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We present a novel computational paradigm for process design in manufacturing processes that incorporates simulation responses to optimize manufacturing process parameters in high-dimensional temporal and spatial design spaces. We developed a differentiable finite element analysis framework using automatic differentiation which allows accurate optimization of challenging process parameters such as time-series laser power. We demonstrate the capability of our proposed method through three illustrative case studies in additive manufacturing for: (i) material and process parameter inference using partial observable data, (ii) controlling time-series thermal behavior, and (iii) stabilizing melt pool depth. This research opens new avenues for high-dimensional manufacturing design using solid mechanics simulation tools such as finite element methods. Our codes are made publicly available for the research community at https://github.com/mojtabamozaffar/differentiable-simulation-am.