Many machine learning and scientific computing tasks, including computer vision and the computational modeling of physical and engineering systems, have intrinsic structures. Empirical studies demonstrate that models incorporating these structures often achieve significantly improved performance. Meanwhile, there is growing interest in discovering structures directly from observational data. In this talk, I will present our recent works on the interplay between structure and data. I will discuss how specific structures can be efficiently embedded into machine learning models and rigorously quantify the resulting performance gains. Furthermore, I will explore techniques for discovering structures, such as conservation laws, integrability, and Lax pairs, from observational physical data.
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