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

Patrick Krüger, Patrick Materne, Werner Krebs, Hanno Gottschalk

TL;DR

This study reframes gas-turbine combustor inverse design as a Bayesian inverse problem and benchmarks three state-of-the-art generative approaches—Invertible Neural Networks, Conditional Flow Matching, and Conditional Wasserstein GANs—against a traditional forward-model Bayesian baseline. Using a CFD-derived combustor dataset of six design parameters mapped to three performance labels, the authors train surrogates to scale data and evaluate accuracy and diversity of retrieved designs as a function of training size. Conditional Flow Matching consistently achieves superior data efficiency and precision while preserving design diversity, outperforming competing models and the Bayesian baseline. The results underscore the potential of continuous-flow, flow-matching approaches for robust, diverse inverse-design in engineering, and they motivate applying these methods to higher-dimensional, real-world problems.

Abstract

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.

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

TL;DR

This study reframes gas-turbine combustor inverse design as a Bayesian inverse problem and benchmarks three state-of-the-art generative approaches—Invertible Neural Networks, Conditional Flow Matching, and Conditional Wasserstein GANs—against a traditional forward-model Bayesian baseline. Using a CFD-derived combustor dataset of six design parameters mapped to three performance labels, the authors train surrogates to scale data and evaluate accuracy and diversity of retrieved designs as a function of training size. Conditional Flow Matching consistently achieves superior data efficiency and precision while preserving design diversity, outperforming competing models and the Bayesian baseline. The results underscore the potential of continuous-flow, flow-matching approaches for robust, diverse inverse-design in engineering, and they motivate applying these methods to higher-dimensional, real-world problems.

Abstract

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.
Paper Structure (24 sections, 42 equations, 11 figures, 5 tables)

This paper contains 24 sections, 42 equations, 11 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Inverse Problem Description/ The Combustor Dataset
  4. Generative Models
  5. Normalizing Flows and Invertible Neural Networks
  6. Neural ODE and Conditional Flow Matching
  7. Conditional Wasserstein Generative Adversarial Networks
  8. Conditional Gan
  9. Conditional Wasserstein Gan
  10. Bayesian Inference for Posterior Sampling in Inverse Problems
  11. Implementation
  12. Surrogate Models and Data Augmentation
  13. Generative Models
  14. Conditional Wasserstein GAN Implementation
  15. Flow-Based Models
  16. ...and 9 more sections

Figures (11)

  • Figure 1: Schematic of the parameterized combustor geometry defined by independent variables $X$. Source: krueger2025generative
  • Figure 2: Scatter plot of the combustor dataset $\mathcal{D}$, showing relationships between design parameters $X_i$ and performance metrics $Y_i$. Dashed line: stability boundary $G = 0$. Source: krueger2025generative.
  • Figure 3: Mean absolute errors between target label values and true label values of generated designs for all dataset sizes $d$ and labels $Y_i\in\{U_M,\Delta p_{t,\textrm{rel}},G\}$.
  • Figure 4: Parity plots between target and true label values for $Y_i=U_M$ and $d\in\{100,1\,000, 10\,000, 100\,000\}$.
  • Figure 5: Distributions of the independent parameters $X^{\mathrm{Gen}}$ of generated by the CFM model conditioned target label values $Y_i^{\mathrm{Target}}$ and their respective values (top: $U_M$, center: $\Delta p_{t,\textrm{rel}}$, bottom: $G$).
  • ...and 6 more figures