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

TL;DR

The paper addresses meshfree topology optimization by integrating physics-informed Gaussian processes with a shared mean function to model both state variables and topology indicators. It introduces LC-SMTO, which uses a Parametric Grid Convolutional Attention Network (PGCAN) to parameterize the GP mean and employs localized PDE residual weighting and curriculum training to guide optimization. The method minimizes a multi-component loss that includes the objective $\mathcal{J}$, constraints $C_i$, and PDE residuals $R_i$, solved in a unified framework, demonstrated on three Brinkman/Stokes-flow benchmarks with sharp interfaces and predefined volume targets. On Rugby, Obstacle, and Double pipe problems, LC-SMTO outperforms SMTO in topology quality and convergence while remaining competitive with COMSOL in computational cost, providing sharper interfaces and improved symmetry in optimized designs.

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.

Figures (7)

• Figure 1: Simultaneous and meshfree topology optimization with localized features : It is assumed that the structure has two state variables $\boldsymbol{\mathrm{u}}(\boldsymbol{\mathrm{x}}) = \mathopen{}\left[u_1(\boldsymbol{\mathrm{x}}), u_2(\boldsymbol{\mathrm{x}})\right]\mathclose{}$. The level set function $\psi(\boldsymbol{\mathrm{x}})$ indicates the material phase at any given point where negative/positive values correspond to solid/void. The covariance matrices ensure that the variables satisfy the boundary conditions while the parameters of the PGCAN (mean function) are optimized to minimize Equation \ref{['eq penalty_opt_generic']}. In practice, we fix the length-scale parameters of all the kernels to $10^3$ and only optimize $\boldsymbol{\theta}$.
• Figure 2: Design domain and the imposed boundary conditions: The Rugby and Obstacle domains are square, whereas the Double Pipe domain is a $3\times1$ rectangle.
• Figure 3: COMSOL median optimal topologies: The topologies corresponding to the median $\mathcal{J}$ out of $5$ random initializations are depicted for Rugby, Obstacle, and Double pipe problems.
• Figure 4: Different topologies obtained for the Obstacle problem: Designed topologies by LC-SMTO are imported in COMSOL to obtain $\mathcal{J}$ via FEA. Minimum and maximum values are relatively close but result in different topologies.
• Figure 5: Volume loss history: Each curve represent the median across $5$ training repetitions. For comparison, we also provide the volume loss error obtained by COMSOL for its median optimal topology, see markers at $20k$.