AI 中文总结
本文提出两种神经网络模型直接从域几何学习微分算子谱，实现高效形状优化，在星形域精度达0.2%，景观模型前十特征值误差1%，优于FNO，并恢复经典谱最优形状。
AI 中文摘要
给定一个依赖于微分算子谱的泛函，我们研究寻找一个域以优化该泛函的问题。偏微分方程求解器可用于处理这一优化问题，但其计算成本高昂。我们提出了两种神经网络模型，它们直接从域的几何形状学习谱，并可用于根据一个或多个特征值优化域。我们研究了两种表示方法。第一种通过傅里叶系数和一个轻量级多层感知机（MLP）对域进行编码，该方法在星形几何上高效，达到了0.2%的精度。通过对系数进行重新缩放，所设计的模型满足特征值的标度律。此外，对训练好的代理模型的输出在旋转和反射上取平均，可诱导这些变换下的不变性。第二种模型输入景观函数、指示函数以及景观函数的梯度。通过Gram-Schmidt过程，模型输出正交的特征函数及其相关的特征值。该景观模型在前十个特征值上达到了1%的平均相对误差，而FNO模型的误差为4%。将景观函数替换为符号距离函数（SDF）会恶化预测和优化误差。训练好的模型还能从合成形状泛化到经典图像数据集给出的域。两种方法的代理模型均恢复了经典的谱最优形状，如第一特征值的圆盘或更高特征值的猜想最小值。这证实了我们的模型能够产生关于特征值的准确且可微的估计，可用于涉及谱量的形状优化问题。
英文摘要
Given a functional dependent on the spectrum of a differential operator, we address the problem of finding a domain which optimizes this functional. PDE solvers might be used to tackle this optimization. It is however computationally expensive. We propose two neural network models which learn the spectrum directly from the geometry of the domain and can be used to optimize the domain from one or more eigenvalues. We investigate two representations. The first encodes the domain through Fourier coefficients and a light MLP, which is efficient on star-shaped geometries, achieving a precision of 0.2\%. Through a rescaling of the coefficients the designed models satisfy the scaling law of the eigenvalues. Additionally, averaging the outputs of the trained surrogates over rotations and reflections induces invariance for these transformations. The second is a model that takes the landscape function, the indicator function and the gradient of the landscape function. A Gram-Schmidt process produces orthogonal eigenfunctions as output of the model along with the associated eigenvalues. The landscape model reaches 1\% mean relative error on the first ten eigenvalues, compared with 4\% for an FNO model. Replacing the landscape by an SDF worsened both prediction and optimization errors. The trained model also generalizes from synthetic shapes to domains given as classical image dataset. The resulting surrogates of both approaches recover classical spectral optima such as the disk for the first eigenvalue or the conjectured minima of higher eigenvalues. This confirms that our models produce accurate differentiable estimates of eigenvalues, which can be used in shape optimization problems involving spectral quantities.
