| Summary: | With the recent advancement of modern machine learning methods, there are now many exciting opportunities to use machine learning in scientific research, including for modeling and data analysis. Machine learning has the potential to become an indispensable tool for scientific discovery, but it is often difficult to directly apply to scientific problems. Especially in the case of deep learning approaches, machine learning methods are often lacking in interpretability, robustness, out-of-distribution generalization, and data efficiency—all qualities that are necessary for many scientific and engineering applications. In this thesis, we will illustrate several approaches for addressing these issues using a variety of applications. First, we develop a physics-informed framework for partially observed system identification, showing how combining an encoder with a sparse symbolic model allows us to reconstruct unobserved hidden states as well as the exact governing equations. Then, we design a physics-informed deep representation learning architecture for analyzing spatiotemporal systems and demonstrate its ability to extract interpretable physical parameters, corresponding to uncontrolled variables, from time-series data. Finally, we use tools from optimal transport theory and manifold learning to develop a robust non-parametric method for discovering conservation laws, showing the advantage of using geometric machine learning methods to solve scientific problems. By designing physics-informed architectures and adapting representation learning methods for scientific applications, we can overcome many of the difficulties that are currently preventing machine learning from playing a more important role in scientific discovery and create more useful computational tools for scientists and engineers trying to analyze, understand, and model their data.
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