Solving inverse problems is a very common task in many scientific and engineering fields. The task can be summarized as: "You give me some data, I will give you some probability distributions of parameters that you might care about.". For example, given a time series of gravitational wave strain, find out what is the posterior distribution of the masses of the black holes that caused the gravitational wave. The is a very important process that quantify the uncertainty of a measurement. A measurement without any uncertainty is the same as no measurement.
Solving inverse problems often involves some sampling procedure or optimization procedure, such as Markov Chain Monte Carlo (MCMC), Variational Inference (VI), or more modernly, through simulated-based inference (SBI). There are different trade-offs between these methods. For example, MCMC is regarded as the gold standard for sampling from a posterior distribution, but it can be slow when compared to VI. SBI can be very fast and very accurate, but if the simulation does not represent the true systems well, it is very hard to diagnose where the systematics are coming from. I am interested in composing these methods to exploit their strengths and mitigate their weaknesses.
The interplay between these methods are quite delicate and rather unclear to the community (as of 2024). For me, on one hand I enjoy adding new stuffs into the mix to see what it does, but it is also important to understand the reasons why certain ways of composing these methods work better than others.
I am the main developer and maintainer of an adaptive MCMC sampler flowMC, so I think I have the right to say the tuning procedure of flowMC may not be the most friendly and robust. Part of the reason is flowMC is designed to handle sampling problems that have bad geometries such as multi-modality and local correlation (e.g. a donut), and these problems can be really hard! In fact, global optimization of non-convex problem is known to be NP-complete, and if I tell you there is a generic way for me to handle this class of problems in a provably fast way, you can be sure that is a lie since that will mean I have proven P=NP, which is a way bigger deal.
Now that is not to say this problem is hopeless either. It is very hard to solve the problem in a genral sense, but for a specific usecase which the users have some insights about the complexity of the problem, there could be a guide for the user to tune the sampler to be efficient enough for them. I won't say flowMC is the best solution for your problem, but flowMC should be a reasonably good start. And if flowMC cannot solve your problem, it should provide you insights on where are the difficulties of the problem.