There might be, but very few. Most (like gpt-oss, or Deepseek R1) are open weight, because no one wants to make their model training data public.

Open sourcing a model would likely give away they are using pirated content or scrap internet websites without permission. That's something Anthropic was fined for last year, they were caught using pirated books for model training.

Moreover, training data is one of most lucrative resources in AI race next to GPUs, and AI companies are going all in on finding it, desperately fighting for any possible data. Heck recently they even started buying antique books, digitizing, and burning them.

They invest a whole lot of resources to get quality data, and they don't want to share it with competitors for free.

My personal opinion though is FOSS philosophy doesn't apply that cleanly to LLMs anyway. If compiling Chromium or Linux kernel from source is tedious enough, "compiling" an LLM from source on consumer hardware is straight impossible. It's something only your competitors could do.

Thus you can't verify the LLM is the same as it's source code anyway, so open source label becomes meaningless.

And open weight model is still an improvement over using a cloud model, as it's local and doesn't call home with your chats data.