We introduce in this work the normalizing field flows (NFF) for learning random fields from scattered measurements. More precisely, we construct a bijective transformation (a normalizing flow characterizing by neural networks) between a reference random field (say, a Gaussian random field with the Karhunen-Lo\`eve expansion structure) and the target stochastic field, where the KL expansion coefficients and the invertible networks are trained by maximizing the sum of the log-likelihood on scattered measurements. This NFF model can be used to solve data-driven forward, inverse, and mixed forward/inverse stochastic partial differential equations in a unified framework. We demonstrate the capability of the proposed NFF model for learning Non Gaussian processes, mixed Gaussian processes, and forward & inverse stochastic partial differential equations.

翻译：我们在这项工作中引入了从分散测量中学习随机字段的正常化字段流(NFF) 。 更确切地说, 我们在一个参考随机字段( 例如, 带有Karhunen- Lo ⁇ ⁇ ⁇ éeve扩展结构的高斯随机字段) 和目标随机字段之间构建了双向转换( 通过神经网络网络的正常化流程), 通过将分散测量的日志相似度之和最大化来培训 KL 扩展系数和可视网络。 这个 NFF 模型可用于在统一框架内解决数据驱动的前向、 反向和混合前向/ 反向偏移部分差异方程式。 我们展示了拟议的 NFF 模型在学习非高斯进程、 混合高斯进程和前向和反向偏差部分方程式方面的能力 。