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See the COIN-OR Optimization Services (OS) project for a solver-independent xml format called OSiL. It's functionally equivalent to AMPL .nl format but more human-readable. Unfortunately the osil readers for solvers have more performance problems and bugs than .nl readers, since the AMPL format has been around longer and is commercially supported.

The proprietary part of AMPL is the program that turns human-readable .mod and .dat files into low-level .nl encodings of the problem representation. The .nl format is documented, and there's an MIT licensed library on netlib (and now github too) that reads it and evaluates functions, constraints, gradients, Jacobians, Hessians, etc. (AD is super useful and has been around way longer than the deep learning popularization of it.)

Julia has a low-level solver-independent API called MathProgBase that sits between solver wrappers and modeling languages like JuMP / Convex.jl. It's in the process of being rewritten as MathOptInterface, but it would allow people to write solvers in pure Julia and be usable right away with JuMP.