Understanding systems by forward and inverse modeling is a recurrent topic of research in many domains of science and engineering. In this context, Monte Carlo methods have been widely used as powerful tools for numerical inference and optimization. They require the choice of a suitable proposal density that is crucial for their performance. For this reason, several adaptive importance sampling (AIS) schemes have been proposed in the literature. We here present an AIS framework called Regression-based Adaptive Deep Importance Sampling (RADIS). In RADIS, the key idea is the adaptive construction via regression of a non-parametric proposal density (i.e., an emulator), which mimics the posterior distribution and hence minimizes the mismatch between proposal and target densities. RADIS is based on a deep architecture of two (or more) nested IS schemes, in order to draw samples from the constructed emulator. The algorithm is highly efficient since employs the posterior approximation as proposal density, which can be improved adding more support points. As a consequence, RADIS asymptotically converges to an exact sampler under mild conditions. Additionally, the emulator produced by RADIS can be in turn used as a cheap surrogate model for further studies. We introduce two specific RADIS implementations that use Gaussian Processes (GPs) and Nearest Neighbors (NN) for constructing the emulator. Several numerical experiments and comparisons show the benefits of the proposed schemes. A real-world application in remote sensing model inversion and emulation confirms the validity of the approach.

通过正向和逆向建模了解系统是科学和工程学许多领域中反复出现的研究主题。在这种情况下，蒙特卡洛方法已被广泛用作数值推断和优化的强大工具。他们需要选择对他们的绩效至关重要的合适建议密度。由于这个原因，在文献中已经提出了几种自适应重要性抽样（AIS）方案。我们在这里提出一个称为基于回归的自适应深度重要性抽样（RADIS）的AIS框架。在RADIS中，关键思想是通过非参数投标密度（即仿真器）的回归进行自适应构造），它模仿了后验分布，因此最大程度地减少了建议密度与目标密度之间的不匹配。RADIS基于两个（或多个）嵌套IS方案的深度体系结构，以便从构造的仿真器中提取样本。由于采用后验近似作为提议密度，因此该算法非常高效，可以通过添加更多支持点来进行改进。结果，RADIS在温和条件下渐近收敛到精确的采样器。另外，由RADIS生产的仿真器又可以用作便宜的替代模型，以供进一步研究。我们介绍了两个特定的RADIS实现，这些实现使用高斯进程（GPs）和最近邻居（NN）来构建仿真器。若干数值实验和比较表明了所提方案的好处。