Information geometry of physics-informed statistical manifolds and its use in data assimilation
Information geometry of physics-informed statistical manifolds and its use in data assimilation
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DOI：
10.1016/j.jcp.2022.111438
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发表时间：
2021-03
期刊：
影响因子：
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通讯作者：
F. Boso;D. Tartakovsky
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文献类型：
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作者：
F. Boso;D. Tartakovsky
数据感知分布方法（DAMD）是一种低维数据同化方法，用于预测由微分方程描述的动力系统的行为。DAMD的核心是在分布项中最小化观测和预测之间的距离，先验分布和后验分布被约束到由分布方法（MD）定义的统计流形。我们利用统计流形的信息几何特性，通过数据同化来减少预测的不确定性。具体来说，我们利用的信息几何结构引起的两个差异度量，Kullback-Leibler分歧和Wasserstein距离，明确地产生自然梯度下降。使用深度神经网络作为MD的代理模型可以实现自动微分，进一步加速优化。流形的几何形状在没有采样的情况下被量化，从而产生梯度下降方向的精确近似。我们的数值实验表明，占流形的几何形状显着降低了数据同化的计算成本，既方便梯度的计算，减少所需的迭代次数。
The data-aware method of distributions (DAMD) is a low-dimensional data assimilation procedure to forecast the behavior of dynamical systems described by differential equations. The core of DAMD is the minimization of a distance between an observation and a prediction in distributional terms, with prior and posterior distributions constrained to a statistical manifold defined by the method of distributions (MD). We leverage the information-geometric properties of the statistical manifold to reduce predictive uncertainty via data assimilation. Specifically, we exploit the information-geometric structures induced by two discrepancy metrics, the Kullback-Leibler divergence and the Wasserstein distance, which explicitly yield natural gradient descent. The use of a deep neural network as a surrogate model for MD enables automatic differentiation, further accelerating optimization. The manifold's geometry is quantified without sampling, yielding an accurate approximation of the gradient descent direction. Our numerical experiments demonstrate that accounting for the manifold's geometry significantly reduces the computational cost of data assimilation by both facilitating the calculation of gradients and reducing the number of required iterations.