发表机构
King Abdullah University of Science and Technology; Politecnico di Milano（阿卜杜拉国王科技大学; 米兰理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
ADEx-FNO是适配不同几何结构的FNO统一框架，无需可训练几何模块，在CFD算例中可初始化求解器以减少迭代次数、提升效率，在多类问题上表现优异。
AI 中文摘要
傅里叶神经算子（FNO）可实现高效的非局部谱学习，但仍难以适配不同几何结构及独立选择的离散化方式。我们提出环境域扩展傅里叶神经算子（ADEx-FNO），这是一种确定性框架，无需修改定义傅里叶算子层即可融入几何信息。每个物理域被嵌入固定的环境超立方体中，并用符号距离函数表示；输入及解场被确定性扩展至环境域，转移至通用的、可能非均匀的矩形潜网格，经FNO处理后，插值至独立选择的目标离散化方式并限制在物理域内。所有几何转移操作均位于优化流程之外，无需可训练的图、点云、变形或几何解码模块。ADEx-FNO在保留的二维和三维光滑域非线性泊松方程、平流-反应-扩散问题上实现了0.32%-0.77%的相对L2误差，还在未见过的非光滑几何上进行了评估。随后，单次ADEx-FNO推理结果被用于初始化传统CFD求解器：在所有29个收敛的二维和三维RANS算例中，伪时间迭代次数减少，二维和三维的平均减少率分别为44.17%和43.03%，三种网格分辨率均获得相当的增益；URANS算例的后窗口物理时间推进减少了18.52%-27.51%。在从二维URANS训练数据迁移至不同马赫数和雷诺数下的DNS任务时，根据目标统计量的不同，自举区间减少了23.47%-48.21%。在所有CFD测试中，ADEx-FNO仅提供初始场，控制后续解的是控制方程求解器，物理或统计一致性评估与计算节省分开进行。
英文摘要
Fourier neural operators (FNOs) provide efficient nonlocal spectral learning, but varying geometries and independently chosen discretizations remain difficult to accommodate. We introduce the ambient-domain extension Fourier neural operator (ADEx-FNO), a deterministic framework that incorporates geometry without modifying the defining Fourier-operator layers. Each physical domain is embedded in a fixed ambient hypercube and represented by a signed distance function. Inputs and solution fields are deterministically extended to the ambient domain, transferred to a common, potentially nonuniform rectilinear latent grid, processed by the FNO, then interpolated to an independently chosen target discretization and restricted to the physical domain. All geometry-transfer operations lie outside the optimization procedure and require no trainable graph, point-cloud, deformation, or geometry-decoding modules. ADEx-FNO achieves relative l2 errors of 0.32%-0.77% on held-out smooth-domain nonlinear Poisson and advection-reaction-diffusion problems in 2D and 3D, and is also evaluated on unseen nonsmooth geometries. A single ADEx-FNO inference is then used to initialize conventional CFD solvers. For all 29 converged 2D and 3D RANS cases, pseudo-time iterations decrease, with mean reductions of 44.17% and 43.03%, respectively, with comparable gains across three mesh resolutions. URANS cases reduce post-window physical-time advances by 18.52%-27.51%. In transfer from 2D URANS training data to DNS at different Mach and Reynolds numbers, the bootstrap interval decreases by 23.47%-48.21%, depending on the target statistic. In all CFD tests, ADEx-FNO provides only the initial field; the governing-equation solver controls the subsequent solution, while physical or statistical consistency is assessed separately from computational savings.
