论文标题
粘性流体力学的数据驱动方法 - 固定案例
A data-driven approach to viscous fluid mechanics -- the stationary case
论文作者
论文摘要
我们引入了一种数据驱动的方法，以进行粘性流体力学的建模和分析。我们建议直接使用实验数据。只有一组从第一原理得出的差异约束，并保留了经典PDE模型的边界条件，并与数据集结合使用。数学框架建立在最近引入的固体机械方法的数据驱动方法上[KO16，CMO18]。我们构建了最佳数据驱动的解决方案，这些解决方案是无物质模型的，从某种意义上说，从数据中制定或推断了对流体的流动性行为的假设。流体力学的差异约束是在恒定等级差异操作员的语言中重铸。适应较低血管持续性的抽象结果和$ \ Mathscr {a} $ - Quasiconvexity，我们显示了在数据驱动的流体机械问题中产生的功能的$γ$ - 连接结果。该理论扩展到紧凑的非线性扰动，因此我们的结果适用于有限雷诺数的惯性流体和流动。数据驱动的解决方案提供了一种新的放松解决方案概念。我们证明，如果数据集具有单调组成型关系的形式，则构建的数据驱动的解决方案与流体力学经典PDE的解决方案是一致的。
We introduce a data-driven approach to the modelling and analysis of viscous fluid mechanics. Instead of including constitutive laws for the fluid's viscosity in the mathematical model, we suggest to directly use experimental data. Only a set of differential constraints, derived from first principles, and boundary conditions are kept of the classical PDE model and are combined with a data set. The mathematical framework builds on the recently introduced data-driven approach to solid-mechanics [KO16,CMO18]. We construct optimal data-driven solutions that are material model free in the sense that no assumptions on the rheological behaviour of the fluid are made or extrapolated from the data. The differential constraints of fluid mechanics are recast in the language of constant rank differential operators. Adapting abstract results on lower-semicontinuity and $\mathscr{A}$-quasiconvexity, we show a $Γ$-convergence result for the functionals arising in the data-driven fluid mechanical problem. The theory is extended to compact nonlinear perturbations, whence our results apply to both inertialess fluids and flows with finite Reynolds number. Data-driven solutions provide a new relaxed solution concept. We prove that the constructed data-driven solutions are consistent with solutions to the classical PDEs of fluid mechanics if the data sets have the form of a monotone constitutive relation.
