Modern techniques for physical simulations rely on numerical schemes and mesh-refinement methods to address trade-offs between precision and complexity, but these handcrafted solutions are tedious and require high computational power. Data-driven methods based on large-scale machine learning promise high adaptivity by integrating long-range dependencies more directly and efficiently. In this work, we focus on fluid dynamics and address the shortcomings of a large part of the literature, which are based on fixed support for computations and predictions in the form of regular or irregular grids. We propose a novel setup to perform predictions in a continuous spatial and temporal domain while being trained on sparse observations. We formulate the task as a double observation problem and propose a solution with two interlinked dynamical systems defined on, respectively, the sparse positions and the continuous domain, which allows to forecast and interpolate a solution from the initial condition. Our practical implementation involves recurrent GNNs and a spatio-temporal attention observer capable of interpolating the solution at arbitrary locations. Our model not only generalizes to new initial conditions (as standard auto-regressive models do) but also performs evaluation at arbitrary space and time locations. We evaluate on three standard datasets in fluid dynamics and compare to strong baselines, which are outperformed both in classical settings and in the extended new task requiring continuous predictions.

翻译：现代物理模拟技术依赖数值方案和网格细化方法来平衡精度与复杂度，但这些手工设计的解决方案既繁琐又需要高计算能力。基于大规模机器学习的数驱方法通过更直接高效地整合长程依赖关系，展现出强大的自适应性。本研究聚焦流体动力学领域，针对文献中大量基于规则/非规则网格进行固定支撑计算与预测的局限性，提出一种新型框架：在稀疏观测数据训练下实现连续时空域内的预测。我们将该任务构建为双观测问题，并提出由稀疏位置域和连续域上定义的两个相互关联的动力学系统构成的解决方案，从而能够基于初始条件预测并插值得到连续解。实际实现中，我们采用循环图神经网络与时空注意力观测器，可在任意位置插值求解。该模型不仅能够像标准自回归模型一样泛化至新初始条件，还可实现任意时空位置的高效评估。我们在流体动力学领域的三个标准数据集上进行了验证，结果表明：无论是经典设置还是需要连续预测的新扩展任务场景，本模型均显著超越强基线方法。