The nonlinear Poisson-Boltzmann equation (NPBE) is an elliptic partial differential equation used in applications such as protein interactions and biophysical chemistry (among many others). It describes the nonlinear electrostatic potential of charged bodies submerged in an ionic solution. The kinetic presence of the solvent molecules introduces randomness to the shape of a protein, and thus a more accurate model that incorporates these random perturbations of the domain is analyzed to compute the statistics of quantities of interest of the solution. When the parameterization of the random perturbations is high-dimensional, this calculation is intractable as it is subject to the curse of dimensionality. However, if the solution of the NPBE varies analytically with respect to the random parameters, the problem becomes amenable to techniques such as sparse grids and deep neural networks. In this paper, we show analyticity of the solution of the NPBE with respect to analytic perturbations of the domain by using the analytic implicit function theorem and the domain mapping method. Previous works have shown analyticity of solutions to linear elliptic equations but not for nonlinear problems. We further show how to derive \emph{a priori} bounds on the size of the region of analyticity. This method is applied to the trypsin molecule to demonstrate that the convergence rates of the quantity of interest are consistent with the analyticity result. Furthermore, the approach developed here is sufficiently general enough to be applied to other nonlinear problems in uncertainty quantification.

翻译：非线性泊松-玻尔兹曼方程（NPBE）是一种椭圆型偏微分方程，广泛应用于蛋白质相互作用与生物物理化学等领域。该方程描述了浸入离子溶液中带电体产生的非线性静电势。溶剂分子的动力学存在导致蛋白质形状呈现随机性，因此本文分析了纳入这些随机区域扰动的更精确模型，以计算解的感兴趣量的统计特性。当随机扰动的参数化维度较高时，该计算因维度灾难而难以处理。然而，若NPBE的解关于随机参数呈解析变化，则该问题可通过稀疏网格和深度神经网络等技术加以解决。本文利用解析隐函数定理和区域映射方法，证明了NPBE的解关于区域解析扰动的解析性。已有研究证明了线性椭圆方程解的解析性，但未涉及非线性问题。我们进一步推导了解析区域大小的先验界。该方法应用于胰蛋白酶分子，验证了感兴趣量的收敛速率与解析性结果的一致性。此外，本文所建立的方法具有足够的一般性，可应用于不确定性量化中的其他非线性问题。