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In information systems, data analysis plays a crucial role in uncovering hidden patterns and insights. Visualizing data behavior enables researchers to examine its dynamics, strengths, and limitations. For pure mathematical problems, traditional approaches rely on mathematical tools for problem-solving. However, adopting a data-driven approach---where relevant data is generated within the problem's scope and analyzed intuitively---allows for alternative perspectives and solutions. In this article, we present an analytical and visual framework for addressing mathematical and engineering problems. By developing a novel manifold learning-based algorithm, we examine these problems from a unique perspective. We demonstrate the effectiveness of this approach through various applications, including approximate solutions to partial differential equations (PDEs) and classical mathematical problems such as studying the distribution and behavior of prime numbers. Our results show that even pure mathematical problems can benefit from this methodology. This framework can also be applied to other scientific and engineering disciplines. We aim to provide innovative perspectives on diverse challenges across mathematics, engineering, and the sciences.