"Forward problem" is a method of predicting observations (predicted data) based on certain general principles or models, and a series of known specific conditions related to the problem being addressed.
"Inversion problem", roughly speaking, is to deal with the opposite problem, that is, to determine the estimated values of model parameters from the data and some general principles or models, the purpose is to provide information about unknown parameters in the model . The relationship between model parameters and observations is somehow called a "model." It is usually one or several formulas that make the data and model parameters meet these formulas, and can provide a way to evaluate whether a given model is correct, or to distinguish among several possible models, which is correct.
In general, because the forward model is deterministic and the model parameters are complete and noiseless, the solution to the forward problem is unique. In the inversion problem, the actual data always contains noise, and in most cases, it is difficult to observe all the information, causing the inversion problem to have multiple solutions.