Physics-informed convolutional neural networks (PICNNs) have emerged as a powerful extension of physics-informed neural networks (PINNs), offering superior generalization and efficiency for solving systems of partial differential equations (PDEs) modeling complex physical systems. However, current PICNNs, which rely on manual designs of both network architectures and loss functions tailored to specific PDEs, may fail to solve other types of PDEs that require different architectural designs and loss function formulations. To address these limitations, we propose leveraging automated machine learning (AutoML) to efficiently search for optimal network architectures and loss functions tailored to specific physical problems. By designing specialized search spaces and proposing a two-stage search strategy, our automated method substantially outperforms manually designed state-of-the-art models. It achieves up to a 59.8-fold reduction in prediction error and an average 13.31-fold error reduction across six diverse PDE systems spanning heat transfer, free fluid flow, and porous media flow, significantly enhancing the modeling capabilities for physical systems governed by steady or unsteady PDEs. An advantage of our automated method is that it can help researchers, without expertise in designing neural networks, develop the best models for their scientific problems. Physics-informed convolutional neural networks are effective extensions of physics-informed neural networks for solving systems of partial differential equations modeling complex physical systems. Using automated machine learning, this study offers a framework to achieve optimal network architectures and loss functions that outperform manually designed state-of-the-art models, with a 59.8-fold reduction in prediction error across six systems including heat transfer, incompressible fluid flow, and porous media flow.