Conducting an accurate yet simple mathematical description of large-scale dynamical networks, such as biological networks, is an essential yet challenging step toward enabling design and control of such complex systems. The extraordinary growth of data-rich biology further calls for discovery of transformative mathematical techniques to facilitate data-driven research on dynamical systems. For example, daily rhythms in physiology allow organisms to anticipate reliable environmental events and adapt to changing environmental cues. The suprachiasmatic nucleus (SCN) generating these precise circadian oscillations and adapting to environmental changes is comprised of thousands of circadian neurons in the brain. The long-term goals of this research are to: 1) create a rigorous kernel method that generates a simple yet accurate representation for modeling complex nonlinear dynamical processes; 2) leverage this method to design interpretable and principled data-driven techniques for system control and learning of dynamic networks; 3) resolve longstanding biological questions about deciphering the underlying dynamics of coupled circadian clocks; and 4) identify enhanced environmental cues to produce desired behavior patterns such as chronotype setting and sleep consolidation. This proposal aims to overcome the significant mathematical challenges to solve important problems in circadian biology. In Aim 1, we will establish the moment kernel machine (MKM) that generates lossless compression models equivalent to the first-principles state-space models of dynamic network systems. In Aim 2, we will use the predictive power of MKM modeling to design signals that create desired synchronization patterns in oscillator networks. In Aim 3, we will demonstrate the versatility of the MKM technique with applications to circadian biology. This will involve dynamic learning of the SCN gene expression from measurements; and learning and shaping the output of SCN networks by designing dynamic light protocols for enforced biphasic sleep in young animals and consolidation of sleep fragmentation in aged animals, and assigning phases of entrainment (e.g., onset of daily activities) to animals of different genotypes and chronobiological properties. The integration of novel mathematical and biological tools will provide insight and guidelines for the theory- inspired experimental designs and lead to a comprehensive understanding of complex network behaviors such as coupled circadian oscillators producing daily patterns of sleep and wake.