Whether deterministic or stochastic, models can be viewed as functions designed to approximate a specific quantity of interest. We propose a data-driven framework that aggregates predictions from diverse models into a single, more accurate output. This aggregation approach exploits each model's strengths to enhance overall accuracy. It is non-intrusive - treating models as black-box functions - model-agnostic, requires minimal assumptions, and can combine outputs from a wide range of models, including those from machine learning and numerical solvers. We argue that the aggregation process should be point-wise linear and propose two methods to find an optimal aggregate: Minimal Error Aggregation (MEA), which minimizes the aggregate's prediction error, and Minimal Variance Aggregation (MVA), which minimizes its variance. While MEA is inherently more accurate when correlations between models and the target quantity are perfectly known, Minimal Empirical Variance Aggregation (MEVA), an empirical version of MVA - consistently outperforms Minimal Empirical Error Aggregation (MEEA), the empirical counterpart of MEA, when these correlations must be estimated from data. The key difference is that MEVA constructs an aggregate by estimating model errors, while MEEA treats the models as features for direct interpolation of the quantity of interest. This makes MEEA more susceptible to overfitting and poor generalization, where the aggregate may underperform individual models during testing. We demonstrate the versatility and effectiveness of our framework in various applications, such as data science and partial differential equations, showing how it successfully integrates traditional solvers with machine learning models to improve both robustness and accuracy.

翻译：无论确定性还是随机性模型，均可视为旨在逼近特定目标量的函数。我们提出一种数据驱动框架，将来自不同模型的预测聚合为单一且更精确的输出。该聚合方法通过利用各模型的优势来提升整体精度。它具有非侵入性——将模型视为黑箱函数、模型无关性、所需假设极少，并且能够整合来自广泛模型（包括机器学习和数值求解器）的输出。我们认为聚合过程应是逐点线性的，并提出两种寻找最优聚合的方法：最小误差聚合（MEA），其最小化聚合的预测误差；以及最小方差聚合（MVA），其最小化聚合的方差。虽然当模型与目标量之间的相关性完全已知时，MEA本质上更精确，但在这些相关性必须从数据中估计的情况下，最小经验方差聚合（MEVA）——MVA的经验版本——始终优于最小经验误差聚合（MEEA），即MEA的经验对应方法。关键区别在于，MEVA通过估计模型误差来构建聚合，而MEEA则将模型视为直接插值目标量的特征。这使得MEEA更容易出现过拟合和泛化能力差的问题，导致聚合结果在测试时可能逊于单个模型。我们在数据科学和偏微分方程等多种应用中展示了该框架的通用性和有效性，说明了它如何成功地将传统求解器与机器学习模型相结合，从而同时提升鲁棒性和精度。