Scatter due to interaction of photons with the imaged object is a fundamental problem in X-ray Computed Tomography (CT). It manifests as various artifacts in the reconstruction, making its abatement or correction critical for image quality. Despite success in specific settings, hardware-based methods require modification in the hardware, or increase in the scan time or dose. This accounts for the great interest in software-based methods, including Monte-Carlo based scatter estimation, analytical-numerical, and kernel-based methods, with data-driven learning-based approaches demonstrated recently. In this work, two novel physics-inspired deep-learning-based methods, PhILSCAT and OV-PhILSCAT, are proposed. The methods estimate and correct for the scatter in the acquired projection measurements. Different from previous works, they incorporate both an initial reconstruction of the object of interest and the scatter-corrupted measurements related to it, and use a deep neural network architecture and cost function, both specifically tailored to the problem. Numerical experiments with data generated by Monte-Carlo simulations of the imaging of phantoms reveal consistent improvement over a recent purely projection-domain deep neural network scatter correction method.