MHD的物理约束低维模型：第一原理和数据驱动方法
Physics-constrained, low-dimensional models for MHD: First-principles and data-driven approaches
摘要：
等离子体是高度非线性和多尺度的，激发了一系列模型来理解和描述它们的行为。然而，有一个较低的保真度比磁流体力学（MHD）的等离子体模型的稀缺，虽然这些减少模型持有的承诺，了解关键的物理机制，有效的计算，实时优化和控制。通过将MHD方程投影到截断模态基础上获得的Galerkin模型，以及通过现代机器学习和系统识别获得的数据驱动模型，可以在模型层次结构的较低级别中填补这一空白。这项工作为可压缩等离子体开发了一个降阶建模框架，利用了数十年来在基于投影和数据驱动的流体建模方面的进展。我们首先形式化的非线性MHD系统的基于投影的模型降阶。为了避免单独的模态分解的磁场，速度和压力场，我们引入了一个能量内积合成到一个维度一致的，降阶的基础上的所有领域。接着，我们通过将Hall-MHD方程投影到这些模式上，得到了一个解析模型。我们说明了全局守恒定律如何约束模型参数，揭示了可以在数据驱动模型中强制执行的对称性，将这些模型直接连接到底层物理。我们证明了这种方法的有效性，从高保真数值模拟的3D spheromak实验的数据。这份手稿在流体力学中建立了一座通往广泛的伽辽金文献的桥梁，并促进了等离子体基于投影和数据驱动模型的未来原则性发展。
英文摘要：
Plasmas are highly nonlinear and multi-scale, motivating a hierarchy of models to understand and describe their behavior. However, there is a scarcity of plasma models of lower fidelity than magnetohydrodynamics (MHD), although these reduced models hold promise for understanding key physical mechanisms, efficient computation, and real-time optimization and control. Galerkin models, obtained by projection of the MHD equations onto a truncated modal basis, and data-driven models, obtained by modern machine learning and system identification, can furnish this gap in the lower levels of the model hierarchy. This work develops a reduced-order modeling framework for compressible plasmas, leveraging decades of progress in projection-based and data-driven modeling of fluids. We begin by formalizing projection-based model reduction for nonlinear MHD systems. To avoid separate modal decompositions for the magnetic, velocity, and pressure fields, we introduce an energy inner product to synthesize all of the fields into a dimensionally-consistent, reduced-order basis. Next, we obtain an analytic model by Galerkin projection of the Hall-MHD equations onto these modes. We illustrate how global conservation laws constrain the model parameters, revealing symmetries that can be enforced in data-driven models, directly connecting these models to the underlying physics. We demonstrate the effectiveness of this approach on data from high-fidelity numerical simulations of a 3D spheromak experiment. This manuscript builds a bridge to the extensive Galerkin literature in fluid mechanics, and facilitates future principled development of projection-based and data-driven models for plasmas.
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