AI 中文总结
该研究从贝叶斯非参数视角研究非线性PDE求解的概率数值方法，建立了后验在索伯列夫空间及一致拓扑下的高斯近似，还证明高斯拉普拉斯近似在√N尺度下渐近等价于真实后验。
AI 中文摘要
我们从贝叶斯非参数视角研究求解非线性偏微分方程（PDE）的概率数值方法。给定随机配置点上的带噪评估值，我们对未知解施加截断高斯级数先验，并建立了极小极大非参数速率下的收缩性，该速率差一个对数因子。我们的主要结果给出了正阶索伯列夫空间中后验的高斯近似，且在合适条件下还能给出一致拓扑下的后验高斯近似。这与经典不适定逆问题形成对比，经典不适定逆问题中伯恩斯坦-米塞斯定理通常要求弱得多的拓扑。此处观测算子是微分型而非平滑型，其线性化的逆运算可提升正则性，从而使这些强拓扑结果成为可能。后验可以后验均值或后验模式为中心。我们进一步证明，高斯拉普拉斯近似在√N尺度下渐近等价于真实后验。
英文摘要
We study probabilistic numerical methods for solving nonlinear PDEs from a Bayesian nonparametric perspective. Given noisy evaluations at random collocation points, we place a truncated Gaussian series prior on the unknown solution and establish contraction at the minimax nonparametric rate, up to a logarithmic factor. Our main results give Gaussian approximations of the posterior in positive-order Sobolev spaces and, under suitable conditions, in the uniform topology. This contrasts with classical ill-posed inverse problems, where Bernstein--von Mises theorems typically require substantially weaker topologies. Here, the observation operator is differential rather than smoothing, and inversion of its linearisation gains regularity, making these strong-topology results possible. The posterior may be centred at either the posterior mean or the posterior mode. We further prove that the Gaussian Laplace approximation is asymptotically equivalent to the true posterior at a $\sqrt{N}$-scale.
Comments51 pages, 2 figures
