标题： 通过物理信息神经网络对PDE模型中的参数不确定性进行建模 显示英文标题

标题： Modelling parametric uncertainty in PDEs models via Physics-Informed Neural Networks

摘要： 我们提供一种方法，使能够使用物理信息神经网络（PINNs）进行不确定性量化。 我们的方法适用于观测数据稀少（甚至缺乏）的系统，这些是与地下水流体相关的典型情况。 我们在不确定性下的新型物理信息神经网络（PINN-UU）将过程发生的时空域和不确定参数空间整合到一个计算域中。 然后，PINN-UU 被训练以满足定义输入域中的相关物理原理（例如，质量守恒）。 我们采用阶段训练方法通过迁移学习来适应高维解空间。 我们在与多孔介质中反应传输相关的场景中展示了 PINN-UU 的有效性，展示了其可靠性、效率以及对敏感性分析的适用性。 PINN-UU 成为一种有前途的工具，用于稳健的不确定性量化，并广泛适用于地下水系统。 因此，它可以被视为传统方法（如基于直接求解器的多实现实蒙特卡洛模拟或黑盒代理模型）的一种有价值的替代方法。 显示英文摘要

摘要： We provide an approach enabling one to employ physics-informed neural networks (PINNs) for uncertainty quantification. Our approach is applicable to systems where observations are scarce (or even lacking), these being typical situations associated with subsurface water bodies. Our novel physics-informed neural network under uncertainty (PINN-UU) integrates the space-time domain across which processes take place and uncertain parameter spaces within a unique computational domain. PINN-UU is then trained to satisfy the relevant physical principles (e.g., mass conservation) in the defined input domain. We employ a stage training approach via transfer learning to accommodate high-dimensional solution spaces. We demonstrate the effectiveness of PINN-UU in a scenario associated with reactive transport in porous media, showcasing its reliability, efficiency, and applicability to sensitivity analysis. PINN-UU emerges as a promising tool for robust uncertainty quantification, with broad applicability to groundwater systems. As such, it can be considered as a valuable alternative to traditional methods such as multi-realization Monte Carlo simulations based on direct solvers or black-box surrogate models.