This work formulates a new approach to reduced modeling of parameterized, time-dependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning and physics-based modeling. The parametric structure of the governing equations is embedded directly into the reduced-order model, and parameterized reduced-order operators are learned via a data-driven linear regression problem. The result is a reduced-order model that can be solved rapidly to map parameter values to approximate PDE solutions. Such parameterized reduced-order models may be used as physics-based surrogates for uncertainty quantification and inverse problems that require many forward solves of parametric PDEs. Numerical issues such as well-posedness and the need for appropriate regularization in the learning problem are considered, and an algorithm for hyperparameter selection is presented. The method is illustrated for a parametric heat equation and demonstrated for the FitzHugh-Nagumo neuron model.
翻译:这项工作为减少参数化、时间依赖部分差异方程式的建模制定了新的方法。该方法采用操作员推断法,这是一个科学机器学习框架,将数据驱动的学习和物理建模相结合。管辖方程式的参数结构直接嵌入减序模型,通过数据驱动的线性回归问题来学习减序操作员的参数。其结果是一个减序模型,可以快速解析,以绘制参数值接近PDE解决方案。这种减序参数模型可以用作基于物理的代孕,用于不确定性的量化和反向问题,需要许多远方参数PDE的解决方案。考虑了数值问题,如保有问题和学习问题的适当正规化需要,并提出了超参数选择的算法。该方法用于参数热方程式,为FitzHugh-Nagumo神经模型演示。</s>