We present a method for learning neural representations of flow maps from time-varying vector field data. The flow map is pervasive within the area of flow visualization, as it is foundational to numerous visualization techniques, e.g. integral curve computation for pathlines or streaklines, as well as computing separation/attraction structures within the flow field. Yet bottlenecks in flow map computation, namely the numerical integration of vector fields, can easily inhibit their use within interactive visualization settings. In response, in our work we seek neural representations of flow maps that are efficient to evaluate, while remaining scalable to optimize, both in computation cost and data requirements. A key aspect of our approach is that we can frame the process of representation learning not in optimizing for samples of the flow map, but rather, a self-consistency criterion on flow map derivatives that eliminates the need for flow map samples, and thus numerical integration, altogether. Central to realizing this is a novel neural network design for flow maps, coupled with an optimization scheme, wherein our representation only requires the time-varying vector field for learning, encoded as instantaneous velocity. We show the benefits of our method over prior works in terms of accuracy and efficiency across a range of 2D and 3D time-varying vector fields, while showing how our neural representation of flow maps can benefit unsteady flow visualization techniques such as streaklines, and the finite-time Lyapunov exponent.

翻译：我们提出了一种从时变向量场数据中学习流映射神经表示的方法。流映射在流场可视化领域应用广泛，它是众多可视化技术的基础，例如用于流线或脉线的积分曲线计算，以及流场内分离/吸引结构的提取。然而，流映射计算中的瓶颈，即向量场的数值积分，很容易阻碍其在交互式可视化环境中的应用。为此，在我们的工作中，我们寻求计算高效的流映射神经表示，同时在计算成本和数据需求方面保持可扩展性以进行优化。我们方法的一个关键方面在于，我们可以将表示学习过程构建为并非针对流映射样本的优化，而是基于流映射导数的自洽性准则，该准则完全消除了对流映射样本以及数值积分的需求。实现这一点的核心是为流映射设计的新型神经网络，并结合一种优化方案，其中我们的表示仅需要用于学习的时变向量场（编码为瞬时速度）。我们展示了我们的方法在多种二维和三维时变向量场上相较于先前工作在准确性和效率方面的优势，同时阐明了我们的流映射神经表示如何有益于非稳态流场可视化技术（如脉线）以及有限时间李雅普诺夫指数。