Table of contents

1. Mesh Generation in Computational Science

2. Deep Neural Networks (DNNs) for mesh-based simulations
3. Graph Neural Networks (GNNs) for mesh-based simulations
4. MESHNET: GNN for mesh generation
5. GRAPHNET: GNN for mesh adaptation
6. ADAPTNET: Framework for automated mesh generation and adaptation

Meshes in simulation

Mesh-based finite element (FE) simulations are the 1^st for many problems across science & engineering.

"The finite element method (FEM) is the most widely used method for solving problems of engineering and mathematical models".

Meshes in simulation

High quality mesh generation framework

Meshes in simulation

High quality mesh generation framework

Parameters for Mesh Generation

Mesh generation depends on several critical parameters:

• Local Metrics: Metrics at specific points in the mesh.
• Element Sizes: Maximum and minimum sizes of elements.
• Hausdorff Coefficient: A measure of domain complexity.
• Metric Gradation: Acceptable gradation of the metric.
• Others: Additional parameters influencing the mesh.

These parameters play a vital role in determining mesh quality and efficiency.

Parameters for Mesh Generation

• Local Metrics: Metrics at specific points in the mesh.

Parameters for Mesh Generation

• Local Metrics: Metrics at specific points in the mesh.

						//+
						Mesh.MshFileVersion = 2.2;
						//+
						SetFactory("OpenCASCADE");
						//+
						Box(1) = {0.045462, -3.922459, 0.000000, 35.722251, 9.127825, 1.000000};
						//+
						MeshSize {1} = 2.518574;
						//+
						MeshSize {2} = 2.518574;
						//+
						MeshSize {3} = 2.409870;
						//+
						MeshSize {4} = 2.409870;
						//+
						MeshSize {5} = 0.597988;
						//+
						MeshSize {6} = 0.597988;
						//+
						MeshSize {7} = 0.700752;
						//+
						MeshSize {8} = 0.700752;
						//+
						Cylinder(2) = {27.432369, -0.761381, 0.000000, 0.000000, 0.000000, 1.000000, 1.138991, 2*Pi};
						//+
						MeshSize {9} = 0.113899;
						//+
						MeshSize {10} = 0.113899;
						//+
						Cylinder(3) = {17.813682, 1.160731, 0.000000, 0.000000, 0.000000, 1.000000, 0.789672, 2*Pi};
						//+
						MeshSize {11} = 0.078967;
						//+
						MeshSize {12} = 0.078967;
						//+
						BooleanDifference{ Volume{1}; Delete; }{ Volume{2}; Volume{3}; Delete; }
						

Parameters for Mesh Generation

• Local Metrics: Metrics at specific points in the mesh.

MESHNET

A framework for learning local metrics parameters using graph neural networks.

MESHNET

A framework for learning local metrics parameters using graph neural networks.

						Point(1) = {0.045462, -3.922459, 1};
						Point(2) = {0.045462, -3.922459, 0};
						Point(3) = {0.045462, 5.205366, 1};
						Point(4) = {0.045462, 5.205366, 0};
						Point(5) = {35.767713, -3.922459, 0};
						Point(6) = {35.767713, -3.922459, 1};
						Point(7) = {35.767713, 5.205366, 1};
						Point(8) = {28.57136, -0.7613810000000002, 1};
						Point(9) = {18.603354, 1.160731, 1};
						Point(10) = {35.767713, 5.205366, 0};
						Point(11) = {28.57136, -0.7613810000000002, 0};
						Point(12) = {18.603354, 1.160731, 0};
						Line(1) = {2, 1};
						Line(2) = {1, 3};
						Line(3) = {4, 3};
						Line(4) = {2, 4};
						Line(5) = {2, 5};
						Line(6) = {5, 6};
						Line(7) = {1, 6};
						Line(8) = {3, 7};
						Line(9) = {6, 7};
						Spline(10) = {8, ...
						

MESHNET

A framework for learning local metrics parameters using graph neural networks.

						cl__1 = 2.518574;
						cl__2 = 2.40987;
						cl__3 = 0.597988;
						cl__4 = 0.700752;
						cl__5 = 0.113899;
						cl__6 = 0.078967;
						Point(1) = {0.045462, -3.922459, 1, cl__1};
						Point(2) = {0.045462, -3.922459, 0, cl__1};
						Point(3) = {0.045462, 5.205366, 1, cl__2};
						Point(4) = {0.045462, 5.205366, 0, cl__2};
						Point(5) = {35.767713, -3.922459, 0, cl__3};
						Point(6) = {35.767713, -3.922459, 1, cl__3};
						Point(7) = {35.767713, 5.205366, 1, cl__4};
						Point(8) = {28.57136, -0.7613810000000002, 1, cl__5};
						Point(9) = {18.603354, 1.160731, 1, cl__6};
						Point(10) = {35.767713, 5.205366, 0, cl__4};
						Point(11) = {28.57136, -0.7613810000000002, 0, cl__5};
						Point(12) = {18.603354, 1.160731, 0, cl__6};
						Line(1) = {2, 1};
						Line(2) = {1, 3};
						Line(3) = {4, 3};
						Line(4) = {2, 4};
						Line(5) = {2, 5};
						Line(6) = {5, 6};
						Line(7) = {1, 6};
						Line(8) = {3, 7};
						Line(9) = {6, 7};
						Spline(10) = {8, ...
						

