AI 中文总结
研究线性偏微分方程随机特征配置问题，构建算子感知离散化，结合残差-克里斯托费尔密度等方法，通过实验表明该方法能产生条件数和迭代次数更小的满秩变换系统，为稳定随机特征配置提供有原则设计。
AI 中文摘要
随机特征配置固定随机生成的试验空间，并通过线性最小二乘系统确定其系数。稳定性取决于采样的残差方程是否代表微分算子诱导的几何结构。我们构建了一种算子感知离散化，其中算子应用的特征决定了配置度量和系数白化映射。随机方案将残差-克里斯托费尔密度与逆密度权重相结合，而确定性标量行替代方案则最大化连续正则化对数行列式增量。在已实现的试验空间条件下，采样的白化内部格拉姆矩阵是保留残差空间上参考格拉姆矩阵的谱近似，样本复杂度在保留维度上线性增长，最多有一个对数因子。对于均匀解析的残差核，相关算子具有拉伸指数衰减的特征值和在逆岭尺度上为多对数的岭有效维度。对标量和向量方程、不同几何结构以及一到三个空间维度的实验表明，残差空间采样和白化产生数值上满秩的变换系统，条件数和迭代次数显著更小。确定性构造在最小标量样本大小下达到最低误差。因此，残差空间几何结构为稳定的强形式随机特征配置提供了一种有原则的设计。
英文摘要
Random feature collocation fixes a randomly generated trial space and determines its coefficients from a linear least-squares system. Stability then depends on whether the sampled residual equations represent the geometry induced by the differential operator. We construct an operator-aware discretization in which the operator-applied features determine both the collocation measure and a coefficient whitening map. The randomized scheme combines a residual-Christoffel density with inverse-density weights, while a deterministic scalar-row alternative maximizes successive regularized log-determinant increments. Conditional on the realized trial space, the sampled whitened interior Gram is a spectral approximation to the reference Gram on the retained residual space, with sample complexity linear in the retained dimension up to a logarithmic factor. For uniformly analytic residual kernels, the associated operator has stretched-exponentially decaying eigenvalues and ridge effective dimension that is polylogarithmic in the inverse ridge scale. Experiments on scalar and vector equations, varied geometries, and one to three spatial dimensions show that residual-space sampling and whitening produce numerically full-rank transformed systems with substantially smaller condition numbers and iteration counts. The deterministic construction attains the lowest errors at the smallest scalar sample sizes. Residual-space geometry therefore yields a principled design for stable strong-form random feature collocation.
