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.

💡 Research Summary

This paper investigates the capability of modern generative models to solve Bayesian inverse problems, using the concrete task of gas‑turbine combustor design as a benchmark. The authors compare a traditional Bayesian inverse approach—forward regression combined with a prior sampled via Markov Chain Monte Carlo (MCMC)—against three state‑of‑the‑art conditional generative models: Conditional Wasserstein‑GAN (cWGAN), Invertible Neural Networks (INN), and Conditional Flow Matching (CFM).

The inverse problem is defined as follows: six independent design parameters (X) determine three performance metrics (Y) through a costly CFD simulation s(·). Observations are noisy, y = s(x) + ε, with ε ∼ N(0,σ²I). The goal is, given a target performance vector y*, to generate design vectors x that satisfy s(x) ≈ y*. Traditional Bayesian inference models the posterior p(x|y) ∝ p(y|x)p_X(x) and draws samples with MCMC, but this requires careful tuning and yields only a single design per optimization run.

Generative approaches aim to learn the conditional distribution p(x|y) directly from paired data {(x^{(j)}, y^{(j)})}. The dataset consists of 1,295 designs generated by Latin‑Hypercube Sampling across the six‑dimensional design space, each evaluated with CFD to obtain unmixedness (U_M), pressure drop (Δp_t,rel), and thermoacoustic growth rate (G). The authors propose a suite of evaluation metrics: (1) label reconstruction error (RMSE between generated design’s predicted labels and the target), (2) diversity of generated designs measured by average minimum distance and coverage of the target posterior region, and (3) data efficiency (performance as a function of training set size).

Model specifics: cWGAN uses a Wasserstein loss with a critic conditioned on y; INN builds a stack of invertible blocks to map x ↔ latent z while conditioning on y; CFM employs a continuous‑time Neural ODE whose vector field is conditioned on y, trained by minimizing the flow‑matching loss. All models are trained with comparable architectures and hyper‑parameters, and experiments are repeated with full data (1,295 samples) and reduced subsets (200, 400, 800 samples).

Results show that CFM consistently outperforms the other methods. With the full dataset, CFM achieves an RMSE of 0.042 on the three performance labels, a diversity score (average minimum distance) of 0.87, and covers 94 % of the target posterior region. cWGAN attains an RMSE of 0.118, diversity 0.62, and coverage 71 %; INN reaches RMSE 0.067, diversity 0.78, and coverage 82 %. When training data are limited to 200 samples, CFM’s RMSE only rises to 0.058 and coverage remains at 89 %, whereas cWGAN and INN degrade sharply (RMSE 0.132 and 0.081, respectively). The traditional Bayesian MCMC approach yields an RMSE around 0.075 but suffers from high computational cost and lack of automation.

The authors conclude that conditional flow‑matching offers the best trade‑off between label fidelity, design diversity, and data efficiency, making it the most practical tool for automated inverse design in engineering contexts. They note that CFM’s performance depends on ODE solver choices and time‑step settings, which may become limiting in very high‑dimensional design spaces. Future work is suggested on integrating active learning, Bayesian optimization, and hybrid CFD‑generative pipelines to further accelerate and robustify inverse design workflows.