Effective dimension reduction is a key factor in facilitating large-scale simulation of high-dimensional dynamical systems. The behaviour of low-dimensional surrogate models often relies on accurate reconstruction of quantities that can be nonlinear functions of the original parameters. For instance, in low-dimensional combustion models, source terms representative of complex chemical kinetics must be modelled accurately in the reduced dimensional space in order to yield accurate predictions. Features such as sharp gradients or non-uniqueness in a quantity of interest (QoI) may be introduced through parameterisation and pose difficulties for reconstruction techniques. Many existing manifold quality assessments do not consider these features and limit examination to the original parameters and low-dimensional embedding. We have developed a technique for quantitatively assessing manifold quality through characterising the feature size of QoIs by monitoring the change in variance over an increasing filter width. Through the identification of variance at small scales, this technique detects undesirable sharp gradients and non-uniqueness of QoIs. Our technique is not limited to a specific reduction method and can be used to compare or assess manifold parameterisations in arbitrary dimensions. We demonstrate our technique on combustion data from both simulation and experiment.

有效降维是促进高维动力系统大规模模拟的关键因素。低维代理模型的行为通常依赖于对原始参数的非线性函数数量的准确重构。例如，在低维燃烧模型中，代表复杂化学动力学的源项必须在降维空间中准确建模，以便产生准确的预测。诸如陡峭梯度或感兴趣量 (QoI) 的非唯一性等特征可能会通过参数化引入，并对重建技术造成困难。许多现有的流形质量评估没有考虑这些特征，并将检查限制为原始参数和低维嵌入。我们开发了一种技术，通过监测随滤波器宽度增加而变化的变化来表征 QoI 的特征大小，从而定量评估流形质量。通过在小尺度上识别方差，该技术可以检测到不受欢迎的陡峭梯度和 QoI 的非唯一性。我们的技术不限于特定的归约方法，可用于比较或评估任意维度的流形参数化。我们展示了我们的模拟和实验燃烧数据技术。这种技术可以检测到不受欢迎的陡峭梯度和 QoI 的非唯一性。我们的技术不限于特定的减少方法，可用于比较或评估任意维度的流形参数化。我们展示了我们的模拟和实验燃烧数据技术。这种技术可以检测到不受欢迎的陡峭梯度和 QoI 的非唯一性。我们的技术不限于特定的归约方法，可用于比较或评估任意维度的流形参数化。我们展示了我们的模拟和实验燃烧数据技术。