GenMod：一种用于随机输入偏微分方程谱表示的生成式建模方法
GenMod: A generative modeling approach for spectral representation of PDEs with random inputs
摘要：
我们提出了一种方法来量化高维PDE系统的随机参数，其中的解的评估数量是小的不确定性。参数偏微分方程的解通常使用基于多项式混沌展开的谱分解来近似。对于我们考虑的系统类别（即，具有有限解评估的高维数），系数由回归公式中的欠定线性系统给出。这意味着需要额外的假设，例如系数向量的稀疏性，来近似解。在这里，我们提出了一种方法，其中我们假设系数接近生成模型的范围，该生成模型从系数的低维空间映射到高维空间。我们的方法的灵感来自最近的工作，研究如何生成模型可以用于压缩感知系统中的随机高斯测量矩阵。利用偏微分方程理论的系数衰减率的结果，我们构建了一个显式的生成模型，预测多项式混沌系数的大小。我们开发的算法来找到系数，我们称之为GenMod，由两个主要步骤组成。首先，我们使用正交匹配追踪预测系数符号。然后，我们假设系数在符号调整生成模型的范围的稀疏偏差内。这允许我们通过求解生成模型的输入空间和稀疏向量空间上的非凸优化问题来找到系数。我们得到理论上的恢复结果的Lipschitz连续生成模型和一个更具体的生成模型，基于系数衰减率界。我们研究了三个高维问题，并表明，对于所有三个例子，生成模型的方法优于稀疏促进方法在小样本量。
英文摘要：
We propose a method for quantifying uncertainty in high-dimensional PDE systems with random parameters, where the number of solution evaluations is small. Parametric PDE solutions are often approximated using a spectral decomposition based on polynomial chaos expansions. For the class of systems we consider (i.e., high dimensional with limited solution evaluations) the coefficients are given by an underdetermined linear system in a regression formulation. This implies additional assumptions, such as sparsity of the coefficient vector, are needed to approximate the solution. Here, we present an approach where we assume the coefficients are close to the range of a generative model that maps from a low to a high dimensional space of coefficients. Our approach is inspired be recent work examining how generative models can be used for compressed sensing in systems with random Gaussian measurement matrices. Using results from PDE theory on coefficient decay rates, we construct an explicit generative model that predicts the polynomial chaos coefficient magnitudes. The algorithm we developed to find the coefficients, which we call GenMod, is composed of two main steps. First, we predict the coefficient signs using Orthogonal Matching Pursuit. Then, we assume the coefficients are within a sparse deviation from the range of a sign-adjusted generative model. This allows us to find the coefficients by solving a nonconvex optimization problem, over the input space of the generative model and the space of sparse vectors. We obtain theoretical recovery results for a Lipschitz continuous generative model and for a more specific generative model, based on coefficient decay rate bounds. We examine three high-dimensional problems and show that, for all three examples, the generative model approach outperforms sparsity promoting methods at small sample sizes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
