Efficient downscaling of large ensembles of coarse-scale information is crucial in several applications, such as oceanic and atmospheric modeling. The determining form map is a theoretical lifting function from the low-resolution solution trajectories of a dissipative dynamical system to their corresponding fine-scale counterparts. Recently, a physics-informed deep neural network ("CDAnet") was introduced, providing a surrogate of the determining form map for efficient downscaling. CDAnet was demonstrated to efficiently downscale noise-free coarse-scale data in a deterministic setting. Herein, the performance of well-trained CDAnet models is analyzed in a stochastic setting involving (i) observational noise, (ii) model noise, and (iii) a combination of observational and model noises. The analysis is performed employing the Rayleigh-Benard convection paradigm, under three training conditions, namely, training with perfect, noisy, or downscaled data. Furthermore, the effects of noises, Rayleigh number, and spatial and temporal resolutions of the input coarse-scale information on the downscaled fields are examined. The results suggest that the expected l2-error of CDAnet behaves quadratically in terms of the standard deviations of the observational and model noises. The results also suggest that CDAnet responds to uncertainties similar to the theorized and numerically-validated CDA behavior with an additional error overhead due to CDAnet being a surrogate model of the determining form map.

翻译：高效降尺度处理大量粗尺度信息集成数据，在海洋与大气建模等诸多应用中至关重要。确定形式映射是一种理论提升函数，能将耗散动力系统的低分辨率解轨迹映射至相应的精细尺度解轨迹。近期，一种基于物理信息的深度神经网络"CDAnet"被提出，可作为确定形式映射的代理模型实现高效降尺度。该网络已被证实能在确定性条件下有效降尺度处理无噪声粗尺度数据。本文在随机条件下分析训练完备的CDAnet模型性能，具体涉及：(i)观测噪声、(ii)模型噪声、及(iii)观测与模型噪声组合。分析基于Rayleigh-Benard对流范例，在三种训练条件下展开，即使用完美数据、含噪数据或降尺度数据进行训练。此外，本研究还考察了噪声、Rayleigh数、以及输入粗尺度信息的时空分辨率对降尺度场的影响。结果表明，CDAnet的期望l2误差与观测噪声和模型噪声的标准差呈二次函数关系。同时发现，CDAnet对不确定性的响应特征与经理论推导及数值验证的CDA行为相似，但作为确定形式映射的代理模型，其存在额外的误差开销。