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check out this, it's not really a framework but good enough to start with
I would suggest you to look into Julia ecosystem instead of C++. Julia is almost identical to Python in terms of how you use it but it's still very fast. You should look into flux.jl package for Julia.
However, if you are writing numerical code in Fortran and want to be able to better interface it it with machine learning tools and methods, the number one thing I can recommend is to look into Fortran-based automatic differentiation tools. This will enable you to calculate exact derivatives of your code, which are useful to have for training and optimization loops. You can also look into f2py and f90wrap for generating Python wrappers for your Fortran codes, as well as Julia for hooking into it directly.
However, if you are writing numerical code in Fortran and want to be able to better interface it it with machine learning tools and methods, the number one thing I can recommend is to look into Fortran-based automatic differentiation tools. This will enable you to calculate exact derivatives of your code, which are useful to have for training and optimization loops. You can also look into f2py and f90wrap for generating Python wrappers for your Fortran codes, as well as Julia for hooking into it directly.
The hints in place ("there is more infrastructure already available outside Fortran, consider using them instead"). Beliavsky's compilation Fortran code on GitHub with its section about neural networks and machine learning still may be worth a visit e.g. how let Fortran reach out for the implementations in other languages.
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