De Florio Mario, Zou Zongren, Schiavazzi Daniele E, Karniadakis George Em
Division of Applied Mathematics, Brown University, Providence, 02906 RI, USA.
Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, 46556 IN, USA.
ArXiv. 2024 Aug 13:arXiv:2408.07201v1.
When predicting physical phenomena through simulation, quantification of the total uncertainty due to multiple sources is as crucial as making sure the underlying numerical model is accurate. Possible sources include irreducible uncertainty due to noise in the data, uncertainty induced by insufficient data or inadequate parameterization, and uncertainty related to the use of misspecified model equations. In addition, recently proposed approaches provide flexible ways to combine information from data with full or partial satisfaction of equations that typically encode physical principles. Physics-based regularization interacts in nontrivial ways with aleatoric, epistemic and model-form uncertainty and their combination, and a better understanding of this interaction is needed to improve the predictive performance of physics-informed digital twins that operate under real conditions. To better understand this interaction, with a specific focus on biological and physiological models, this study investigates the decomposition of total uncertainty in the estimation of states and parameters of a differential system simulated with MC X-TFC, a new physics-informed approach for uncertainty quantification based on random projections and Monte-Carlo sampling. After an introductory comparison between approaches for physics-informed estimation, MC X-TFC is applied to a six-compartment stiff ODE system, the CVSim-6 model, developed in the context of human physiology. The system is first analyzed by progressively removing data while estimating an increasing number of parameters, and subsequently by investigating total uncertainty under model-form misspecification of non-linear resistance in the pulmonary compartment. In particular, we focus on the interaction between the formulation of the discrepancy term and quantification of model-form uncertainty, and show how additional physics can help in the estimation process. The method demonstrates robustness and efficiency in estimating unknown states and parameters, even with limited, sparse, and noisy data. It also offers great flexibility in integrating data with physics for improved estimation, even in cases of model misspecification.
通过模拟预测物理现象时，量化多个来源导致的总不确定性与确保基础数值模型的准确性同样重要。可能的来源包括数据噪声引起的不可约不确定性、数据不足或参数化不充分导致的不确定性，以及与错误指定的模型方程使用相关的不确定性。此外，最近提出的方法提供了灵活的方式来结合来自数据的信息，并完全或部分满足通常编码物理原理的方程。基于物理的正则化以复杂的方式与偶然不确定性、认知不确定性和模型形式不确定性及其组合相互作用，需要更好地理解这种相互作用，以提高在实际条件下运行的物理信息数字孪生体的预测性能。为了更好地理解这种相互作用，本研究特别关注生物和生理模型，研究了使用MC X-TFC模拟的微分系统状态和参数估计中总不确定性的分解，MC X-TFC是一种基于随机投影和蒙特卡洛采样的用于不确定性量化的新的物理信息方法。在对物理信息估计方法进行介绍性比较之后，将MC X-TFC应用于在人体生理学背景下开发的六室刚性常微分方程系统CVSim-6模型。该系统首先通过在估计越来越多参数的同时逐步去除数据进行分析，随后通过研究肺室中非线性阻力的模型形式错误指定下的总不确定性进行分析。特别是，我们关注差异项的公式化与模型形式不确定性量化之间的相互作用，并展示额外的物理知识如何在估计过程中提供帮助。该方法在估计未知状态和参数时表现出稳健性和效率，即使数据有限、稀疏且有噪声。即使在模型错误指定的情况下，它在将数据与物理知识集成以改进估计方面也具有很大的灵活性。
