollow beam with a circular hole.
• Figure 4: Comparison of final designs: For both approaches we visualize the topologies corresponding to the median compliance in each example. The cell vertices of PGCAN are also shown to demonstrate the effect of $Res$ on partitioning the design domain via PGCAN's encoder.
• Figure 5: Training dynamics in the MBB example: (a) evolution of the three loss terms, (b) convergence of the equilibrium condition shown by the agreement between $2\times$strain energy and the external work, (c) evolution of volume fraction and the gray area fraction. Similar trajectories are observed for other four benchmark examples.