Surrogate-Based Differentiable Pipeline for Shape Optimization

TL;DR

This work tackles the bottleneck of non-differentiable components in CAE workflows that impede gradient-based optimization. It proposes an end-to-end differentiable pipeline built by replacing non-differentiable steps (mesh generation and CFD) with a data-driven surrogate, specifically a 3D U-Net that maps the signed distance field (SDF) of a geometry to flow fields, all within a modular Tesseract framework. The authors demonstrate the approach on an aerodynamic shape optimization problem, replacing OpenFOAM with the U-Net surrogate and optimizing the design parameters by backpropagating through the pipeline to maximize the objective $\Theta = \mathrm{mean}(U_x)$. Results show gradient-based optimization converging in about 14 iterations, with the design aligning to the flow and reducing frontal area, indicating the practicality of surrogate-based differentiation for CAE tasks. However, the approach requires substantial upfront data generation and training and introduces model risk from approximation errors, necessitating validation against high-fidelity simulations and future work extending to more complex geometries and objectives.

Abstract

Gradient-based optimization of engineering designs is limited by non-differentiable components in the typical computer-aided engineering (CAE) workflow, which calculates performance metrics from design parameters. While gradient-based methods could provide noticeable speed-ups in high-dimensional design spaces, codes for meshing, physical simulations, and other common components are not differentiable even if the math or physics underneath them is. We propose replacing non-differentiable pipeline components with surrogate models which are inherently differentiable. Using a toy example of aerodynamic shape optimization, we demonstrate an end-to-end differentiable pipeline where a 3D U-Net full-field surrogate replaces both meshing and simulation steps by training it on the mapping between the signed distance field (SDF) of the shape and the fields of interest. This approach enables gradient-based shape optimization without the need for differentiable solvers, which can be useful in situations where adjoint methods are unavailable and/or hard to implement.

Figures (6)

• Figure 1: Pipeline architecture for surrogate-based shape optimization. Arrow colors indicate workflow stages: blue for data generation, orange for surrogate training, green for optimization. Solid arrows represent forward primal values, while dashed arrows represent backward gradient flow. All components that are written as Tesseracts are displayed as black boxes.
• Figure 2: Example of the primitives we use for shape optimization
• Figure 3: Optimization results: (left) convergence of the objective function over 14 iterations; (right) design evolution showing the shape aligning with flow and reducing volume.
• Figure 4: Example normalized target and prediction slices in xy, yz, and xz planes for trained baseline model.
• Figure 5: Training and validation loss curves for the ablation study. Solid lines indicate training loss, dashed lines indicate validation loss. All models converge to similar performance, with the baseline configuration achieving the lowest validation loss.