We present a novel reduced-order Model (ROM) that leverages optimal transport (OT) theory and displacement interpolation to enhance the representation of nonlinear dynamics in complex systems. While traditional ROM techniques face challenges in this scenario, especially when data (i.e., observational snapshots) is limited, our method addresses these issues by introducing a data augmentation strategy based on OT principles. The proposed framework generates interpolated solutions tracing geodesic paths in the space of probability distributions, enriching the training dataset for the ROM. A key feature of our approach is its ability to provide a continuous representation of the solution's dynamics by exploiting a virtual-to-real time mapping. This enables the reconstruction of solutions at finer temporal scales than those provided by the original data. To further improve prediction accuracy, we employ Gaussian Process Regression to learn the residual and correct the representation between the interpolated snapshots and the physical solution. We demonstrate the effectiveness of our methodology with atmospheric mesoscale benchmarks characterized by highly nonlinear, advection-dominated dynamics. Our results show improved accuracy and efficiency in predicting complex system behaviors, indicating the potential of this approach for a wide range of applications in computational physics and engineering.

翻译：本文提出了一种新颖的降阶模型（ROM），该模型利用最优传输（OT）理论和位移插值来增强复杂系统中非线性动力学的表示能力。传统ROM技术在此类场景下面临挑战，尤其是在数据（即观测快照）有限的情况下。我们的方法通过引入基于OT原理的数据增强策略来解决这些问题。所提出的框架生成沿概率分布空间中测地线路径的插值解，从而丰富了ROM的训练数据集。该方法的一个关键特性在于，通过利用虚拟时间到真实时间的映射，能够提供解动力学的连续表示。这使得我们能够在比原始数据更精细的时间尺度上重建解。为了进一步提高预测精度，我们采用高斯过程回归来学习残差，并校正插值快照与物理解之间的表示误差。我们通过以高度非线性、平流主导动力学为特征的大气中尺度基准测试验证了所提方法的有效性。结果表明，该方法在预测复杂系统行为方面具有更高的精度和效率，显示出该方法在计算物理和工程领域的广泛应用潜力。