Scientific machine learning has demonstrated significant potential in advancing computational science by integrating data-driven methods with established physical models. This talk presents an analysis-driven, structured framework for learning models from data, focusing on key aspects such as uniqueness, mathematical well-posedness, and interpretability of the learned models. In contrast to purely data-driven approaches, the proposed framework augments existing physical models by integrating approximation schemes for unknown physical model components that preserve consistency with the underlying physical principles. Central to this methodology is the careful design of the approximation scheme (e.g., neural-network based) and regularization strategies for the unknown components of the physical model. When properly chosen, these elements enable reliable identification of governing dynamics even in the presence of noisy or incomplete data. The framework can also incorporate physical constraints, such as conservation laws, directly into the learning process. This guarantees that the learned models respect fundamental scientific laws, providing solutions that are both mathematically well-posed and physically meaningful, e.g. in reaction-diffusion systems. By employing symbolic models designed for interpretability, the learned models provide human-interpretable insights into the underlying physics. A key message of this talk is that many desirable properties of learned physical models, such as well-posedness, consistency, and interpretability, can be achieved through an analysis-driven design of the learning process, including the underlying approximation schemes and regularization. While challenges remain in scaling the approach to high-dimensional systems and improving computational efficiency, these limitations highlight important directions for future research in hybrid methods for scientific computing. References: Erion Morina and Martin Holler. On uniqueness in structured model learning. ArXiv preprint arXiv:2410.22009, 2024. doi: 10.48550/arXiv.2410.22009 Erion Morina and Martin Holler. Physically Consistent Model Learning for Reaction-Diffusion Systems. ArXiv preprint arXiv:2512.14240, 2025. doi: 10.48550/arXiv.2512.14240