Surrogate-based active learning for parameter inference in geosciences via Bayesian sparse² multi-adaptivity enhanced by information theory

水-粮食-能源之间的关系是可持续发展的核心。社会需要更好地了解环境，以便进行有效和安全的互动，在资源管理中实现最大的福利和可持续性。采用校准良好的模型进行模拟，为预测地下系统的多方面行为提供了一种独特的方法，例如裂缝介质中的多相流输运、耦合水力系统模型和多孔介质中的多组分反应输运等。由于数值模拟的粗糙性或缺乏可用数据以及计算成本高，这类问题对于不确定性量化和机器学习（ML）的现代方法仍然是具有挑战性的。拟议的项目试图以地质构造中的二氧化碳储存建模为例应对这一挑战。CO2存储问题是地下问题的广泛类别的代表性情况，因为它涉及多相流，其中CO2驱替前沿是非常非线性的，这可能是难以捕获的。ML技术在科学界越来越受欢迎，似乎是解决此类非线性问题的合适候选人。经典的ML方法需要来自模型参数以及模型响应的大量数据。不幸的是，地球科学中解决的许多问题只能提供非常稀疏的数据。数据稀疏是由于缺乏可用的测量和高计算成本的数值模拟的现实模型。在当前的项目中，我们建议开发一种ML方法，该方法能够自适应地处理物理问题的非线性，同时考虑到可用数据的稀疏性。该项目旨在以目标为导向的方式探索贝叶斯推理和信息论之间的联系，以根据可用的观测数据和计算资源自适应地定位物理问题的非线性。我们将遵循随机模型简化的最新趋势，并根据观测数据，使用原始CO2模型的有限（稀疏）信息训练数学上最优的响应曲面。 在这里，我们建议开发的多项式混沌稀疏重建的基础上贝叶斯理论伴随着信息论的参数的多适应性。将贝叶斯推理与信息论相结合，将有助于自适应地改进响应面，同时迭代地将相关信息纳入自适应响应面。有了新的贝叶斯稀疏2多适应性，它将有可能校准高度非线性模型，大大降低计算成本和量化的校准后不确定性。我们还预计，所提出的贝叶斯稀疏2多适应性概念将为其他ML方法开辟一条基于物理的途径，并将对各种环境问题非常有益。

The water-food-energy nexus is central to sustainable development. Society needs a better understanding of the environment in order to have an efficient and safe interaction for the sake of maximized welfare and sustainability in resources management. Simulations with well-calibrated models offer a unique way to predict the multifaceted behavior of subsurface systems, such as multiphase flow transport in fractured media, coupled hydrosystem model and multi-species reactive transport in porous media, etc. Due to the roughness or lack of available data and high computational costs of the numerical simulation, this class of problems is still challenging for modern methods of uncertainty quantification and Machine Learning (ML). The proposed project seeks to address this challenge on the example of modelling carbon dioxide (CO2) storage in geological formations. The CO2 storage problem is a representative case for the broad class of subsurface problems, because it involves multiphase flow where the CO2 displacement fronts are very non-linearity, which can be difficult capture. ML techniques are become increasing popular in the scientific community and seem to be very suitable candidates for such non-linear problems. Classical ML approaches require huge amounts of data coming from model parameters, as well as model response. Unfortunately, many problems addressed in geosciences can only provide very sparse data. Data sparsity is caused by lack of available measurements and high computational costs of numerical simulation of realistic models. In the current project, we propose to develop a ML approach that will be able to treat the non-linearity of the physical problem adaptively taking into account the sparse nature of the available data. The project intend to explore the link between the Bayesian inference and information theory in a goal-oriented fashion to localize non-linearity of the physical problem adaptively according to available observation data and computational resources. We will follow the recent trend in stochastic model reduction and will train a mathematically optimal response surface using limited (sparse) information from the original CO2 model in light of observed data. Here, we suggest to develop the multi-adaptivity for polynomial chaos employing sparse reconstruction based on Bayesian theory accompanied by the information-theoretic arguments. Combining Bayesian inference with information theory will help adaptively improvement of the response surface, while iteratively including relevant information into the adaptive response surface. With the novel Bayesian sparse2 multi-adaptivity, it will be possible to calibrate highly non-linear models at strongly reduced computational costs and with quantified post-calibration uncertainty. We also expect that the suggested Bayesian sparse2 multi-adaptivity concept will open a physics-based pathway for other ML approaches and will be very beneficial for various environmental problems.