Question: I wonder what are the open problems , where computational experiments might me helpful? (Setting some bounds, excluding some cases, shaping some expectations ).
Grant program: The context of the question is the following - in some sense there is a way to crowdsource some (quite specific, but still) computational experiments - the platform for data science challenges Kaggle (Google's subsidiary) - offers grants to researchers in various fields to organize the challenges. Properly organized it might attract thousands participants, some of whom are extremely skilled in solving computational challenges (like NVIDIA grandmaster's team - 6 super-skillful guys who are payed by NVIDIA just to participate in challenges (and promote some NVIDIA tools) - kind of profi-sportsmen). There already many research oriented challenges took place - mainly around in bioinformatics (e.g. CAFA5, Stanford-Ribonanza, Single Cell Perturbations, etc.. ).
Example: The one quite related to mathematics - is ongoing challenge on finite permutation groups - the task is to find as short paths as possible for sets of given points on Cayley graphs. The groups include N-dimensional Rubik's cube and other puzzle motivated groups like "Globe". For example even the diameters of the N-dimensional Rubik's cube are still unknown, so computational experiments may shed light on these questions. The other questions - like better understanding of the random walks, growth patterns, patterns of mixing, fast scrambling algorithms - seems to be of interest from both sides - machine learning and mathematical. So fruitful ideas exchange and interconnections might be possible. It would be great if some mathoverflow colleagues experienced in such kind of permutation group questions may share experience - please post on forum e.g. here or/and contact e.g. me.
PS
I am a kind of fan of many sides of Kaggle - it managed to create community of people who not only competing but sharing lots of experience. Gamified "reputation" scores - similar to mathoverflow - play a role, moreover Kaggle rankings are appreciated in data science world, so can help to find a better job - another stimulus for many to participate, in addition to networking. Many people think that the best way to learn practical data science - participate in Kaggle challenges. It is not only for competition, but a free cloud service - like Google Colab, but more devoted to team work - offering free computing power, storage, sharing, version control, team work. Would be great to have something similar for mathematical community or to use Kaggle itself for mathematical community - one may think of a competition as a kind of mini-"Polymath" project. Where all the infrastructure - is ready - forum, community, free computing power - the only thing one needs - is to create a challenge. Kaggle offers everybody to create a challenge for free, but only "official" challenges attract most attention.