标题： 微米力学的通用傅里叶神经算子 显示英文标题

标题： Universal Fourier Neural Operators for Micromechanics

摘要： 求解均质化中的单元问题很困难，现有的深度学习框架无法达到传统计算框架的速度和通用性。 更具体地说，通常不清楚机器学习方法能期望什么，更不用说挑出哪些方法有前景了。 在本研究中，我们倡导使用傅里叶神经算子（FNOs）来解决微力学问题，并通过基于快速傅里叶变换（FFT）的计算微力学方法的见解来增强它们。 我们构建了一个FNO代理模型，模仿基于FFT方法的基本方案，并表明该结果算子仅在材料对比度约束下，就能以所需的精度预测任意刚度分布的单元问题的解。 特别是，对于材料对称性（如各向同性）、相数以及材料之间界面的几何形状没有任何限制。 此外，提供的保真度是精确且均匀的，提供了明确的保证，利用了我们对FNOs的物理增强。 为了展示所需的普遍逼近性质，我们显式构建了一个无需训练即可开始的FNO。 尽管如此，所获得的神经算子仍然符合基本方案的内存要求，并且运行时间与经典FFT求解器成比例。 特别是，可以轻松处理超过1亿个体素的大规模问题。 本工作的目标是强调FNOs在解决微力学问题方面的潜力，并将基于FFT的方法与FNOs联系起来。 这种联系预计将在两者之间提供一个富有成效的交流。 显示英文摘要

摘要： Solving cell problems in homogenization is hard, and available deep-learning frameworks fail to match the speed and generality of traditional computational frameworks. More to the point, it is generally unclear what to expect of machine-learning approaches, let alone single out which approaches are promising. In the work at hand, we advocate Fourier Neural Operators (FNOs) for micromechanics, empowering them by insights from computational micromechanics methods based on the fast Fourier transform (FFT). We construct an FNO surrogate mimicking the basic scheme foundational for FFT-based methods and show that the resulting operator predicts solutions to cell problems with arbitrary stiffness distribution only subject to a material-contrast constraint up to a desired accuracy. In particular, there are no restrictions on the material symmetry like isotropy, on the number of phases and on the geometry of the interfaces between materials. Also, the provided fidelity is sharp and uniform, providing explicit guarantees leveraging our physical empowerment of FNOs. To show the desired universal approximation property, we construct an FNO explicitly that requires no training to begin with. Still, the obtained neural operator complies with the same memory requirements as the basic scheme and comes with runtimes proportional to classical FFT solvers. In particular, large-scale problems with more than 100 million voxels are readily handled. The goal of this work is to underline the potential of FNOs for solving micromechanical problems, linking FFT-based methods to FNOs. This connection is expected to provide a fruitful exchange between both worlds.