In this study, we present the mechanism-informed reinforcement learning
framework for airfoil shape optimization. By leveraging the twin delayed deep
deterministic policy gradient algorithm for its notable stability, our approach
addresses the complexities of optimizing shapes governed by fluid dynamics. The
PDEs-based solver is adopted for its accuracy even when the configurations and
geometries are extraordinarily changed during the exploration. Dual-weighted
residual-based mesh refinement strategy is applied to ensure the accurate
calculation of target functionals. To streamline the iterative optimization
process and handle geometric deformations, our approach integrates Laplacian
smoothing, adaptive refinement, and a B\'ezier fitting strategy. This
combination not only remits mesh tangling but also guarantees a precise
manipulation of the airfoil geometry. Our neural network architecture leverages
B\'ezier curves for efficient dimensionality reduction, thereby enhancing the
learning process and ensuring the geometric accuracy of the airfoil shapes. An
attention mechanism is embedded within the network to calculate potential
action on the state as well. Furthermore, we have introduced different reward
and penalty mechanisms tailored to the specific challenges of airfoil
optimization. This algorithm is designed to support the optimization task,
facilitating a more targeted and effective approach for airfoil shape
optimization.

研究中，我们介绍了一种基于机制的增强学习框架，用于空气动力学翼型形状优化，借助于稳定性较强的双延迟深度确定性策略梯度算法，解决了涉及流体动力学的形状优化的复杂问题。该算法支持翼型形状优化任务，实现了更有针对性和有效的方法。

面向机理的强化学习框架在机翼气动形状优化中的应用

A mechanism-informed reinforcement learning framework for shape  optimization of airfoils

We address the problem of semantic correspondence, that is, establishing a
dense flow field between images depicting different instances of the same
object or scene category. We propose to use images annotated with binary
foreground masks and subjected to synthetic geometric deformations to train a
convolutional neural network (CNN) for this task. Using these masks as part of
the supervisory signal offers a good compromise between semantic flow methods,
where the amount of training data is limited by the cost of manually selecting
point correspondences, and semantic alignment ones, where the regression of a
single global geometric transformation between images may be sensitive to
image-specific details such as background clutter. We propose a new CNN
architecture, dubbed SFNet, which implements this idea. It leverages a new and
differentiable version of the argmax function for end-to-end training, with a
loss that combines mask and flow consistency with smoothness terms.
Experimental results demonstrate the effectiveness of our approach, which
significantly outperforms the state of the art on standard benchmarks.

通过使用二进制前景蒙版和几何变形训练卷积神经网络（CNN）实现语义对应的密集流场，提出了一种新的 CNN 框架 SFNet，并在各项基准测试中表现出比现有方案更好的泛化性能。

SFNet: 学习物体感知的语义对应性

SFNet: Learning Object-aware Semantic Correspondence

Graph matching is an important and persistent problem in computer vision and
pattern recognition for finding node-to-node correspondence between
graph-structured data. However, as widely used, graph matching that
incorporates pairwise constraints can be formulated as a quadratic assignment
problem (QAP), which is NP-complete and results in intrinsic computational
difficulties. In this paper, we present a functional representation for graph
matching (FRGM) that aims to provide more geometric insights on the problem and
reduce the space and time complexities of corresponding algorithms. To achieve
these goals, we represent a graph endowed with edge attributes by a linear
function space equipped with a functional such as inner product or metric, that
has an explicit geometric meaning. Consequently, the correspondence between
graphs can be represented as a linear representation map of that functional.
Specifically, we reformulate the linear functional representation map as a new
parameterization for Euclidean graph matching, which is associative with
geometric parameters for graphs under rigid or nonrigid deformations. This
allows us to estimate the correspondence and geometric deformations
simultaneously. The use of the representation of edge attributes rather than
the affinity matrix enables us to reduce the space complexity by two orders of
magnitudes. Furthermore, we propose an efficient optimization strategy with low
time complexity to optimize the objective function. The experimental results on
both synthetic and real-world datasets demonstrate that the proposed FRGM can
achieve state-of-the-art performance.

本研究提出一种新的基于功能表征的图匹配方法，能够降低相关算法的时间和空间复杂度，并且具有比现有方法更准确的匹配性能。