NeuROM-Py 3.1.18

NeuROM, a NN-PGD architecture based on the HiDeNN Framework (Finite Element Neural Network Interpolation)

NeuROM, a NN-PGD architecture based on the HiDeNN Framework

NeuROM provides an interpretable deep neural network architecture that uses tensor decomposition to give a parametric reduced-order model. This network interpretability comes from the HiDeNN architecture that offers an interpolation framework through a deep neural network in which weights and biases are constrained so that the interpolation matches a finite element interpolation (P1 or P2). The first hidden layers play the role of the shape functions, while the last layer, called the interpolation layer in the remainder of the document, utilises the output of the shape functions to interpolate the output. Training the weights of that last hidden layer is the same as solving a FEM problem on a fixed mesh. The weights of the interpolation layer directly correspond to the nodal values associated with each shape function. Therefore, prescribing Dirichlet boundary conditions is straightforward by freezing the weights associated with the prescribed values of fixed DoFs. However, learning the parameters related to the first layers accounts for mesh adaptation.

This code implements a Finite Element Neural Network Interpolation (FENNI) based on the HiDeNN framework. The layer's input is the coordinate $\underline{x}$ where the output is evaluated and the value of the parameters $\underline{\mu}$ for which the output is computed. In this case, the output of the network is the displacement $\underline{u}\left(\underline{x},\underline{\mu}\right)$

Using the code

Executing the code is as easy as running

neurom -cf Configurations/your_config.toml where the option -cf stands for configuration file and allow the user to specify which configuration file should be used.

The code must be lauched from a folder containing two subfolders

• Configurations
  • where one or more configuration files are stored
• Geometries
  • where the gmsh .geo files are stored

The folder from which the code is launched should have the following architecture.

.
│
├── Geometries
│   └── Geometry.geo
└── Configurations
    └── your_config.toml

Repository's structure

.
├── neurom
│       ├── HiDeNN_PDE.py
|       |
│       ├── src
│       |   ├── PDE_Library.py
│       |   ├── Pre_processing.py
│       |   └── Training.py
|       |
|       ├── Post
|       │   └── Plots.py
|       |
│       └── main.py
│   
├── Geometries
│   ├── Beam.geo
│   ├── Hole_3.geo
│   └── Hole.geo
│
│
├── Illustrations
│   └── NeuROM_1D.gif
|
│
├── README.md
│
└── TrainedModels
    | 
    └── 2D_Bi_Parameters_el_0.2
 

Architecture of the NN

There are two levels of modelling involved. The space interpolation in itself can be trained to get a solution for a given problem. Such interpolation is achieved using the model MeshNN. Building on that interpolation model and using a similar interpolation for the parametric space, a reduced-order model can also be trained using the NeuROM class.

Data entry

A config file needs to be filled with the job parameters (name of the geometry, solvers' parameters, post-precessing required, etc.) A default file is specified at the top of the main.py script, but any other file can be passed as an argument when running the main script as neurom -cf 'Configuration/config.toml'.

Pre-processing

Mesh class that builds the Geometry and the Mesh_object from the Inputs:

• config["geometry"]["Name"]:
• config["interpolation"]["order"],
• config["interpolation"]["dimension"]

The method .AddBCs then allows the definition of Dirichlet boundary conditions and specifies the volume element of the structure that should be considered. Inputs:

• List of BCs (Physical surface, value, normal direction)
• volume index

Then the methods

• .MeshGeo()
• .ReadMesh()
• .AssemblyMatrix() (if 1D) respectively mesh the geometry (if the mesh does not already exist), parse the mesh file, assemble the matrix later used for the assembly of each shape functions

Space interpolation

Given a mesh object created using the Mesh class, MeshNN gives a space interpolation of the quantity of interest using first or second-order FE shape functions.

• MeshNN class that, given a mesh, "assemble" the shape functions accordingly
  • model = MeshNN_2D(Mesh_object) (or model = MeshNN_1D(Mesh_object) in 1D) Creates an interpolation model
  • Mesh_object being the previously created mesh

Reduced-order modelling

Given a hypercube ParameterHypercube defining the parametric space, the Space dirichlet boundary conditions a mesh and the maximum number of requested modes, a reduced-order model of the parametric field is built

ROM_model = NeuROM(Mesh_object,ParameterHypercube,ConfigFile)

Illustration of the ROM parametric solution

Training the NN

The Volumic forces are accounted for in the loss function through the right-hand side (RHS) function, and the loss function is the potential energy.

The trainable parameters can be changed on the fly.

• model.Freeze_Mesh() Freezes the mesh so that only the nodal values are trained

• model.UnFreeze_Mesh() Unfreezes the mesh so that the coordinates values can be trained

• model.Freeze_FEM() Freezes the nodal values so that only the coordinates are trained

• model.UnFreeze_FEM() Unfreezes the nodal so that FEM problem can be solved

Reduced-order modelling

The Volumic forces are accounted for in the loss function through the right-hand side (RHS) function, and the loss function is the potential energy.

The trainable parameters can be changed on the fly.

• ROM_model.Freeze_Mesh() Freezes the space mesh so that only the nodal values are trained

• ROM_model.UnFreeze_Mesh() Unfreezes the space mesh so that the coordinates values can be trained

• ROM_model.Freeze_Space() Freezes the space nodal values so that only the coordinates are trained

• ROM_model.UnFreeze_Space() Unfreezes the space nodal so that FEM problem can be solved

• ROM_model.Freeze_MeshPara() Freezes the parametric mesh so that only the nodal values are trained

• ROM_model.UnFreeze_MeshPara() Unfreezes the parametric mesh so that the coordinates values can be trained

• ROM_model.Freeze_Para() Freezes the parametric nodal values so that only the coordinates are trained

• ROM_model.UnFreeze_Para() Unfreezes the parametric nodal so that FEM problem can be solved

Licensing

Copyright (C) 2024, UMR 7649, École Polytechnique, IPP, CNRS, MΞDISIM Team, INRIA

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see https://www.gnu.org/licenses/.