Simulation is a powerful tool to better understand physical systems, but generally requires computationally expensive numerical methods. Downstream applications of such simulations can become computationally infeasible if they require many forward solves, for example in the case of inverse design with many degrees of freedom. In this work, we investigate and extend neural PDE solvers as a tool to aid in scaling simulations for two-phase flow problems, and simulations of oil expulsion from a pore specifically. We extend existing numerical methods for this problem to a more complex setting involving varying geometries of the domain to generate a challenging dataset. Further, we investigate three prominent neural PDE solver methods, namely the UNet, DRN and U-FNO, and extend them for characteristics of the oil-expulsion problem: (1) spatial conditioning on the geometry; (2) periodicity in the boundary; (3) approximate mass conservation. We scale all methods and benchmark their speed-accuracy trade-off, evaluate qualitative properties, and perform an ablation study. We find that the investigated methods can accurately model the droplet dynamics with up to three orders of magnitude speed-up, that our extensions improve performance over the baselines, and that the introduced varying geometries constitute a significantly more challenging setting over the previously considered oil expulsion problem.

翻译：模拟是深入理解物理系统的有力工具，但通常需要计算成本高昂的数值方法。若下游应用需大量正向求解（例如具有多自由度的逆向设计场景），此类模拟的计算可能变得不可行。本研究探讨并拓展了神经偏微分方程求解器作为辅助工具，以提升两相流问题（特别是孔隙中石油驱替过程）模拟的扩展能力。我们针对该问题扩展了现有数值方法，构建了涉及变化几何域的更复杂场景，从而生成具有挑战性的数据集。进一步地，我们研究了三种主流的神经偏微分方程求解方法（即UNet、DRN和U-FNO），并针对石油驱替问题的特征进行拓展：（1）几何结构的空间条件编码；（2）边界周期性；（3）近似质量守恒。我们对所有方法进行规模化测试，评估其速度-精度权衡关系，检验定性特性，并开展消融实验。研究发现：所考察的方法能以最高三个数量级的加速比精确模拟液滴动力学；我们的扩展方案较基线模型性能更优；引入的变化几何结构相较于先前考虑的石油驱替问题构成了显著更具挑战性的场景。