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

arXiv: https://arxiv.org/abs/2405.01975 · HTML: https://mlsanity.com/paper/2405.01975/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 42 references that are in the map.
- First posted on arXiv 3 May 2024.

## Builds on (most cited first, 14 of 14)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Survey of multifidelity methods in uncertainty propagation, inference, and optimization](https://mlsanity.com/paper/1806.10761/) (2018)
- [Transfer learning based multi-fidelity physics informed deep neural network](https://mlsanity.com/paper/2005.10614/) (2020)
- [Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics](https://mlsanity.com/paper/2205.03990/) (2022)
- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [Learning in latent spaces improves the predictive accuracy of deep neural operators](https://mlsanity.com/paper/2304.07599/) (2023)
- [Analysis of three dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis](https://mlsanity.com/paper/2010.12060/) (2020)
- [A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformations](https://mlsanity.com/paper/2404.00074/) (2024)
- [Multi-fidelity surrogate modeling for temperature field prediction using deep convolution neural network](https://mlsanity.com/paper/2301.06674/) (2023)
- [A Data-Driven Approach to Full-Field Damage and Failure Pattern Prediction in Microstructure-Dependent Composites using Deep Learning](https://mlsanity.com/paper/2104.04485/) (2021)
- [Integration of physics-informed operator learning and finite element method for parametric learning of partial differential equations](https://mlsanity.com/paper/2401.02363/) (2024)

## Built on by (most cited first, 3 of 3)

- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries](https://mlsanity.com/paper/2504.02459/) (2025)
- [Multi-fidelity aerodynamic data fusion by autoencoder transfer learning](https://mlsanity.com/paper/2512.13069/) (2025)

## Abstract (arXiv)

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.