Abstract:The simulation of fluid dynamics, typically by numerically solving partial differential equations, is an essential tool in many areas of science and engineering. However, the high computational cost can limit application in practice and may prohibit exploring large parameter spaces. Recent deep-learning approaches have demonstrated the potential to yield surrogate models for the simulation of fluid dynamics. While such models exhibit lower accuracy in comparison, their low runtime makes them appealing for desi… Show more
“…1(a)). There are two alternative solutions to the matrix power enhancement that ensure conservation of the connectivity at coarser layers: 1) Lino et al (2022a;2022b) build the coarser graph by projecting the finer nodes to the nearby background grids (Fig. 1(b)); 2) Fortunato et al (2022) create coarser meshes for the same domain (Fig.…”
Learning physical systems on unstructured meshes by flat Graph neural networks (GNNs) faces the challenge of modeling the long-range interactions due to the scaling complexity w.r.t. the number of nodes, limiting the generalization under mesh refinement. On regular grids, the convolutional neural networks (CNNs) with a U-net structure can resolve this challenge by efficient stride, pooling, and upsampling operations. Nonetheless, these tools are much less developed for graph neural networks (GNNs), especially when GNNs are employed for learning large-scale mesh-based physics. The challenges arise from the highly irregular meshes and the lack of effective ways to construct the multi-level structure without losing connectivity. Inspired by the bipartite graph determination algorithm, we introduce Bi-Stride Multi-Scale Graph Neural Network (BSMS-GNN) by proposing bi-stride as a simple pooling strategy for building the multi-level GNN. Bi-stride pools nodes by striding every other BFS frontier; it 1) works robustly on any challenging mesh in the wild, 2) avoids using a mesh generator at coarser levels, 3) avoids the spatial proximity for building coarser levels, and 4) uses non-parametrized aggregating/returning instead of MLPs during pooling and unpooling. Experiments show that our framework significantly outperforms the state-of-the-art method's computational efficiency in representative physics-based simulation cases.
“…1(a)). There are two alternative solutions to the matrix power enhancement that ensure conservation of the connectivity at coarser layers: 1) Lino et al (2022a;2022b) build the coarser graph by projecting the finer nodes to the nearby background grids (Fig. 1(b)); 2) Fortunato et al (2022) create coarser meshes for the same domain (Fig.…”
Learning physical systems on unstructured meshes by flat Graph neural networks (GNNs) faces the challenge of modeling the long-range interactions due to the scaling complexity w.r.t. the number of nodes, limiting the generalization under mesh refinement. On regular grids, the convolutional neural networks (CNNs) with a U-net structure can resolve this challenge by efficient stride, pooling, and upsampling operations. Nonetheless, these tools are much less developed for graph neural networks (GNNs), especially when GNNs are employed for learning large-scale mesh-based physics. The challenges arise from the highly irregular meshes and the lack of effective ways to construct the multi-level structure without losing connectivity. Inspired by the bipartite graph determination algorithm, we introduce Bi-Stride Multi-Scale Graph Neural Network (BSMS-GNN) by proposing bi-stride as a simple pooling strategy for building the multi-level GNN. Bi-stride pools nodes by striding every other BFS frontier; it 1) works robustly on any challenging mesh in the wild, 2) avoids using a mesh generator at coarser levels, 3) avoids the spatial proximity for building coarser levels, and 4) uses non-parametrized aggregating/returning instead of MLPs during pooling and unpooling. Experiments show that our framework significantly outperforms the state-of-the-art method's computational efficiency in representative physics-based simulation cases.
“…GNNs often consist of sequential MP and activation layers [92,93]. These networks have already proved successful in learning to simulate a variety of physics [87,88,94], including fluid dynamics [53,54,93]. Figure 4( a ) Diagram of a graph with nodes V=falsefalse{1,2,3,4,5falsefalse} and edges E=falsefalse{false(1,2false),false(2,1false),false(2,3false),false(3,2false),false(3,4false),false(4,1false),false(5,4false)falsefalse}.…”
Section: Deep-learning Fundamentalsmentioning
confidence: 99%
“…In fluid problems, it penalizes spurious oscillations in the velocity and pressure fields and favours smooth solutions. This loss function subsequently became popular in unsteady simulations [42,52,75]. Bhatnagar et al.…”
Over the last decade, deep learning (DL), a branch of machine learning, has experienced rapid progress. Powerful tools for tasks that have been traditionally complex to automate have been developed, such as image synthesis and natural language processing. In the context of simulating fluid dynamics, this has led to a series of novel DL methods for replacing or augmenting conventional numerical solvers. We broadly classify these methods into physics- and data-driven methods. Physics-driven methods, generally, tune a DL model to provide an analytical and differentiable solution to a given fluid dynamics problem by minimizing the residuals of the governing partial differential equations. Data-driven methods provide a fast and approximate solution to any fluid dynamics problem that shares some physical properties with the observations used when tuning the DL model’s parameters. Meanwhile, the symbiosis of numerical solvers and DL has led to promising results in turbulence modelling and accelerating iterative solvers. However, these methods present some challenges. Exclusively data-driven flow simulators often suffer from poor extrapolation, error accumulation in time-dependent simulations, as well as difficulties in training against turbulent flows. Substantial effort is, therefore, being invested into approaches that may improve the current state of the art.
“…Tumanov et al [TKC21] uses a sub‐pixel convolution to learn particle based fluid simulations with obstacles. Lino et al [LCBF21] use a multi‐scale graph network to learn 2D fluid on an unstructured point set. Park et al [PLL21] use time‐wise point net to learn physics of 2D deformable objects.…”
We propose a hierarchical graph for learning physics and a novel way to handle obstacles. The finest level of the graph consist of the particles itself. Coarser levels consist of the cells of sparse grids with successively doubling cell sizes covering the volume occupied by the particles. The hierarchical structure allows for the information to propagate at great distance in a single message passing iteration. The novel obstacle handling allows the simulation to be obstacle aware without the need for ghost particles. We train the network to predict effective acceleration produced by multiple sub‐steps of 3D multi‐material material point method (MPM) simulation consisting of water, sand and snow with complex obstacles. Our network produces lower error, trains up to 7.0X faster and inferences up to 11.3X faster than [SGGP*20]. It is also, on average, about 3.7X faster compared to Taichi Elements simulation running on the same hardware in our tests.
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