物理信息统计流形的信息几何及其在资料同化中的应用
Information-geometry of physics-informed statistical manifolds and its use in data assimilation
摘要：
数据感知分布方法（DA-MD）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。它将序贯贝叶斯更新与MD相结合，使得前者利用可用的观测值，而后者传播不确定性系统状态的（联合）概率分布。DA-MD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约束在由MD定义的统计流形上。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。具体而言，我们利用两个差异度量，Kullback-Leibler发散和Wasserstein距离，这明确地产生自然梯度下降引起的信息几何结构。为了进一步加速优化，我们构建了一个深度神经网络作为MD的代理模型，以实现自动微分。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，占流形的信息几何显着降低了数据同化的计算成本，通过方便梯度的计算和减少所需的迭代次数。存储需求和计算成本都取决于统计流形的维数，这通常是小的MD建设。当达到收敛时，Kullback-Leibler和$L_2$ Wasserstein度量具有相似的性能，前者对先验的不良选择更敏感。
英文摘要：
The data-aware method of distributions (DA-MD) is a low-dimension data assimilation procedure to forecast the behavior of dynamical systems described by differential equations. It combines sequential Bayesian update with the MD, such that the former utilizes available observations while the latter propagates the (joint) probability distribution of the uncertain system state(s). The core of DA-MD is the minimization of a distance between an observation and a prediction in distributional terms, with prior and posterior distributions constrained on a statistical manifold defined by the 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. To further accelerate optimization, we build a deep neural network as a surrogate model for the MD that enables automatic differentiation. 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 information-geometry of the manifold significantly reduces the computational cost of data assimilation by facilitating the calculation of gradients and by reducing the number of required iterations. Both storage needs and computational cost depend on the dimensionality of a statistical manifold, which is typically small by MD construction. When convergence is achieved, the Kullback-Leibler and $L_2$ Wasserstein metrics have similar performances, with the former being more sensitive to poor choices of the prior.
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