Adaptive Mesh Refinement

Adapting the precision of the numerical computation based on the requirements of a computation problem in specific areas.

Adaptive Mesh Refinement

Adapting the precision of the numerical computation based on the requirements of a computation problem in specific areas.

Hessian-based metric

Adaptive Mesh Refinement

Hessian-based metric

GRAPHNET

A framework for learning mesh-based simulations using graph neural networks (GNNs)

GRAPHNET

A framework for learning mesh-based simulations using graph neural networks (GNNs)

ADAPTNET

A framework for automated mesh generation and adaptation

Table of contents

1. Mesh Generation in Computational Science

2. Deep Neural Networks (DNNs) for mesh-based simulations

3. Graph Neural Networks (GNNs) for mesh-based simulations
4. MESHNET: GNN for mesh generation
5. GRAPHNET: GNN for mesh adaptation
6. ADAPTNET: Framework for automated mesh generation and adaptation

Deep Neural Networks for mesh-based simulations

Most work on predicting high-dimensional physical systems focuses on Cartesian grids, owing to the popularity and hardware support for CNN architectures.

Deep Neural Networks for mesh-based simulations

Most work on predicting high-dimensional physical systems focuses on Cartesian grids, owing to the popularity and hardware support for CNN architectures.

Zoomed-in view of the discrete structured C-mesh representation of an airfoil (S814) (in black) on a Cartesian grid (in red). The airfoil boundary points are in blue.

A signed distance function contour plotted for the S814 airfoil in a 150x150 Cartesian grid. The magnitude of the SDF values on the Cartesian grid equals the shortest distance to the airfoil. The airfoil boundary points are in white. [1]

Deep Neural Networks for mesh-based simulations

Most work on predicting high-dimensional physical systems focuses on Cartesian grids, owing to the popularity and hardware support for CNN architectures.

Signed-fixed distance

Prediction

Ground truth

Deep Neural Networks for mesh-based simulations

Most work on predicting high-dimensional physical systems focuses on Cartesian grids, owing to the popularity and hardware support for CNN architectures.

Signed-fixed distance

Prediction

Ground truth

This limitation significantly undermines the applicability of these surrogates to field-scale models with more complex geometries.

Table of contents

1. Mesh Generation in Computational Science
2. Deep Neural Networks (DNNs) for mesh-based simulations

3. Graph Neural Networks (GNNs) for mesh-based simulations

4. MESHNET: GNN for mesh generation
5. GRAPHNET: GNN for mesh adaptation
6. ADAPTNET: Framework for automated mesh generation and adaptation

Graph Neural Networks for mesh-based simulations

The key feature of GNNs is to represent unstructured simulation data as graphs with nodes and edges to achieve one of three tasks:

• Node classification: predicting unknown quantities for the nodes of the graph
• Link classification: predicting the existence of missing links between nodes
• Graph classification: predicting unknown for the entire graph

Message passing Graph Neural Networks

Whichever the task, the key idea in GNNs is to learn how to propagate information into nodes from their local neighbourhoods across interconnecting edges for all nodes in the graph.

An example graph used to illustrate node classification

Message passing Graph Neural Netwroks

An example graph used to illustrate node classification

• Message computation: for each node, we compute a message that it passes to neighboring nodes in the network.

\[ m_u^{(l)} = \textbf{MSG}^{(l)}(h_u^{(l-1)}) \]

Embedding h of node u at the previous level of the GNN are transformed by the message function MSG into message m.

Message passing Graph Neural Netwroks

An example graph used to illustrate node classification

• Message computation
• Aggregation: for each node, a function aggregates all of the messages received from neighbours.

\[ h_v^{(l)} = \textbf{AGG}_{u \in N(v)}^{(l)}(m_u^{(l)}) \]

Function used to aggregate the messages from neighboring nodes v for node u.

Message passing Graph Neural Netwroks

An example graph used to illustrate node classification

• Message computation


• Aggregation
• Updating: given the aggregated messages from neighbouring nodes, the embedding of each node is updated using a processor function.

\[ h_v^{(l)} = \textbf{Processor}^{(l)}(h_v^{(l)}) \]

Embedding of node v is updated using a processor function given the aggregated information from neighbouring nodes.

Message passing Graph Neural Netwroks

Each node has its own computational graph defined by its local neighborhood.

An example graph used to illustrate node classification

The general computational graph for the update of the blue node

MeshGraphNets (MGN)

Learning mesh-based simulation with Graph Networks [2]

MeshGraphNets (MGN)

Dataset

Simulations of incompressible Navier-Stokes flow trajectories over a circular cylinder over time.

