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A computationally demanding physical simulation often presents a significant impediment to scientific and technological progress. Fortunately, recent advancements in machine learning (ML) and artificial intelligence have given rise to data-driven methods that can expedite these simulations. For instance, a well-trained 2D convolutional deep neural network can provide a 100,000-fold acceleration in solving complex problems like Richtmyer-Meshkov instability [1]. However, conventional black-box ML models lack the integration of fundamental physics principles, such as the conservation of mass, momentum, and energy. Consequently, they often run afoul of critical physical laws, raising concerns among physicists. These models attempt to compensate for the absence of physics information by relying on vast amounts of data. Additionally, they suffer from various drawbacks, including a lack of structure-preservation, computationally intensive training phases, reduced interpretability, and susceptibility to extrapolation issues. To address these shortcomings, we propose an approach that incorporates physics into the data-driven framework. This integration occurs at different stages of the modeling process, including the sampling and model-building phases. A physics-informed greedy sampling procedure minimizes the necessary training data while maintaining target accuracy [2]. A physics-guided data-driven model not only preserves the underlying physical structure more effectively but also demonstrates greater robustness in extrapolation compared to traditional black-box ML models. We will showcase numerical results in areas such as hydrodynamics [3,4], particle transport [5], plasma physics, pore-collapse, and 3D printing to highlight the efficacy of these data-driven approaches. The advantages of these methods will also become apparent in multi-query decision-making applications, such as design optimization [6,7].
More info: https://mfem.org/seminar/