We introduce Lagrangian Flow Networks (LFlows) for modeling fluid densities and velocities continuously in space and time. The proposed LFlows satisfy by construction the continuity equation, a PDE describing mass conservation in its differentiable form. Our model is based on the insight that solutions to the continuity equation can be expressed as time-dependent density transformations via differentiable and invertible maps. This follows from classical theory of existence and uniqueness of Lagrangian flows for smooth vector fields. Hence, we model fluid densities by transforming a base density with parameterized diffeomorphisms conditioned on time. The key benefit compared to methods relying on Neural-ODE or PINNs is that the analytic expression of the velocity is always consistent with the density. Furthermore, there is no need for expensive numerical solvers, nor for enforcing the PDE with penalty methods. Lagrangian Flow Networks show improved predictive accuracy on synthetic density modeling tasks compared to competing models in both 2D and 3D. We conclude with a real-world application of modeling bird migration based on sparse weather radar measurements.

翻译：我们提出拉格朗日流网络（LFlows），用于在空间和时间上连续建模流体密度与速度。所提出的LFlows通过构造满足连续性方程（以可微形式描述质量守恒的偏微分方程）。该模型基于以下洞见：连续性方程的解可表示为通过可微可逆映射的时变密度变换。这源于光滑向量场拉格朗日流存在唯一性的经典理论。因此，我们通过条件依赖时间的参数化微分同胚变换基密度来建模流体密度。与依赖神经ODE或PINNs的方法相比，关键优势在于速度的解析表达式始终与密度保持一致。此外，该方法既不需要昂贵的数值求解器，也无需通过惩罚方法强制满足偏微分方程。在二维和三维合成密度建模任务中，拉格朗日流网络相较于竞争模型展现出更优的预测精度。我们最后展示了基于稀疏天气雷达测量数据的鸟类迁徙建模这一实际应用。