Symmetric Basis Convolutions for Learning Lagrangian Fluid Mechanics
This work addresses the computational expense and inverse problem challenges in fluid mechanics simulations for researchers and practitioners in machine learning and computational physics, representing an incremental improvement with a broad evaluation of basis functions.
The authors tackled the problem of learning physical simulations for Lagrangian fluid mechanics by proposing a general formulation for continuous convolutions using separable basis functions, and demonstrated that Fourier-based continuous convolutions outperform other architectures in accuracy and generalization, with specific evaluations on compressible and incompressible SPH simulations.
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.