The abundance of observed data in recent years has increased the number of statistical augmentations to complex models across science and engineering. By augmentation we mean coherent statistical methods that incorporate measurements upon arrival and adjust the model accordingly. However, in this research area methodological developments tend to be central, with important assessments of model fidelity often taking second place. Recently, the statistical finite element method (statFEM) has been posited as a potential solution to the problem of model misspecification when the data are believed to be generated from an underlying partial differential equation system. Bayes nonlinear filtering permits data driven finite element discretised solutions that are updated to give a posterior distribution which quantifies the uncertainty over model solutions. The statFEM has shown great promise in systems subject to mild misspecification but its ability to handle scenarios of severe model misspecification has not yet been presented. In this paper we fill this gap, studying statFEM in the context of shallow water equations chosen for their oceanographic relevance. By deliberately misspecifying the governing equations, via linearisation, viscosity, and bathymetry, we systematically analyse misspecification through studying how the resultant approximate posterior distribution is affected, under additional regimes of decreasing spatiotemporal observational frequency. Results show that statFEM performs well with reasonable accuracy, as measured by theoretically sound proper scoring rules.

翻译：近年来观测数据的丰富使得科学与工程领域中复杂模型的统计增广方法日益增多。所谓增广，是指通过连贯的统计方法在新测量数据到达时将其融入模型并相应调整。然而该研究领域的方法论发展常占据主导地位，而对模型保真度的关键评估往往退居次席。近期提出的统计有限元方法（statFEM）被认为可解决当数据被认为由底层偏微分方程系统生成时的模型错误设定问题。贝叶斯非线性滤波技术允许数据驱动的有限元离散解通过更新获得后验分布，从而量化模型解的不确定性。statFEM在处理轻度错误设定系统时展现出巨大潜力，但其应对严重模型错误设定场景的能力尚未得到论证。本文填补了这一空白，选择具有海洋学相关性的浅水方程作为研究对象。通过线性化、黏性项及水深地形三种方式故意错误设定控制方程，我们在降低时空观测频率的附加条件下，系统分析近似后验分布所受影响来研究错误设定。结果表明，通过理论上合理的适当评分规则衡量，statFEM在保持合理精度的同时表现优异。