AI 中文总结
针对具有不等内部和边界观测值的椭圆型偏微分方程，提出算子分裂贝叶斯学习框架，为域内源和边界值分配独立BNN先验，经椭圆解算子得到收缩后验，结合极小极大下界得出近似上界，并通过数值实验展示相关影响。
AI 中文摘要
我们为具有不等数量内部和边界观测值的二阶一致椭圆狄利克雷问题提出了一种算子分裂贝叶斯学习框架。数据由域内源的噪声测量值和边界值的噪声测量值组成。为这两个量分配独立的贝叶斯神经网络（BNN）先验，并通过椭圆解算子推进得到的乘积后验。我们证明这种构造诱导的后验在真解周围收缩。收缩半径将由二阶椭圆算子控制的域贡献与由边界固有维度控制的边界贡献分开。结合文献[ZhaoLu2026]的极小极大下界，这产生了一个直到对数因子的近似极小极大上界。我们的数值实验说明了源和边界不确定性的传播以及不等采样预算对后验重建的影响。
英文摘要
We propose an operator-split Bayesian learning framework for second-order uniformly elliptic Dirichlet problems with unequal numbers of interior and boundary observations. The data consist of noisy measurements of the source in the domain and noisy measurements of the boundary values. Independent Bayesian neural-network (BNN) priors are assigned to these two quantities, and the resulting product posterior is pushed forward through the elliptic solution operator. We prove that the posterior induced by this construction contracts around the true solution. The contraction radius separates a domain contribution, governed by the second-order elliptic operator, from a boundary contribution, governed by the intrinsic dimension of the boundary. Together with the minimax lower bound of \cite{ZhaoLu2026}, this yields a near-minimax upper bound up to logarithmic factors. Our numerical experiments illustrate the propagation of source and boundary uncertainty and the effects of unequal sampling budgets on the posterior reconstruction.
Comments28 pages, 7 figures, 3 tables
