Generative Inverse Design with Abstention via Diagonal Flow Matching

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.

How well do generative models solve inverse problems? A benchmark study

Generative learning generates high dimensional data based on low dimensional conditions, also called prompts. Therefore, generative learning algorithms are eligible for solving (Bayesian) inverse problems. In this article we compare a traditional Bayesian inverse approach based on a forward regression model and a prior sampled with the Markov Chain Monte Carlo method with three state of the art generative learning models, namely conditional Generative Adversarial Networks, Invertible Neural Networks and Conditional Flow Matching. We apply them to a problem of gas turbine combustor design where we map six independent design parameters to three performance labels. We propose several metrics for the evaluation of this inverse design approaches and measure the accuracy of the labels of the generated designs along with the diversity. We also study the performance as a function of the training dataset size. Our benchmark has a clear winner, as Conditional Flow Matching consistently outperforms all competing approaches.