Compliance Minimization via Physics-Informed Gaussian Processes

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.