As far as I’m aware, our understanding of the strong interaction and strong nuclear force is pretty good. We can explain how particles move and stick together with good models. But from what I can find, I gather we do not have a mathematical theory that explains this in a precise manner. For example, for electromagnetism and gravitation, we have well established models that predict the phenomena precisely. My question is, is there anything lacking, such as a big hole in our knowledge, or is there any big problem or issue that has stopped us from establishing a mathematical model for the strong interaction?
Most of the calculations in quantum mechanics are made using the perturbation calculus. That is, we assume that the theory with no interactions that describes a free particle is an approximation of the full theory, take the classical solution of the free equations, and then add corrections. There are usually infinitely many quantum corrections that need to be calculated, but they often form a series in which every next term is much smaller than the previous one, and calculating just few initial terms gives a good approximation of a full result.
The problem with strong interaction is that the interaction is so strong we cannot reasonably approximate the real physical states with free-particle states. That means we fail at the first step, and we cannot use the perturbation approach that works in other cases.
Anther problem with QCD is that gluon fields interact directly with itself, making even the classical equations are non-linear, as opposed to linear equations of QED (in which electromagnetic field does not interact directly with itself, but only with charged matter). And we don't have the right mathematical tools to fully solve non-linear equations. This is similar to Generla Relativity, which also has non-linear equations. Because of that in GR we only know very few exact solutions. QCD has the same problem, and because of that we need to relay more on the simulations than on analytical solutions.
Wikipedia has mentioned that in the high energy range which is the domain of particle physics the perturbation theory works fine which can result in sort-of-analytical form of predictions which match the experiment results with good precision which is the main success of the theory of strong interaction come from for now.
But in the low energy range when quarks bind to together to form hadrons and hadron bind toether to form atomic nucleus, the perturbation theory fails and the only thing valid is sort-of-pure-numerical-framework solutions called lattice QCD.
With LQCD and enough computing power you can predict properties of hadrons and atomic nucleus and even nuclear reactions with high precision from the theory of strong interaction. But for now due to limited computation power LQCD can only predict hadron with some acceptable precision.
Due to the near to the end of the Moore's Law, maybe only small atomic nucleus can be predicted finally in the future without model/algorithm/hardware breakthrough. Another breakthrough may come from Quantum Computing.
Another route is phenomenological model like the many models of atomic nucleus which has little to do with the theory of strong interaction.
For example, for electromagnetism and gravitation, we have well established models that predict the phenomena precisely
This is because the classical model - the Maxwell' equation which deal only with EM field in vacuum or simple material which can be treated as a continous bulk with some properties like dialetric constant which can be used in the equation. Once you begain to deal with "true EM problem" - like predicting structures and properties atom/molecule/liquid/solid from quantum mechanics, it soon become not that precise too, a lot of trade off and approximation is needed to make it computable on current generation of computers.
For gravity, there is not even a "true gravity"(quantum gravity) theory for now and you treat star/galaxy/Universe as a continuous bulk using GR, like in classical EM.