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In this paper we show that our Machine Learning (ML) approach, CoMLSim (Composable Machine Learning Simulator), can simulate PDEs on highly-resolved grids with higher accuracy and generalization to out-of-distribution source terms and geometries than traditional ML baselines. Our unique approach combines key principles of traditional PDE solvers with local-learning and low-dimensional manifold techniques to iteratively simulate PDEs on large computational domains. The proposed approach is validated on more than 5 steady-state PDEs across different PDE conditions on highly-resolved grids and comparisons are made with the commercial solver, Ansys Fluent as well as 4 other state-of-the-art ML methods. The numerical experiments show that our approach outperforms ML baselines in terms of 1) accuracy across quantitative metrics and 2) generalization to out-of-distribution conditions as well as domain sizes. Additionally, we provide results for a large number of ablations experiments conducted to highlight components of our approach that strongly influence the results. We conclude that our local-learning and iterative-inferencing approach reduces the challenge of generalization that most ML models face.