Learning physical simulations has been an essential and central aspect of many recent research efforts in machine learning, particularly for Navier-Stokes-based fluid mechanics. Classic numerical solvers have traditionally been computationally expensive and challenging to use in inverse problems, whereas Neural solvers aim to address both concerns through machine learning. We propose a general formulation for continuous convolutions using separable basis functions as a superset of existing methods and evaluate a large set of basis functions in the context of (a) a compressible 1D SPH simulation, (b) a weakly compressible 2D SPH simulation, and (c) an incompressible 2D SPH Simulation. We demonstrate that even and odd symmetries included in the basis functions are key aspects of stability and accuracy. Our broad evaluation shows that Fourier-based continuous convolutions outperform all other architectures regarding accuracy and generalization. Finally, using these Fourier-based networks, we show that prior inductive biases, such as window functions, are no longer necessary. An implementation of our approach, as well as complete datasets and solver implementations, is available at https://github.com/tum-pbs/SFBC.

翻译：学习物理模拟一直是机器学习领域许多近期研究的核心和关键方面，特别是针对基于纳维-斯托克斯方程的流体力学。传统的数值求解器通常计算成本高昂且难以应用于逆问题，而神经求解器旨在通过机器学习解决这两个问题。我们提出了一种使用可分离基函数的连续卷积的通用公式，该公式是现有方法的超集，并在（a）一维可压缩SPH模拟、（b）二维弱可压缩SPH模拟和（c）二维不可压缩SPH模拟的背景下评估了大量基函数。我们证明了包含在基函数中的偶对称性和奇对称性是稳定性和准确性的关键因素。我们的广泛评估表明，基于傅里叶的连续卷积在准确性和泛化能力方面优于所有其他架构。最后，利用这些基于傅里叶的网络，我们证明先验的归纳偏差（如窗函数）已不再是必要的。我们的方法实现以及完整的数据集和求解器实现可在 https://github.com/tum-pbs/SFBC 获取。