用可微物理和通用微分方程反推粘弹性本构模型，提升流体建模精度。
Finding the Underlying Viscoelastic Constitutive Equation via Universal Differential Equations and Differentiable Physics
- 融合神经网络与微分方程，自动补全粘弹性模型缺失项。
- 在四种模型上准确预测剪切与法向应力，但ePTT模型表现受限。
- 可提取简化模型，适合流变学建模与复杂系统分析。
本研究采用通用微分方程（UDEs）结合可微物理方法，对粘弹性流体进行建模，将传统微分方程、神经网络与数值方法融合，以重构本构模型中的缺失项。研究聚焦于四种粘弹性模型：上卷绕麦克斯韦（UCM）、约翰逊-塞格尔曼、吉泽克斯及指数型范-蒂恩-滕纳（ePTT），使用合成数据集进行验证。方法在振荡流与启动流等不同实验条件下均被测试。尽管UDE框架在多数模型中能有效预测剪切应力与法向应力，但在ePTT模型上仍存在局限性。结果表明，UDE在流体力学中有巨大潜力，同时揭示了方法改进的关键方向。此外，通过模型蒸馏方法从复杂模型中提取简化模型，进一步凸显了UDE在流变学建模中的灵活性与鲁棒性。
原文摘要 · Abstract (English)
This research employs Universal Differential Equations (UDEs) alongside differentiable physics to model viscoelastic fluids, merging conventional differential equations, neural networks and numerical methods to reconstruct missing terms in constitutive models. This study focuses on analyzing four viscoelastic models: Upper Convected Maxwell (UCM), Johnson-Segalman, Giesekus, and Exponential Phan-Thien-Tanner (ePTT), through the use of synthetic datasets. The methodology was tested across different experimental conditions, including oscillatory and startup flows. While the UDE framework effectively predicts shear and normal stresses for most models, it demonstrates some limitations when applied to the ePTT model. The findings underscore the potential of UDEs in fluid mechanics while identifying critical areas for methodological improvement. Also, a model distillation approach was employed to extract simplified models from complex ones, emphasizing the versatility and robustness of UDEs in rheological modeling.
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