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Introducing a microstructure-embedded autoencoder approach for reconstructing high-resolution solution field data from a reduced parametric space

Rasoul Najafi Koopas, Shahed Rezaei, Natalie Rauter, Richard Ostwald, Rolf Lammering

TL;DR

The paper tackles the computational burden of obtaining high-resolution solution fields for PDEs in heterogeneous microstructures by proposing a Microstructure-Embedded Autoencoder (MEA) that upscales low-fidelity results. MEA fuses multi-resolution conductivity maps into the decoder and uses a coarse-grid, physics-informed solver (FOL) to generate a low-fidelity field, which is then refined to a high-fidelity map, preserving sharp interfaces with high accuracy and reduced training data. Key contributions include the introduction of microstructure-informed decoder concatenation, a comparative study against interpolation, FFNN, and standard U-Net, and demonstrated reductions in computation time (up to 280x versus FEM) and data requirements, with robust performance on out-of-distribution cases. The approach is poised to augment neural operator frameworks and adaptable multi-fidelity workflows, with clear paths to 3D extensions and coupling to other solvers like FEM, FFT, or DeepOnet.

Abstract

In this study, we develop a novel multi-fidelity deep learning approach that transforms low-fidelity solution maps into high-fidelity ones by incorporating parametric space information into a standard autoencoder architecture. This method's integration of parametric space information significantly reduces the need for training data to effectively predict high-fidelity solutions from low-fidelity ones. In this study, we examine a two-dimensional steady-state heat transfer analysis within a highly heterogeneous materials microstructure. The heat conductivity coefficients for two different materials are condensed from a 101 x 101 grid to smaller grids. We then solve the boundary value problem on the coarsest grid using a pre-trained physics-informed neural operator network known as Finite Operator Learning (FOL). The resulting low-fidelity solution is subsequently upscaled back to a 101 x 101 grid using a newly designed enhanced autoencoder. The novelty of the developed enhanced autoencoder lies in the concatenation of heat conductivity maps of different resolutions to the decoder segment in distinct steps. Hence the developed algorithm is named microstructure-embedded autoencoder (MEA). We compare the MEA outcomes with those from finite element methods, the standard U-Net, and various other upscaling techniques, including interpolation functions and feedforward neural networks (FFNN). Our analysis shows that MEA outperforms these methods in terms of computational efficiency and error on test cases. As a result, the MEA serves as a potential supplement to neural operator networks, effectively upscaling low-fidelity solutions to high fidelity while preserving critical details often lost in traditional upscaling methods, particularly at sharp interfaces like those seen with interpolation.

Introducing a microstructure-embedded autoencoder approach for reconstructing high-resolution solution field data from a reduced parametric space

TL;DR

The paper tackles the computational burden of obtaining high-resolution solution fields for PDEs in heterogeneous microstructures by proposing a Microstructure-Embedded Autoencoder (MEA) that upscales low-fidelity results. MEA fuses multi-resolution conductivity maps into the decoder and uses a coarse-grid, physics-informed solver (FOL) to generate a low-fidelity field, which is then refined to a high-fidelity map, preserving sharp interfaces with high accuracy and reduced training data. Key contributions include the introduction of microstructure-informed decoder concatenation, a comparative study against interpolation, FFNN, and standard U-Net, and demonstrated reductions in computation time (up to 280x versus FEM) and data requirements, with robust performance on out-of-distribution cases. The approach is poised to augment neural operator frameworks and adaptable multi-fidelity workflows, with clear paths to 3D extensions and coupling to other solvers like FEM, FFT, or DeepOnet.

Abstract

In this study, we develop a novel multi-fidelity deep learning approach that transforms low-fidelity solution maps into high-fidelity ones by incorporating parametric space information into a standard autoencoder architecture. This method's integration of parametric space information significantly reduces the need for training data to effectively predict high-fidelity solutions from low-fidelity ones. In this study, we examine a two-dimensional steady-state heat transfer analysis within a highly heterogeneous materials microstructure. The heat conductivity coefficients for two different materials are condensed from a 101 x 101 grid to smaller grids. We then solve the boundary value problem on the coarsest grid using a pre-trained physics-informed neural operator network known as Finite Operator Learning (FOL). The resulting low-fidelity solution is subsequently upscaled back to a 101 x 101 grid using a newly designed enhanced autoencoder. The novelty of the developed enhanced autoencoder lies in the concatenation of heat conductivity maps of different resolutions to the decoder segment in distinct steps. Hence the developed algorithm is named microstructure-embedded autoencoder (MEA). We compare the MEA outcomes with those from finite element methods, the standard U-Net, and various other upscaling techniques, including interpolation functions and feedforward neural networks (FFNN). Our analysis shows that MEA outperforms these methods in terms of computational efficiency and error on test cases. As a result, the MEA serves as a potential supplement to neural operator networks, effectively upscaling low-fidelity solutions to high fidelity while preserving critical details often lost in traditional upscaling methods, particularly at sharp interfaces like those seen with interpolation.
Paper Structure (18 sections, 8 equations, 30 figures, 4 tables)

This paper contains 18 sections, 8 equations, 30 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem description
  3. Necessity of reducing parameter space
  4. Problem statement
  5. Finite Element Modeling
  6. Finite operator learning
  7. Sample generation
  8. Enhancing resolution of solution: A methodological Study
  9. Interpolation approach
  10. Feedforward neural network approach
  11. Standard U-Net
  12. Microstructure-embedded autoencoder (MEA)
  13. Results
  14. Hyperparameter analysis
  15. Quality of high-fidelity solutions from studied architectures
  16. ...and 3 more sections

Figures (30)

  • Figure 1: In the developed MEA architecture, the initial step involves condensing a high-resolution heat conductivity map into various lower-resolution maps. In the next step, the coarsest grid—defined in this study as an $11 \times 11$ grid—is utilized to solve the boundary value problem. For this task, one of several techniques, including the Finite Element Method (FEM), Finite Difference Method (FDM), Fast Fourier Transform (FFT), Finite Operator Learning (FOL), or Physics-Informed Neural Networks (PINNs), among others can be implemented. Following the resolution of the boundary value problem at this lower scale, the resultant low-resolution output undergoes an upscaling process through the use of an enhanced autoencoder.
  • Figure 2: Condensation of high-resolution parametric space into lower-resolution parametric spaces using the MaxPool function from the SciPy library.
  • Figure 3: Description of problem setup, boundary conditions, and finite element meshes. The information of the domain is conveyed through a finite number of sensor points or nodes.
  • Figure 4: Network architecture for finite operator learning, where information about the input parameter at each grid point goes in and the The output layer is evaluated by training the network based on the discretized weak form. By satisfying the residual or energy form of the problem through the optimization process in the deep learning model, physical outcomes or solutions are obtained for a parametric input space.
  • Figure 5: Randomly selected training samples from a dataset of 5,670 illustrate the spatial conductivity maps of two-phase materials within the training set.
  • ...and 25 more figures