[Paper Review] An autoencoder-based reduced-order model for eigenvalue problems with application to neutron diffusion

Toby R. F. Phillips, Claire E. Heaney+2 more|arXiv (Cornell University)|Aug 15, 2020

Model Reduction and Neural Networks62 references67 citations

TL;DR

This paper proposes a novel projection-based reduced-order model (ROM) for eigenvalue problems in neutron diffusion using an autoencoder for nonlinear dimensionality reduction, outperforming traditional Proper Orthogonal Decomposition (POD) methods. The hybrid SVD-autoencoder approach reduces input complexity, enabling higher accuracy—especially in k-eff prediction—achieving errors an order of magnitude lower than POD and standard autoencoder ROMs in 2D test cases.

ABSTRACT

Using an autoencoder for dimensionality reduction, this paper presents a novel projection-based reduced-order model for eigenvalue problems. Reduced-order modelling relies on finding suitable basis functions which define a low-dimensional space in which a high-dimensional system is approximated. Proper orthogonal decomposition (POD) and singular value decomposition (SVD) are often used for this purpose and yield an optimal linear subspace. Autoencoders provide a nonlinear alternative to POD/SVD, that may capture, more efficiently, features or patterns in the high-fidelity model results. Reduced-order models based on an autoencoder and a novel hybrid SVD-autoencoder are developed. These methods are compared with the standard POD-Galerkin approach and are applied to two test cases taken from the field of nuclear reactor physics.

Motivation & Objective

• . To develop a projection-based reduced-order model using autoencoders for nonlinear dimensionality reduction in eigenvalue problems.
• . To compare autoencoder-based ROMs with standard POD-Galerkin methods in terms of accuracy and computational efficiency.
• . To introduce and evaluate a novel hybrid SVD-autoencoder method that reduces input dimensionality and improves model robustness.
• . To assess performance on realistic reactor physics test cases involving neutron diffusion with varying material properties.
• . To demonstrate that nonlinear autoencoders can capture complex solution features more efficiently than linear POD in reduced-order modeling.

Proposed method

• . Uses an autoencoder to learn a nonlinear low-dimensional latent space from high-fidelity model (HFM) snapshots, replacing linear POD/SVD for basis function generation.
• . Implements a hybrid SVD-autoencoder: SVD is applied to snapshots first to reduce dimensionality before feeding into the autoencoder, lowering network complexity.
• . Projects the full governing equations (neutron diffusion) onto the learned latent space via Galerkin projection, forming a reduced system of equations.
• . Solves the reduced system for both seen and unseen parameter values, with online evaluation using the trained autoencoder and reduced matrices.
• . Compares results with standard POD-Galerkin ROMs and standard autoencoder-based ROMs using metrics like compression error, flux profile error, and k-eff error.
• . Employs a two-stage offline process: (1) HFM solves for training parameters, (2) autoencoder and SVD are trained on snapshots to generate basis functions and latent representations.

Experimental results

Research questions

• RQ1 . Can an autoencoder-based ROM achieve better accuracy than POD-Galerkin ROMs in solving eigenvalue problems for neutron diffusion?
• RQ2 . How does the performance of a hybrid SVD-autoencoder compare to standard autoencoder and POD-based ROMs in terms of compression and solution error?
• RQ3 . Does the nonlinear embedding of an autoencoder capture solution features more efficiently than linear POD, especially for complex or nonlinear systems?
• RQ4 . Can the hybrid SVD-autoencoder reduce overfitting and improve generalization to unseen parameter values compared to standard autoencoder ROMs?
• RQ5 . What is the impact of reduced input dimensionality (via SVD pre-processing) on the accuracy and training stability of the autoencoder in the context of model reduction?

Key findings

• . The hybrid SVD-autoencoder-based ROM achieved the lowest k-eff error—1.7117 × 10⁻⁴—for unseen data in the 2D test case, an order of magnitude lower than both POD and standard autoencoder ROMs.
• . For the 2D test case, the SVD-autoencoder ROM reduced k-eff error by a factor of 10 compared to the POD-based ROM, which had an error of 1.9947 × 10⁻³.
• . The SVD-autoencoder ROM showed consistently lower compression and flux profile errors across both seen and unseen cases, with compression error of 1.7734 × 10⁻² compared to 5.7756 × 10⁻² for POD.
• . In the 1D test case, the autoencoder-based ROM reduced k-eff error by one order of magnitude compared to the POD-based ROM, demonstrating superior information capture per latent variable.
• . The hybrid SVD-autoencoder approach reduced input dimensionality to the autoencoder, decreasing network complexity and likelihood of overfitting, leading to improved generalization.
• . The latent variables generated by the SVD-autoencoder spanned a stable and informative range across training data, indicating robust and meaningful low-dimensional representation of the solution manifold.