Symmetric Basis Convolutions for Learning Lagrangian Fluid Mechanics

TL;DR

This paper introduces symmetric basis convolutions using Fourier-based continuous convolutions for fluid mechanics simulations, demonstrating superior accuracy and stability over existing methods without needing prior biases.

Contribution

The paper proposes a novel Fourier-based continuous convolution framework with symmetric basis functions, improving stability and accuracy in fluid mechanics simulations.

Findings

Fourier-based convolutions outperform other architectures in accuracy.

Symmetric basis functions enhance stability and generalization.

Prior inductive biases like window functions are unnecessary with the new approach.

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.