This article addresses the inference of physics models from data, from the perspectives of inverse problems and model reduction. These fields develop formulations that integrate data into physics-based models while exploiting the fact that many mathematical models of natural and engineered systems exhibit an intrinsically low-dimensional solution manifold. In inverse problems, we seek to infer uncertain components of the inputs from observations of the outputs, while in model reduction we seek low-dimensional models that explicitly capture the salient features of the input–output map through approximation in a low-dimensional subspace. In both cases, the result is a predictive model that reflects data-driven learning yet deeply embeds the underlying physics, and thus can be used for design, control and decision-making, often with quantified uncertainties. We highlight recent developments in scalable and efficient algorithms for inverse problems and model reduction governed by large-scale models in the form of partial differential equations. Several illustrative applications to large-scale complex problems across different domains of science and engineering are provided.

中文翻译：

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从数据中学习基于物理的模型：逆问题和模型简化的观点

本文从逆问题和模型简化的角度讨论从数据中推断物理模型。这些领域开发了将数据集成到基于物理的模型中的公式，同时利用了许多自然和工程系统的数学模型表现出本质上低维解流形的事实。在逆问题中，我们寻求从对输出的观察中推断输入的不确定成分，而在模型简化中，我们寻求低维模型，通过在低维子空间中的逼近来明确捕获输入-输出图的显着特征。在这两种情况下，结果都是一个预测模型，它反映了数据驱动的学习，但深度嵌入了底层物理，因此可用于设计、控制和决策，通常带有量化的不确定性。我们重点介绍了可扩展且高效的逆问题算法和由偏微分方程形式的大规模模型控制的模型简化的最新进展。提供了跨不同科学和工程领域的大规模复杂问题的几个说明性应用。