• 1200 trajectories


• 600 time steps


• Different simulation domains

MeshGraphNets (MGN)

Data associated with each simulation

• Node type: a 5-dimensional one-hot vector corresponding to node location in fluid, wall, inflow, or outflow regions.
• Mesh topology: a each node contains the 2D position vector of its location in the two dimensional space that is being simulated.
• Node attributes: 2D velocity vector of the fluid, and the scalar pressure.

MeshGraphNets (MGN)

Graph signature in PyTorch Geometric (PyG)

MeshGraphNets (MGN)

Graph signature in PyTorch Geometric (PyG)

						'''
						For each node i in the graph with neighbor node j:
						x - (node features) ground truth 2D velocities &
						5D node type one-hot vector for all nodes [num_nodes x 7]
						edge_index - connectivity of the graph. [2 x num_edges]
						edge_attr - (edge features) 2D position vector between connecting nodes
						& 2-norm of the position vector. [num_edges x 3]
						y - (node outputs) ground truth 2D velocities at next time step.
						[num_nodes x 2]
						p - pressure scalar, used for validation [num_nodes x 1]
						cells and mesh_pos: these attributes contain no new information.
						'''
						Data(x=x, edge_index=edge_index, edge_attr=edge_attr, y=y, p=p,
															cells=cells, mesh_pos=mesh_pos)
						

MeshGraphNets (MGN)

Netwrok architecture: Encoding - Processing - Decoding

MeshGraphNets (MGN)

Netwrok architecture: Encoding - Processing - Decoding

1. Encoding

MeshGraphNets (MGN)

Netwrok architecture: Encoding - Processing - Decoding

2. Processing: Message Passing, Aggregation, Updating

MeshGraphNets (MGN)

Netwrok architecture: Encoding - Processing - Decoding

3. Decode

MeshGraphNets (MGN)

Learning mesh-based simulation with Graph Networks

Table of contents

1. Mesh Generation in Computational Science
2. Deep Neural Networks (DNNs) for mesh-based simulations
3. Graph Neural Networks (GNNs) for mesh-based simulations

4. MESHNET: GNN for mesh generation

5. GRAPHNET: GNN for mesh adaptation
6. ADAPTNET: Framework for automated mesh generation and adaptation

MESHNET: learning local metrics parameters

MESHNET: learning local metrics parameters

Dataset associated with each simulation

MESHNET: learning local metrics parameters

Graph signature in PyTorch Geometric (PyG)

MESHNET: learning local metrics parameters

Different datasets

MESHNET: learning local metrics parameters

Networks

MESHNET: learning local metrics parameters

MESHNET: learning local metrics parameters

Dataset	Duration
(1)	3h23min
(2)	3h25min
(3)	3h27min

MESHNET: learning local metrics parameters

MESHNET: learning local metrics parameters

MESHNET: learning local metrics parameters

Type	Nodes	Cells
ground truth	5387	10778
prediction	5191	10386

MESHNET: learning local metrics parameters

MESHNET: learning local metrics parameters

Type	Nodes	Cells
ground truth	3853	7710
prediction	3916	7836

Table of contents

1. Mesh Generation in Computational Science
2. Deep Neural Networks (DNNs) for mesh-based simulations
3. Graph Neural Networks (GNNs) for mesh-based simulations
4. MESHNET: GNN for mesh generation

5. GRAPHNET: GNN for mesh adaptation

6. ADAPTNET: Framework for automated mesh generation and adaptation

GRAPHNET: direct prediction of steady-state flow field

GRAPHNET: direct prediction of steady-state flow field

Stokes equation:

GRAPHNET: direct prediction of steady-state flow field

Dataset associated with each simulation

GRAPHNET: direct prediction of steady-state flow field

Graph signature in PyTorch Geometric (PyG)

GRAPHNET: direct prediction of steady-state flow field

Datasets

GRAPHNET: direct prediction of steady-state flow field

Networks

GRAPHNET: direct prediction of steady-state flow field

GRAPHNET: direct prediction of steady-state flow field

Dataset	Duration
(1)	19h04min
(2)	27h56min
(3)	25h36min

GRAPHNET: direct prediction of steady-state flow field

GRAPHNET: direct prediction of steady-state flow field

GRAPHNET: direct prediction of steady-state flow field

Type	Nodes	Cells
initial	4118	21143
ground truth	2845	11452
prediction	2838	11405

Table of contents

1. Mesh Generation in Computational Science
2. Deep Neural Networks (DNNs) for mesh-based simulations
3. Graph Neural Networks (GNNs) for mesh-based simulations
4. MESHNET: GNN for mesh generation
5. GRAPHNET: GNN for mesh adaptation

6. ADAPTNET: Framework for automated mesh generation and adaptation

ADAPTNET: automated mesh generation & adaptation pipeline

ADAPTNET: automated mesh generation & adaptation pipeline