We introduce Lagrangian Flow Networks (LFlows) for modeling fluid densities and velocities continuously in space and time. By construction, the proposed LFlows satisfy 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 the 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 numerical ODE solvers or PINNs is that the analytic expression of the velocity is always consistent with changes in density. Furthermore, we require neither expensive numerical solvers, nor additional penalties to enforce the PDE. LFlows show higher predictive accuracy in density modeling tasks compared to competing models in 2D and 3D, while being computationally efficient. As a real-world application, we model bird migration based on sparse weather radar measurements.

翻译：我们提出拉格朗日流网络（LFlows），用于在时空连续建模流体密度和速度。通过构建，所提出的LFlows满足连续性方程，这是一道描述质量守恒的偏微分方程（PDE），并以可微形式呈现。我们的模型基于如下洞见：连续性方程的解可通过可微可逆映射表示为时间相关的密度变换。这遵循了光滑向量场拉格朗日流存在唯一性的经典理论。因此，我们通过将基础密度与参数化、基于时间条件的微分同胚进行变换来建模流体密度。与依赖数值ODE求解器或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变化保持一致。此外，我们既不需要昂贵的数值求解器，也无需额外惩罚项来强制执行PDE。在2D和3D密度建模任务中，LFlows相比竞争模型表现出更高的预测精度，同时保持计算高效。作为实际应用，我们基于稀疏气象雷达测量数据对鸟类迁徙进行建模。