# Conditional Flow Matching for ML-Based Inverse Design Problems

arXiv: https://arxiv.org/abs/2609.00863 · HTML: https://mlsanity.com/paper/2609.00863/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 22 of its 41 references that are in the map.
- First posted on arXiv 1 Sep 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 22 of 22)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Deep Generative Models in Engineering Design: A Review](https://mlsanity.com/paper/2110.10863/) (2021)
- [Diffusion Models Beat GANs on Topology Optimization](https://mlsanity.com/paper/2208.09591/) (2022)
- [TopologyGAN: Topology Optimization Using Generative Adversarial Networks Based on Physical Fields Over the Initial Domain](https://mlsanity.com/paper/2003.04685/) (2020)
- [Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching](https://mlsanity.com/paper/2405.18816/) (2024)
- [Aligning Optimization Trajectories with Diffusion Models for Constrained Design Generation](https://mlsanity.com/paper/2305.18470/) (2023)
- [Data-driven topology design using a deep generative model](https://mlsanity.com/paper/2006.04559/) (2020)
- [EngiBench: A Framework for Data-Driven Engineering Design Research](https://mlsanity.com/paper/2508.00831/) (2025)
- [Latent Diffusion Models for Structural Component Design](https://mlsanity.com/paper/2309.11601/) (2023)
- [On the Generalization in Topology Optimization via Sensitivity-Conditioned Bernoulli Flow Matching](https://mlsanity.com/paper/2606.02179/) (2026)
- [How well do generative models solve inverse problems? A benchmark study](https://mlsanity.com/paper/2601.23238/) (2026)
- [Inverse design with conditional cascaded diffusion models](https://mlsanity.com/paper/2408.08526/) (2024)
- [Flow Matching and Diffusion Models via PointNet for Generating Fluid Fields on Irregular Geometries](https://mlsanity.com/paper/2601.03030/) (2026)
- [Generative Design of Ship Propellers using Conditional Flow Matching](https://mlsanity.com/paper/2601.21637/) (2026)
- [Generative Inverse Design with Abstention via Diagonal Flow Matching](https://mlsanity.com/paper/2603.15925/) (2026)
- [Trajectory-Aware Flow Matching for Topology Optimisation](https://mlsanity.com/paper/2607.14652/) (2026)

## Abstract (arXiv)

Engineering inverse design is often limited by the high computational cost of iterative solvers for optimization problems constrained by partial differential equations (PDEs) and by their sensitivity to initialization. Deep generative models can produce candidate designs without rerunning the simulator at inference time. Generative adversarial networks (GANs) sample in one forward pass, whereas diffusion models require iterative reverse-time integration. In this work, we add conditional flow matching (CFM) to EngiOpt and compare it with a conditional diffusion model and a conditional generative adversarial network (cGAN) on structural (beams2d) and thermal (heatconduction2d) benchmarks from EngiBench using the same downstream optimization protocol. We use cumulative optimality gap (COG) and final optimality gap (FOG) as the primary metrics for evaluating the generated designs as warm starts for gradient-based refinement. On the evaluated EngiOpt implementations and two EngiBench tasks, CFM achieves the lowest measured COG, FOG, maximum mean discrepancy (MMD), and volume-fraction deviation on both tasks. CFM has mean volume-fraction deviations of 0.4% and 1.0% on beams2d and heatconduction2d, respectively, compared with 3.8% and 11.2% for diffusion. At Euler s = 16, CFM achieves 53.2 samples/s on beams2d, about 66 times the measured throughput of the evaluated diffusion baseline using 1000 network evaluations under the same timing protocol, with COG 1.182 +/- 3.126, compared with 1.173 +/- 3.100 for Euler s = 32. Across the two tasks, CFM produces warm starts with lower measured COG than both baselines and uses fewer network evaluations than diffusion.