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CcDPM: A Continuous Conditional Diffusion Probabilistic Model for Inverse Design
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Proposes CcDPM, a continuous-conditional diffusion model for engineering inverse design that injects vicinal risk minimization, accelerates vicinal loss with a k-d tree, and enables multi-point design via a new sampling scheme.
Problem
Inverse engineering design with continuous performance requirements, where cGAN-style methods assume discrete conditions, struggle with training stability and mode collapse, cannot handle multiple working conditions at once, and have slow training from vicinal loss over large datasets.
Approach
Combine vicinal risk minimization (soft vicinal estimate loss) with a conditional diffusion model, use a k-d tree to find neighbors for the vicinal loss, and sample multi-point designs by combining per-condition noise predictions with classifier-free-style guidance weights.
Key results
- CcDPM achieves the lowest label error on all single-point datasets, e.g., 0.0148 on Uneven and 0.0459 on Xfoil versus 0.0174 and 0.0501 for DPM.
- The k-d tree speeds up training by 2.7x on Uneven and up to 353x on Rocket (1h30min vs 530h).
- On multi-point design for CFD and Rocket, CcDPM gives label error 0.0986 and 0.0871 vs 0.1010 and 0.0891 for DPM.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add vicinal risk minimization to DPM | DPM · Uneven single-point | label error: 0.0174 -> 0.0148 | Helped |
| add vicinal risk minimization to DPM | DPM · Xfoil single-point | label error: 0.0501 -> 0.0459 | Helped |
| add vicinal risk minimization to DPM | DPM · CFD single-point | label error: 0.0411 -> 0.0376 | Helped |
| add vicinal risk minimization to DPM | DPM · Rocket single-point | label error: 0.0442 -> 0.0425 | Helped |
| add vicinal risk minimization to DPM | DPM · CFD multi-point | label error: 0.1010 -> 0.0986 | Helped |
| add vicinal risk minimization to DPM | DPM · Rocket multi-point | label error: 0.0891 -> 0.0871 | Helped |
| add the k-d tree to accelerate vicinal-loss neighbor search (ablation) | CcDPM · Uneven | speedup: 2.7 | Helped |
| add the k-d tree to accelerate vicinal-loss neighbor search (ablation) | CcDPM · Xfoil | speedup: 23 | Helped |
| add the k-d tree to accelerate vicinal-loss neighbor search (ablation) | CcDPM · CFD | speedup: 192 | Helped |
| add the k-d tree to accelerate vicinal-loss neighbor search (ablation) | CcDPM · Rocket | speedup: 353 | Helped |
Surprising findings
- The paper notes PcDGAN has higher diversity on the CFD case but generates more invalid designs, so diversity alone is misleading.
Limitations
- Bezier parameterization is difficult for diffusion models, so CST parameterization is used instead.
- Relies on a latent-space encoder/decoder, so it cannot handle inverse design where designs cannot be encoded or parameterized, such as topology generation.
- Gains over DPM on single-point errors are modest in several cases.
Experimental setup
Datasets: Synthetic Uneven dataset Xfoil airfoil dataset CFD airfoil dataset Rocket dataset
Benchmarks: Synthetic Uneven dataset Xfoil CFD airfoil Rocket
Metrics: label error (MAE) diversity (log determinant of similarity matrix) training speedup
Baselines: CcGAN PcDGAN CcLSGAN DPM
Builds on: CcGAN denoising diffusion probabilistic model classifier-free guidance vicinal risk minimization
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- Revisiting Diffusion Models: From Generative Pre-training to One-Step Generation (ICML 2025 · similarity 0.84)
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