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.

学习物理模拟一直都是机器学习中许多最新研究努力的一个核心方面，特别是在基于Navier-Stokes的流体力学中。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，并在可压缩的1D SPH模拟、弱可压缩的2D SPH模拟和不可压缩的2D SPH模拟的背景下评估了大量基函数。我们证明了基函数中包含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，基于傅里叶变换的连续卷积在准确性和泛化性能方面优于其他架构。最后，使用这些基于傅里叶变换的网络，我们证明了先前的归纳偏见，如窗函数，不再必要。我们的方法已经实现，并在此https网址提供了完整的数据集和求解器实现。

学习拉格朗日流体力学的对称基准卷积