Generative Inverse Design with Abstention via Diagonal Flow Matching

Abstract

Inverse design aims to find design parameters $x$ achieving target performance $y^*$. Generative approaches learn bidirectional mappings between designs and labels, enabling diverse solution sampling. However, standard conditional flow matching (CFM), when adapted to inverse problems by pairing labels with design parameters, exhibits strong sensitivity to their arbitrary ordering and scaling, leading to unstable training. We introduce Diagonal Flow Matching (Diag-CFM), which resolves this through a zero-anchoring strategy that pairs design coordinates with noise and labels with zero, making the learning problem provably invariant to coordinate permutations. This yields order-of-magnitude improvements in round-trip accuracy over CFM and invertible neural network baselines across design dimensions up to $P{=}100$. We develop two architecture-intrinsic uncertainty metrics, Zero-Deviation and Self-Consistency, that enable three practical capabilities: selecting the best candidate among multiple generations, abstaining from unreliable predictions, and detecting out-of-distribution targets; consistently outperforming ensemble and general-purpose alternatives across all tasks. We validate on airfoil, gas turbine combustor, and an analytical benchmark with scalable design dimension.

Figures (10)

• Figure 1: Per-coordinate targets in the flow-matching loss. Diag--CFM anchors labels to zero and pairs designs with latent noise, removing spurious dependence on coordinate ordering and scale.
• Figure 2: Final round-trip error (log scale) for training runs on the gas turbine dataset with different parameter orderings, comparing Diag--CFM and standard CFM (5 runs per case). CFM performance varies significantly with ordering while Diag--CFM remains stable. The gas turbine dataset is already scale-normalized to $[0, 1]$, so the observed sensitivity is attributable purely to coordinate ordering; for datasets with heterogeneous scales, the effect would likely be more pronounced. Numerical values in Appendix \ref{['appendix:ablation']}.
• Figure 3: Design diversity as a function of round-trip error threshold $\varepsilon$ for the gas turbine combustor dataset. Solid lines show mean design diversity (variance) computed only over samples with error $< \varepsilon$; dashed lines show the average number of valid samples. Diag--CFM (blue) maintains high diversity at strict accuracy thresholds where INN and CFM have few valid samples, demonstrating that its designs are both accurate and diverse.
• Figure 4: Airfoil geometries reconstructed from the 14-dimensional POD basis. (a) Airfoils from the Unifoil dataset. (b) Airfoils generated by Diag--CFM from randomly sampled performance labels and physical conditioning parameters (angle of attack, Mach number). The generated shapes are realistic and diverse, reflecting the model's ability to produce plausible designs for given aerodynamic targets.
• Figure 5: Design diversity as a function of round-trip error threshold $\varepsilon$ for the Unifoil dataset. Solid lines show mean design diversity computed only over samples with error $< \varepsilon$; dashed lines show valid sample counts. Diag--CFM achieves high diversity at strict accuracy thresholds where it has more valid samples than competing methods.

Theorems & Definitions (4)

• Proposition 4.1
• Proposition 4.2
• proof : Proof of Proposition \ref{['prop:diag-cfm-equivariant']}
• proof : Proof of Proposition \ref{['prop:cfm-not-equivariant']}