标题： 非线性反问题的神经网络核变分推理：与贝叶斯神经网络的比较及在拓扑优化中的应用 显示英文标题

标题： Variational Inference for Nonlinear Inverse Problems via Neural Net Kernels: Comparison to Bayesian Neural Networks, Application to Topology Optimization

摘要： 反问题以及从数据推断未知或潜在参数在工程仿真中无处不在。在识别未知参数的主要观点中，贝叶斯推理占主导地位，其中既包括关于参数的先验信息，也包括通过似然评估从观测中获得的信息，并将其纳入推理过程中。本文采用了一种类似的观点，但在数值程序上与标准推理方法略有不同，以提供关于未知潜在参数局部行为的见解。我们提出了一种变分推理方法，该方法主要以逐点方式结合观测数据，即利用正向映射相对于参数的梯度信息，反转有限数量的观测数据，并在正向映射无噪声且一对一的情况下找到潜在参数的真实个体样本。对于统计计算（作为仿真的最终目标），从经过训练的神经网络生成大量样本，该网络充当从先验到后验潜在参数的传输映射。我们开发的神经网络工具，作为推理框架的一部分并称为神经网络核（NNK），基于层次（深度）核，与标准神经网络相比，提供了更大的训练灵活性。我们展示了我们的推理过程在识别双峰和不规则分布方面，与包括马尔可夫链蒙特卡罗采样方法和贝叶斯神经网络方法在内的多种方法相比的有效性。 显示英文摘要

摘要： Inverse problems and, in particular, inferring unknown or latent parameters from data are ubiquitous in engineering simulations. A predominant viewpoint in identifying unknown parameters is Bayesian inference where both prior information about the parameters and the information from the observations via likelihood evaluations are incorporated into the inference process. In this paper, we adopt a similar viewpoint with a slightly different numerical procedure from standard inference approaches to provide insight about the localized behavior of unknown underlying parameters. We present a variational inference approach which mainly incorporates the observation data in a point-wise manner, i.e. we invert a limited number of observation data leveraging the gradient information of the forward map with respect to parameters, and find true individual samples of the latent parameters when the forward map is noise-free and one-to-one. For statistical calculations (as the ultimate goal in simulations), a large number of samples are generated from a trained neural network which serves as a transport map from the prior to posterior latent parameters. Our neural network machinery, developed as part of the inference framework and referred to as Neural Net Kernels (NNK), is based on hierarchical (deep) kernels which provide greater flexibility for training compared to standard neural networks. We showcase the effectiveness of our inference procedure in identifying bimodal and irregular distributions compared to a number of approaches including Markov Chain Monte Carlo sampling approaches and a Bayesian neural network approach.