We describe an approach for incorporating prior knowledge into machine learning algorithms. We aim at applications in physics and signal processing in which we know that certain operations must be embedded into the algorithm. Any operation that allows computation of a gradient or sub-gradient towards its inputs is suited for our framework. We derive a maximal error bound for deep nets that demonstrates that inclusion of prior knowledge results in its reduction. Furthermore, we also show experimentally that known operators reduce the number of free parameters. We apply this approach to various tasks ranging from CT image reconstruction over vessel segmentation to the derivation of previously unknown imaging algorithms. As such the concept is widely applicable for many researchers in physics, imaging, and signal processing. We assume that our analysis will support further investigation of known operators in other fields of physics, imaging, and signal processing.

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and are important in various technological fields such as energy, electronics, medicine, and many more. [1][2][3][4][5] However, as a consequence of industrial processes and man-made pollution, unwanted nanoparticle size distributions and concentrations [6] give rise to concerns with respect to human health and environmental pollution. While the nanoparticles' physicochemical properties (size, shape, surface chemistry, etc.) determine the quality of products, [7,8] such characteristics are also important in order to evaluate the biological impact of nanoparticles at a molecular, cellular, and systemic level for any risk assessment for environmental and human health. [9] Characterizing nanoparticles in a dynamic context and on a case-by-case basis, microscopic imaging techniques including those that use focused electron or ion beams in scanning electron microscopes (SEMs) or helium ion microscopes [10] (HIMs) to generate nanometer scale spatial resolution are frequently applied in the scientific community. Given the substantial information content of digital images, these techniques often benefit from, or require, automated high-throughput data analysis that enables the accurate identification of large numbers of particles in a robust way.Nanoparticles occur in various environments as a consequence of man-made processes, which raises concerns about their impact on the environment and human health. To allow for proper risk assessment, a precise and statistically relevant analysis of particle characteristics (such as size, shape, and composition) is required that would greatly benefit from automated image analysis procedures. While deep learning shows impressive results in object detection tasks, its applicability is limited by the amount of representative, experimentally collected and manually annotated training data. Here, an elegant, flexible, and versatile method to bypass this costly and tedious data acquisition process is presented. It shows that using a rendering software allows to generate realistic, synthetic training data to train a state-of-the art deep neural network. Using this approach, a segmentation accuracy can be derived that is comparable to man-made annotations for toxicologically relevant metal-oxide nanoparticle ensembles which were chosen as examples. The presented study paves the way toward the use of deep learning for automated, highthroughput particle detection in a variety of imaging techniques such as in microscopies and spectroscopies, for a wide range of applications, including the detection of micro-and nanoplastic particles in water and tissue samples.