发表机构
Boston University（波士顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对PDE约束逆问题中PINN估计偏差问题，提出两步去偏估计程序，实现$\sqrt{n}$相合且渐近正态的估计，并扩展至贝叶斯推断，通过数值实验验证了方法的有效性。
AI 中文摘要
我们研究了在含噪观测下，使用物理信息神经网络（PINNs）逼近PDE解时，估计PDE约束逆问题中未知参数的问题。尽管PINNs在实证中取得了显著成功，但现有估计器往往继承了神经网络解的非参数慢收敛速率，导致对有限维参数的推断存在偏差且统计效率低下。为解决这一问题，我们提出了一种两步去偏估计程序，将基于神经网络的非参数估计与基于影响函数的偏差校正相结合。通过消除估计器对干扰函数误差的一阶敏感性，我们的程序在不要求神经网络分量欠平滑的情况下，得到了一个$\sqrt{n}$相合且渐近正态的估计器。我们进一步将该框架扩展到贝叶斯推断，用去偏拟似然替代原始似然，并建立了Bernstein-von Mises定理，表明所得后验以$\sqrt{n}$速率收缩，且其渐近协方差与频率学派估计器相匹配。作为分析的副产品，我们建立了使用神经网络在Sobolev空间中估计非参数回归函数及其导数的近极小极大最优收敛速率。大量数值实验证实了我们的理论发现，并证明了所提出的去偏程序对于PDE约束逆问题中有效统计推断的必要性。
英文摘要
We study the problem of estimating unknown parameters in PDE-constrained inverse problems from noisy observations, where the PDE solution is approximated using Physics-Informed Neural Networks (PINNs). While PINNs have demonstrated remarkable empirical success, existing estimators often inherit the slow nonparametric convergence rate of the neural-network solution, leading to biased and statistically inefficient inference for the finite-dimensional parameters of interest. To address this, we propose a two-step debiased estimation procedure that combines neural-network-based nonparametric estimation with an influence-function-based bias correction. By eliminating the first-order sensitivity of the estimator to errors in the nuisance function, our procedure yields a $\sqrt{n}$-consistent and asymptotically normal estimator without requiring undersmoothing of the neural network component. We further extend this framework to Bayesian inference by replacing the original likelihood with a debiased quasi-likelihood and establish a Bernstein-von Mises theorem showing that the resulting posterior contracts at the $\sqrt{n}$-rate with an asymptotic covariance matching that of the frequentist estimator. As a by-product of our analysis, we establish near-minimax optimal convergence rates for estimating a nonparametric regression function and its derivatives in Sobolev spaces using neural networks. Extensive numerical experiments corroborate our theoretical findings and demonstrate the necessity of the proposed debiasing procedure for valid statistical inference in PDE-constrained inverse problems.
Comments91 pages, 5 figures
