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Inverse design with conditional cascaded diffusion models

Milad Habibi, Mark Fuge

TL;DR

This work introduces a conditional cascaded diffusion (cCDM) framework for multi-resolution inverse design in topology optimization and benchmarks it against a transfer-learning cGAN. By cascading a low-resolution diffusion stage with a high-resolution super-resolution diffusion stage, conditioned on boundary data and physics channels, the approach achieves superior pixel-level fidelity and compliance performance when high-resolution data are plentiful. The study quantifies how performance scales with high-resolution data size, identifying a break-even around 102 samples where cGANs outperform cCDM under data scarcity, while also showing that prioritizing a strong low-resolution predictor markedly improves downstream high-resolution results. Overall, cCDM demonstrates strong potential for efficient, high-fidelity multi-resolution design in engineering, with clear guidance on data requirements and future directions to reduce inference time and extend to 3D problems.

Abstract

Adjoint-based design optimizations are usually computationally expensive and those costs scale with resolution. To address this, researchers have proposed machine learning approaches for inverse design that can predict higher-resolution solutions from lower cost/resolution ones. Due to the recent success of diffusion models over traditional generative models, we extend the use of diffusion models for multi-resolution tasks by proposing the conditional cascaded diffusion model (cCDM). Compared to GANs, cCDM is more stable to train, and each diffusion model within the cCDM can be trained independently, thus each model's parameters can be tuned separately to maximize the performance of the pipeline. Our study compares cCDM against a cGAN model with transfer learning. Our results demonstrate that the cCDM excels in capturing finer details, preserving volume fraction constraints, and minimizing compliance errors in multi-resolution tasks when a sufficient amount of high-resolution training data (more than 102 designs) is available. Furthermore, we explore the impact of training data size on the performance of both models. While both models show decreased performance with reduced high-resolution training data, the cCDM loses its superiority to the cGAN model with transfer learning when training data is limited (less than 102), and we show the break-even point for this transition. Also, we highlight that while the diffusion model may achieve better pixel-wise performance in both low-resolution and high-resolution scenarios, this does not necessarily guarantee that the model produces optimal compliance error or constraint satisfaction.

Inverse design with conditional cascaded diffusion models

TL;DR

This work introduces a conditional cascaded diffusion (cCDM) framework for multi-resolution inverse design in topology optimization and benchmarks it against a transfer-learning cGAN. By cascading a low-resolution diffusion stage with a high-resolution super-resolution diffusion stage, conditioned on boundary data and physics channels, the approach achieves superior pixel-level fidelity and compliance performance when high-resolution data are plentiful. The study quantifies how performance scales with high-resolution data size, identifying a break-even around 102 samples where cGANs outperform cCDM under data scarcity, while also showing that prioritizing a strong low-resolution predictor markedly improves downstream high-resolution results. Overall, cCDM demonstrates strong potential for efficient, high-fidelity multi-resolution design in engineering, with clear guidance on data requirements and future directions to reduce inference time and extend to 3D problems.

Abstract

Adjoint-based design optimizations are usually computationally expensive and those costs scale with resolution. To address this, researchers have proposed machine learning approaches for inverse design that can predict higher-resolution solutions from lower cost/resolution ones. Due to the recent success of diffusion models over traditional generative models, we extend the use of diffusion models for multi-resolution tasks by proposing the conditional cascaded diffusion model (cCDM). Compared to GANs, cCDM is more stable to train, and each diffusion model within the cCDM can be trained independently, thus each model's parameters can be tuned separately to maximize the performance of the pipeline. Our study compares cCDM against a cGAN model with transfer learning. Our results demonstrate that the cCDM excels in capturing finer details, preserving volume fraction constraints, and minimizing compliance errors in multi-resolution tasks when a sufficient amount of high-resolution training data (more than 102 designs) is available. Furthermore, we explore the impact of training data size on the performance of both models. While both models show decreased performance with reduced high-resolution training data, the cCDM loses its superiority to the cGAN model with transfer learning when training data is limited (less than 102), and we show the break-even point for this transition. Also, we highlight that while the diffusion model may achieve better pixel-wise performance in both low-resolution and high-resolution scenarios, this does not necessarily guarantee that the model produces optimal compliance error or constraint satisfaction.
Paper Structure (18 sections, 2 equations, 9 figures, 3 tables)

This paper contains 18 sections, 2 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Inverse Design and Topology Optimization
  4. Generative models for Inverse Design using Super-Resolution
  5. Diffusion Models
  6. Methodology
  7. Conditioned Cascading Pipelines
  8. U-Net Architecture and Diffusion Parameters
  9. Conditional Generative Adversarial Networks and Transfer Learning
  10. Dataset
  11. Evaluation Metrics
  12. EXPERIMENTAL RESULTS AND DISCUSSION
  13. How well do different models perform in low-resolution?
  14. How well do different models perform in high-resolution with limited training data?
  15. How well do models perform in high-resolution as a function of training data size?
  16. ...and 3 more sections

Figures (9)

  • Figure 1: Proposed cCDM pipeline for TO super-resolution. Model 1 is a standard conditional diffusion model for TO. L is the load applied, represented with an arrow on the topology; v is the volume fraction; f represents the physical field including the stress and strain field. Model 2 is a super-resolution model conditioned on boundary conditions and upsampled prediction of Model 1.
  • Figure 2: The physical layout of topology optimization problem of beams and corresponding training and testing boundary condition. Design domain of a cantilever beam with design parameters of force location (h) and direction ($\alpha$) on the right side of the beam. Boundary conditions: (a)-(f) are used for training and testing and (g)–(h) are unseen BCs only used for testing.
  • Figure 3: Comparison of generated structures on randomly selected samples from seens and unseen boundary conditions in the low-resolution data. The GT column represents the ground truth generated by the SIMP method.
  • Figure 4: Comparison of generated structures on randomly selected samples from seens and unseen boundary conditions in the high-resolution data. The GT column represent the ground truth generated by SIMP method.
  • Figure 5: Mean MSE of the cGAN model with transfer learning versus the cascaded diffusion model across test data with seen and unseen boundary conditions as the number of high-resolution training data changes. Error bars represent the 95% confidence intervals around the average MSE values, providing insights into the uncertainty of the measurements.
  • ...and 4 more figures