2D irregular packing is a classic combinatorial optimization problem with
various applications, such as material utilization and texture atlas
generation. This NP-hard problem requires efficient algorithms to optimize
space utilization. Conventional numerical methods suffer from slow convergence
and high computational cost. Existing learning-based methods, such as the
score-based diffusion model, also have limitations, such as no rotation
support, frequent collisions, and poor adaptability to arbitrary boundaries,
and slow inferring. The difficulty of learning from teacher packing is to
capture the complex geometric relationships among packing examples, which
include the spatial (position, orientation) relationships of objects, their
geometric features, and container boundary conditions. Representing these
relationships in latent space is challenging. We propose GFPack++, an
attention-based gradient field learning approach that addresses this challenge.
It consists of two pivotal strategies: \emph{attention-based geometry encoding}
for effective feature encoding and \emph{attention-based relation encoding} for
learning complex relationships. We investigate the utilization distribution
between the teacher and inference data and design a weighting function to
prioritize tighter teacher data during training, enhancing learning
effectiveness. Our diffusion model supports continuous rotation and outperforms
existing methods on various datasets. We achieve higher space utilization over
several widely used baselines, one-order faster than the previous
diffusion-based method, and promising generalization for arbitrary boundaries.
We plan to release our source code and datasets to support further research in
this direction.

GFPack++ 是一个基于注意力机制梯度场学习方法，可以有效地进行 2D 不规则装箱优化，获得更高的空间利用率，支持连续旋转，并拥有较快的运算速度和对任意边界的良好泛化能力。

GFPack++：学习梯度场与注意力提高二维不规则装箱

GFPack++: Improving 2D Irregular Packing by Learning Gradient Field with  Attention

Directly parameterizing and learning gradients of functions has widespread
significance, with specific applications in optimization, generative modeling,
and optimal transport. This paper introduces gradient networks (GradNets):
novel neural network architectures that parameterize gradients of various
function classes. GradNets exhibit specialized architectural constraints that
ensure correspondence to gradient functions. We provide a comprehensive GradNet
design framework that includes methods for transforming GradNets into monotone
gradient networks (mGradNets), which are guaranteed to represent gradients of
convex functions. We establish the approximation capabilities of the proposed
GradNet and mGradNet. Our results demonstrate that these networks universally
approximate the gradients of (convex) functions. Furthermore, these networks
can be customized to correspond to specific spaces of (monotone) gradient
functions, including gradients of transformed sums of (convex) ridge functions.
Our analysis leads to two distinct GradNet architectures, GradNet-C and
GradNet-M, and we describe the corresponding monotone versions, mGradNet-C and
mGradNet-M. Our empirical results show that these architectures offer efficient
parameterizations and outperform popular methods in gradient field learning
tasks.

本文介绍了梯度网络（GradNets）：一种参数化各类函数梯度的新型神经网络架构，通过增加特定约束确保与梯度函数的对应性。我们提供了一个全面的 GradNet 设计框架，包括将 GradNets 转化为保证表示凸函数梯度的单调梯度网络（mGradNets）的方法。我们证明了所提出的 GradNet 和 mGradNet 的近似能力，并且实验结果表明这些架构在梯度场学习任务中提供了高效的参数化，并且优于流行的方法。