Feynman integrals are crucial in calculating scattering amplitudes and related quantities in particle physics. One of the key properties of these integrals is that they obey linear relations, enabling the simplification of scattering amplitude calculations. Specifically, they are used for decomposing the amplitudes into a basis of functions called master integrals (MIs) and evaluating them using differential equations. In this talk, I will delve deeper into the structure of Feynman integrals and show how they can be better understood by utilizing the vector space structure of the integral and Intersection theory. Furthermore, I will demonstrate how evaluating these MIs aids in obtaining higher-order corrections for observables at colliders, which is pivotal to improving theoretical precision. Finally, I will discuss the potential applications of these techniques developed for studying Feynman integrals to compute precise observables for gravitational wave experiments.