Symmetric Basis Convolutions for Learning Lagrangian Fluid Mechanics

TL;DR

This work proposes a general formulation for continuous convolutions using separable basis functions as a superset of existing methods and evaluates a large set of basis functions in the context of a compressible 1D SPH simulation, a weakly compressible 2D SPH simulation, and an incompressible 2D SPH Simulation.

Abstract

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.

Figures (39)

• Figure 1: Visual overview of SFBC.
• Figure 2: Quantitative evaluation of the one-dimensional kernel function (left) and kernel gradient (right) toy problem. Evaluations are based on the $L_2$ error related to the number of base terms.
• Figure 3: A quantitative evaluation of the relationship between parameter count and test error for basis convolutions(left), and a quantitative evaluation of different networks for a fixed layout with four message-passing steps and 32 features per layer (right). Error bars showing lower to upper $5\%$.
• Figure 4: A qualitative comparison after $64$ inference steps with (f.l.t.r.) of the ground truth data, our SFBC approach, LinCConv, a Fourier basis with window, and Chebyshev with window. The particle data is mapped to a regular grid with color mapping indicating current velocity.
• Figure 5: The quantitative results are based on the integrated error over an inference period of $64$ steps regarding four error metrics not included in the training loss.