| Contents | Many real-world systems from weather dynamics to wildfire smoke dispersion are governed by complex partial differential equations (PDEs), yet traditional numerical solvers often struggle in data-sparse, irregular, and topologically complex settings. Physics-Informed Machine Learning (PIML) has emerged as a transformative paradigm that embeds physical laws into machine learning models to ensure consistency, efficiency, and generalizability. In this talk, I will explore the evolution of PIML, beginning with the foundational framework of Neural Ordinary Differential Equations (Neural ODEs), which laid the groundwork for modeling continuous-time dynamics through differentiable solvers. I will then trace key developments in the field highlighting limitations of early approaches such as PINNs and discrete-grid PDE solvers and present how Graph Neural Networks (GNNs) enable the modeling of PDEs over unstructured spatial domains.
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