Inverse problem is the opposite of direct problem. And it is a general viewpoint to recognize problems.
In practical cases, we consider both problems on the same time.
"Model" is a system in a wide sense. We can use the viewpoint of direct problem and inverse problem for all models and systems.
System identification is an inverse problem. We call "inverse problem", when we make models, for that we can confirm "it is same to the phenomenon", using input and output data (or only output data).
System identification is also a target in Time Series Analysis .
Inverse problem is a general concept. Especially, in vibration analysis and structure analysis, where non-linear partial differential equations are models, "inverse problem" is a big target.
The problem in this field is the case "model is identified, output is known and we want to identify input".
When we use data science, we cannot avoid measuring process. And we cannot avoid noise.
Almost all mathematical models are made without considering noise. So people, who believe "mathematical models show exactly the real world", do not understand the influence of noise.
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