[Paper Review] Deep learning-based reduced-order methods for fast transient dynamics

Martina Cracco, Giovanni Stabile+6 more|arXiv (Cornell University)|Dec 15, 2022

Model Reduction and Neural Networks4 citations

TL;DR

This paper proposes a non-intrusive, deep learning-enhanced reduced order model (POD-DL) for fast, real-time simulation of transient blast waves in urban environments. By combining time-domain partitioning, Proper Orthogonal Decomposition (POD), autoencoders for non-linear dimensionality reduction, and deep forward neural networks to map parameters and time to latent space, the method achieves high accuracy in reconstructing pressure fields and key blast metrics like peak overpressure and impulse, outperforming standard POD for non-linear, fast-transient problems.

ABSTRACT

In recent years, large-scale numerical simulations played an essential role in estimating the effects of explosion events in urban environments, for the purpose of ensuring the security and safety of cities. Such simulations are computationally expensive and, often, the time taken for one single computation is large and does not permit parametric studies. The aim of this work is therefore to facilitate real-time and multi-query calculations by employing a non-intrusive Reduced Order Method (ROM). We propose a deep learning-based (DL) ROM scheme able to deal with fast transient dynamics. In the case of blast waves, the parametrised PDEs are time-dependent and non-linear. For such problems, the Proper Orthogonal Decomposition (POD), which relies on a linear superposition of modes, cannot approximate the solutions efficiently. The piecewise POD-DL scheme developed here is a local ROM based on time-domain partitioning and a first dimensionality reduction obtained through the POD. Autoencoders are used as a second and non-linear dimensionality reduction. The latent space obtained is then reconstructed from the time and parameter space through deep forward neural networks. The proposed scheme is applied to an example consisting of a blast wave propagating in air and impacting on the outside of a building. The efficiency of the deep learning-based ROM in approximating the time-dependent pressure field is shown.

Motivation & Objective

• Address the computational infeasibility of full-order simulations for real-time and multi-query parametric studies of blast waves in urban environments.
• Overcome the limitations of traditional Proper Orthogonal Decomposition (POD) in capturing fast, non-linear transient dynamics due to slow singular value decay.
• Develop a non-intrusive, data-driven reduced order model (ROM) that enables fast, accurate prediction of pressure fields and blast metrics for new parameter and time values.
• Enable real-time or near real-time assessment of blast wave effects on buildings and human safety, supporting emergency response planning.
• Demonstrate the effectiveness of combining POD with autoencoders and deep neural networks for efficient, high-fidelity approximation of complex, transient fluid dynamics.

Proposed method

• Partition the time domain into sub-regions to apply a local reduced order model, improving approximation of fast-transient dynamics.
• Construct a local reduced basis using Proper Orthogonal Decomposition (POD) with a large number of modes (N) to capture solution variability.
• Apply autoencoders to perform non-linear dimensionality reduction, mapping the high-dimensional POD space to a lower-dimensional latent space of dimension n.
• Train deep forward neural networks (DFNNs) to learn the mapping from the time and parameter space to the latent space representation.
• Reconstruct the full pressure field from the latent space using the inverse of the autoencoder and the POD basis functions.
• Use high-fidelity snapshots from the EUROPLEXUS solver as training data for the entire ROM pipeline.

Experimental results

Research questions

• RQ1 Can a non-intrusive, data-driven ROM based on deep learning effectively approximate fast, non-linear transient blast wave dynamics where standard POD fails?
• RQ2 How does the combination of time-domain partitioning, POD, autoencoders, and deep neural networks improve reconstruction accuracy compared to POD alone?
• RQ3 To what extent can the proposed piecewise POD-DL method predict key blast metrics such as peak overpressure and impulse with high fidelity?
• RQ4 How do the errors in the reconstructed pressure field vary with respect to the number of POD modes (N) and the latent dimension (n)?
• RQ5 Can the method achieve real-time or near real-time performance for large-scale urban blast simulations, even when the full-order model is computationally prohibitive?

Key findings

• The piecewise POD-DL method significantly outperforms standard POD in reconstructing the time-dependent pressure field, especially in regions of high gradients and fast transients.
• Errors in the $L^2$ and $L^∞$ norms are highest at early times (t ≈ 0), where a high-pressure bubble forms in a small region, challenging the POD approximation even with many modes.
• Error decreases with increasing number of POD modes (N) when the latent dimension (n) is fixed, and also with increasing latent dimension (n) when N is fixed, up to a saturation point.
• At approximately n = 20, the error flattens due to the regression capacity of the deep forward neural network, indicating a practical limit in model accuracy for the current architecture.
• The computational cost of a single ROM prediction remains low (~0.2s per time step), even for large-scale problems, making the method suitable for real-time applications.
• The method enables accurate approximation of critical blast metrics such as peak overpressure and impulse at the final time, which are essential for structural safety and risk assessment.