Here's one using GANs, so not using an explicit likelihood: https://arxiv.org/abs/2203.06481

Here's a workshop paper applying score-based models: https://arxiv.org/abs/2209.14249

There are some papers showing diffusion working better for high-dimensional data in likelihood free inference, even just using an elbo bound. Can dig up later if wanted

See my comment for flows used as an alternative for ABC.

In science/physics flows are the dominant tool for simulation-based inference. The alternative is lengthy rejection sampling. Diffusion-based models are making an entrance in this area as well but are not as well-understood for practitioners to switch.

This seems like the progression of a fourier transform to a fast fourier transform.