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My research aims to understand data-driven models from a mathematical perspective and to use mathematical analysis to inform development of methods for scientific machine learning.
Operator Learning
Operator Learning
Many equations and systems in science and engineering applications involve maps between function spaces. For example, partial differential equations (PDE) such as the Helmholtz equation (above), map a coefficient function over a domain to a PDE solution. Operator learning is a variant of scientific machine learning/scientific AI that is tailored to these maps between function spaces. Much of my work revolves around considerations for data-driven models in these settings.
Limitations of Data
Limitations of Data
When approximating a map of interest from data, limitations in the quality and quantity of the data have implications for the accuracy of the resulting model. My work addresses these limitations from a mathematical and computational standpoint, seeking to answer questions such as: How can we learn a model with access to only a small amount of data?, What sorts of sample approximation rates can we expect to achieve with certain models?, and How can we incorporate additional information to make the most of the data we do have?
Scientific Machine Learning for Engineering
Scientific Machine Learning for Engineering
Ultimately, making scientific machine learning useful in practice means integrating it with established methodologies and trusted domain knowledge. In past work, I have blended homogenization theory and operator learning to construct theory-backed methodologies for modeling history dependence in constitutive materials. My current research continues this broad goal of combining classical, mathematically proven methods for PDEs with data-driven models.
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