This work presents a data-driven method for learning low-dimensional time-dependent physics-based surrogate models whose predictions are endowed with uncertainty estimates. We use the operator inference approach to model reduction that poses the problem of learning low-dimensional model terms as a regression of state space data and corresponding time derivatives by minimizing the residual of reduced system equations. Standard operator inference models perform well with accurate training data that are dense in time, but producing stable and accurate models when the state data are noisy and/or sparse in time remains a challenge. Another challenge is the lack of uncertainty estimation for the predictions from the operator inference models. Our approach addresses these challenges by incorporating Gaussian process surrogates into the operator inference framework to (1) probabilistically describe uncertainties in the state predictions and (2) procure analytical time derivative estimates with quantified uncertainties. The formulation leads to a generalized least-squares regression and, ultimately, reduced-order models that are described probabilistically with a closed-form expression for the posterior distribution of the operators. The resulting probabilistic surrogate model propagates uncertainties from the observed state data to reduced-order predictions. We demonstrate the method is effective for constructing low-dimensional models of two nonlinear partial differential equations representing a compressible flow and a nonlinear diffusion-reaction process, as well as for estimating the parameters of a low-dimensional system of nonlinear ordinary differential equations representing compartmental models in epidemiology.

翻译：本研究提出了一种数据驱动方法，用于学习具有不确定性估计的低维时变物理代理模型。我们采用算子推断的模型降阶方法，将低维模型项的学习问题转化为对状态空间数据及其对应时间导数的回归分析，通过最小化降阶系统方程的残差实现。标准算子推断模型在处理时间密集的精确训练数据时表现良好，但当状态数据存在噪声和/或时间稀疏时，如何生成稳定精确的模型仍是挑战。另一挑战在于算子推断模型缺乏预测的不确定性估计。我们的方法通过将高斯过程代理模型融入算子推断框架来解决这些问题，以（1）概率化描述状态预测中的不确定性，（2）获取具有量化不确定性的解析时间导数估计。该表述导出了广义最小二乘回归问题，并最终得到以概率形式描述的降阶模型，其算子后验分布具有闭式表达式。所得概率代理模型将观测状态数据中的不确定性传播至降阶预测。我们通过两个非线性偏微分方程（代表可压缩流动和非线性扩散-反应过程）的低维建模，以及流行病学中隔室模型对应的非线性常微分方程低维系统的参数估计，验证了该方法的有效性。