AI 中文总结
该研究构建了直接作用于三维密度分布的立方等变神经密度泛函，通过全卷积网络学习相关泛函，在格子流体测试中改进了状态方程等性能，明确了连续流体扩展的核心挑战。
AI 中文摘要
我们构建了一种直接作用于无限制三维密度分布的神经经典密度泛函。作为计算上易于处理的测试平台，我们考虑简单立方格子上边长为3的平行硬立方体。全卷积网络从在随机外势下进行的巨正则蒙特卡洛模拟获得的数据中学习单体直接关联泛函 $c^{(1)}[\ ho]$。训练和推理过程均使用完整分布；输出位点上的随机伯努利掩码使完整分布训练变得有效，无需显式提取和存储重叠的局域密度窗口。对立方点群的全部48种旋转和反射操作平均第一层卷积核，可在不使用数据增强的情况下施加精确的立方等变性。我们将学习到的泛函与独立模拟数据以及Lafuente和Cuesta提出的格子基本度量泛函进行比较。神经泛函显著改进了均匀状态方程和平面硬壁处的密度分布；对于固定粒子周围的各向异性对分布，两种泛函均能重现主要的堆积壳层，其相对精度取决于晶向。这些结果证明了在完整三维分布上进行神经密度泛函计算的可行性，同时确定了准确的全维训练数据、热力学一致性和结构关联是扩展至连续流体的核心挑战。
英文摘要
We construct a neural classical density functional that acts directly on unrestricted three-dimensional density profiles. As a computationally tractable test bed, we consider parallel hard cubes of side length three on a simple cubic lattice. A fully convolutional network learns the one-body direct-correlation functional $c^{(1)}[ρ]$ from data obtained with grand-canonical Monte Carlo simulations in randomized external potentials. Complete profiles are used during both training and inference; a stochastic Bernoulli mask on the output sites makes full-profile training effective without explicitly extracting and storing overlapping local density windows. Averaging the first-layer kernels over all 48 rotations and reflections of the cubic point group additionally imposes exact cubic equivariance without data augmentation. We compare the learned functional with independent simulation data and with the lattice fundamental-measure functional of Lafuente and Cuesta. The neural functional markedly improves the homogeneous equation of state and the density profile at a planar hard wall. For the anisotropic pair distribution around a fixed particle, both functionals reproduce the principal packing shells, with their relative accuracy depending on crystallographic direction. These results demonstrate neural density-functional calculations on complete three-dimensional profiles while also identifying accurate full-dimensional training data, thermodynamic consistency, and structural correlations as the central challenges for extensions to continuum fluids.
