The committor function is a central object for quantifying the transitions
between metastable states of dynamical systems. Recently, a number of
computational methods based on deep neural networks have been developed for
computing the high-dimensional committor function. The success of the methods
relies on sampling adequate data for the transition, which still is a
challenging task for complex systems at low temperatures. In this work, we
propose a deep learning method with two novel adaptive sampling schemes (I and
II). In the two schemes, the data are generated actively with a modified
potential where the bias potential is constructed from the learned committor
function. We theoretically demonstrate the advantages of the sampling schemes
and show that the data in sampling scheme II are uniformly distributed along
the transition tube. This makes a promising method for studying the transition
of complex systems. The efficiency of the method is illustrated in
high-dimensional systems including the alanine dipeptide and a solvated dimer
system.

在这项研究中，我们提出了一种具有两种新型自适应采样方案（I 和 II）的深度学习方法，通过从学习的转变因子函数构建偏执势，实时生成与转变相关的数据，理论上证明了采样方案的优势，并展示了采样方案 II 中的数据均匀分布在转变通道中，这为研究复杂系统的转变提供了一种有前途的方法。该方法的效率已在包括丙氨酸二肽和溶剂化二聚体系统等高维系统中得到验证。

用自适应采样的深度学习方法计算转换概率函数

Deep Learning Method for Computing Committor Functions with Adaptive  Sampling

We study low rank matrix and tensor completion and propose novel algorithms
that employ adaptive sampling schemes to obtain strong performance guarantees.
Our algorithms exploit adaptivity to identify entries that are highly
informative for learning the column space of the matrix (tensor) and
consequently, our results hold even when the row space is highly coherent, in
contrast with previous analyses. In the absence of noise, we show that one can
exactly recover a $n \times n$ matrix of rank $r$ from merely $\Omega(n
r^{3/2}\log(r))$ matrix entries. We also show that one can recover an order $T$
tensor using $\Omega(n r^{T-1/2}T^2 \log(r))$ entries. For noisy recovery, our
algorithm consistently estimates a low rank matrix corrupted with noise using
$\Omega(n r^{3/2} \textrm{polylog}(n))$ entries. We complement our study with
simulations that verify our theory and demonstrate the scalability of our
algorithms.

研究低秩矩阵和张量完成，提出了采用自适应抽样方案的新算法，它们具有强大的性能保证，并能够恢复具有噪声干扰的低秩矩阵。