Investing in applied machine learning (ML) without understanding its mathematical and statistical foundations is like investing in healthcare without understanding biology. Foundational mathematical research has had a tremendous impact on ML, from optimization methodology to privacy protections and from data acquisition strategies to uncertainty quantification. Understanding the foundations of data science has also deepened our insights into longstanding challenges in statistics, numerical linear algebra, optimization theory, and engineering. My research focuses on developing the mathematical and statistical foundations of machine learning and scientific machine learning methodology. Foundational advances are essential for trustworthy ML, including high-quality, reproducible, ML-enabled scientific research. The widespread adoption of ML in the natural sciences has the potential to integrate scientific inquiry with new modes of hypothesis generation, data analysis, simulation, and testing that will transform our capacity to address scientific problems that currently appear intractable. A key, cross-cutting aspect of this endeavor is physics-informed ML, which optimally leverages physical models and experimental or observational data.  Examples include learning improved climate models and forecasters, using
ML together with physics-based models to uncover material structure and properties, and performing medical image reconstruction with unprecedented speed and accuracy.