Complex systems often show macroscopic coherent behavior due to the
interactions of microscopic agents like molecules, cells, or individuals in a
population with their environment. However, simulating such systems poses
several computational challenges during simulation as the underlying dynamics
vary and span wide spatiotemporal scales of interest. To capture the
fast-evolving features, finer time steps are required while ensuring that the
simulation time is long enough to capture the slow-scale behavior, making the
analyses computationally unmanageable. This paper showcases how deep learning
techniques can be used to develop a precise time-stepping approach for
multiscale systems using the joint discovery of coordinates and flow maps.
While the former allows us to represent the multiscale dynamics on a
representative basis, the latter enables the iterative time-stepping estimation
of the reduced variables. The resulting framework achieves state-of-the-art
predictive accuracy while incurring lesser computational costs. We demonstrate
this ability of the proposed scheme on the large-scale Fitzhugh Nagumo neuron
model and the 1D Kuramoto-Sivashinsky equation in the chaotic regime.

利用深度学习技术，本文展示了如何开发一种精确的多尺度系统时间步进方法，通过坐标和流图的联合发现来表示多尺度动态，同时采用迭代时间步进估计减少的变量，实现了最先进的预测准确性，同时减少了计算成本。

利用深度学习坐标和流动图增强多尺度系统中的计算效率

Enhancing Computational Efficiency in Multiscale Systems Using Deep  Learning of Coordinates and Flow Maps

Predictive simulations are essential for applications ranging from weather
forecasting to material design. The veracity of these simulations hinges on
their capacity to capture the effective system dynamics. Massively parallel
simulations predict the systems dynamics by resolving all spatiotemporal
scales, often at a cost that prevents experimentation. On the other hand,
reduced order models are fast but often limited by the linearization of the
system dynamics and the adopted heuristic closures. We propose a novel
systematic framework that bridges large scale simulations and reduced order
models to extract and forecast adaptively the effective dynamics (AdaLED) of
multiscale systems. AdaLED employs an autoencoder to identify reduced-order
representations of the system dynamics and an ensemble of probabilistic
recurrent neural networks (RNNs) as the latent time-stepper. The framework
alternates between the computational solver and the surrogate, accelerating
learned dynamics while leaving yet-to-be-learned dynamics regimes to the
original solver. AdaLED continuously adapts the surrogate to the new dynamics
through online training. The transitions between the surrogate and the
computational solver are determined by monitoring the prediction accuracy and
uncertainty of the surrogate. The effectiveness of AdaLED is demonstrated on
three different systems - a Van der Pol oscillator, a 2D reaction-diffusion
equation, and a 2D Navier-Stokes flow past a cylinder for varying Reynolds
numbers (400 up to 1200), showcasing its ability to learn effective dynamics
online, detect unseen dynamics regimes, and provide net speed-ups. To the best
of our knowledge, AdaLED is the first framework that couples a surrogate model
with a computational solver to achieve online adaptive learning of effective
dynamics. It constitutes a potent tool for applications requiring many
expensive simulations.

提出一种名为 AdaLED 的新型系统框架，以自动编码器为主要工具，利用概率递归神经网络进行时间步进，通过在线训练不断改进，建立预测模型从而实现大规模模拟的高速运行和在线适应性学习，应用在 Van der Pol 振荡器，2D 反应扩散方程和 2D Navier-Stokes 圆柱流中，具有较高的准确性和鲁棒性。

自适应学习有效动态：适应复杂系统的实时在线建模

Adaptive learning of effective dynamics: Adaptive real-time, online  modeling for complex systems

We present a novel approach to the detection and characterization of edges,
ridges, and blobs in two-dimensional images which exploits the symmetry
properties of directionally sensitive analyzing functions in multiscale systems
that are constructed in the framework of alpha-molecules. The proposed feature
detectors are inspired by the notion of phase congruency, stable in the
presence of noise, and by definition invariant to changes in contrast. We also
show how the behavior of coefficients corresponding to differently scaled and
oriented analyzing functions can be used to obtain a comprehensive
characterization of the geometry of features in terms of local tangent
directions, widths, and heights. The accuracy and robustness of the proposed
measures are validated and compared to various state-of-the-art algorithms in
extensive numerical experiments in which we consider sets of clean and
distorted synthetic images that are associated with reliable ground truths. To
further demonstrate the applicability, we show how the proposed ridge measure
can be used to detect and characterize blood vessels in digital retinal images
and how the proposed blob measure can be applied to automatically count the
number of cell colonies in a Petri dish.

该论文提出了一种基于 α- 分子构造的多尺度系统中方向敏感的分析函数的对称性特征检测方法，可以稳定地检测到边缘、脊和斑点等物体特征，并且在噪声存在的情况下依然能保持精度和稳健性。作者在论文中还阐明了分析函数的缩放和方向对应的系数如何用于精确刻画局部切线方向、宽度和高度等特征。在对干净和失真合成图像的大量数值实验中验证了该方法的准确性和鲁棒性，并将其与其他先进算法进行了比较。通过在数字视网膜图像中检测和表征血管特征以及在细胞培养基中自动计数细胞菌落的实验进一步证明了该方法的应用价值。