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Directed energy deposition (DED) is an advanced additive manufacturing technology for the fabrication of near-net-shape metal parts with complex geometries and high performance metrics. Studying the grain structure evolution during the process is pivotal to evaluating and tailoring the as-built products' mechanical properties. However, it is time-consuming to simulate the multi-layer deposition process using the physics-based numerical model to optimize the process parameters for achieving the desired microstructure. In this paper, a physics-informed machine learning algorithm to predict the grain structure in the DED process is proposed. To generate training data for the machine learning algorithm, we use an experimentally validated cellular automaton finite volume method (CAFVM) for DED Inconel 718, where CA is applied to model the grain structure and FVM to simulate the heat transfer. We develop a neural network model to identify the correlation between the local thermal features and their corresponding grain structure characteristics. The inputs and outputs of the neural network (NN) model are selected based on the governing physics, and a novel way to extract them is proposed. The NN model can quickly predict the grain structure characteristics with the local thermal data for thin-wall builds, and the predictions are in good agreement with the numerical simulation results. We expect the proposed method can benefit other metal additive manufacturing technologies to formulate efficient and accurate process-structure relationships and in-process feedback control. [Display omitted] • Predict additively manufactured grain structure with a machine learning method. • An 3D cellular automaton finite volume method for "the ground truth" microstructure. • Strategy to represent the grain size and aspect ratio variation in a neural network. • Simulated thin-wall builds of Inconel 718 by directed energy deposition. [ABSTRACT FROM AUTHOR]