This paper presents a study of gravitational waves (GWs) from inflation in the context of the LISA mission. The authors develop a template-based analysis pipeline for the stochastic gravitational wave background (SGWB) and introduce a template database to describe well-motivated signals from inflation. They classify seven templates based on their signal frequency shape and identify representative fundamental physics models leading to them. By running a template-based analysis, they forecast the accuracy with which LISA can reconstruct the template parameters of representative benchmark signals, with and without galactic and extragalactic foregrounds. They also investigate how their signal reconstructions shed light on fundamental physics models of inflation, such as the couplings of inflationary axions to gauge fields, the graviton mass during inflation, the fluctuation seeds of primordial black holes, the consequences of excited states during inflation, and the presence of small-scale spectral features. The paper outlines the development of a template database for inflationary mechanisms, including power law, log-normal bump, broken power law, double peak, excited states, linear oscillations, and logarithmic resonant oscillations. Each template is associated with a specific inflationary scenario and is used to forecast the accuracy of LISA in reconstructing the parameters of these signals. The authors also discuss the implications of these reconstructions for fundamental physics models, such as the impact on measurements of inflationary axion couplings, the graviton mass during inflation, and the fluctuation seeds of primordial black holes. The study highlights the importance of accurate modeling of LISA noise, foregrounds, and primordial SGWB for the successful reconstruction of these signals. The paper concludes with a discussion of the scientific implications of these findings for the study of inflation and the detection of gravitational waves from the early universe.This paper presents a study of gravitational waves (GWs) from inflation in the context of the LISA mission. The authors develop a template-based analysis pipeline for the stochastic gravitational wave background (SGWB) and introduce a template database to describe well-motivated signals from inflation. They classify seven templates based on their signal frequency shape and identify representative fundamental physics models leading to them. By running a template-based analysis, they forecast the accuracy with which LISA can reconstruct the template parameters of representative benchmark signals, with and without galactic and extragalactic foregrounds. They also investigate how their signal reconstructions shed light on fundamental physics models of inflation, such as the couplings of inflationary axions to gauge fields, the graviton mass during inflation, the fluctuation seeds of primordial black holes, the consequences of excited states during inflation, and the presence of small-scale spectral features. The paper outlines the development of a template database for inflationary mechanisms, including power law, log-normal bump, broken power law, double peak, excited states, linear oscillations, and logarithmic resonant oscillations. Each template is associated with a specific inflationary scenario and is used to forecast the accuracy of LISA in reconstructing the parameters of these signals. The authors also discuss the implications of these reconstructions for fundamental physics models, such as the impact on measurements of inflationary axion couplings, the graviton mass during inflation, and the fluctuation seeds of primordial black holes. The study highlights the importance of accurate modeling of LISA noise, foregrounds, and primordial SGWB for the successful reconstruction of these signals. The paper concludes with a discussion of the scientific implications of these findings for the study of inflation and the detection of gravitational waves from the early universe.