Reduced Order Modeling with Shallow Recurrent Decoder Networks

TL;DR

This work introduces SHRED-ROM, a decoding-only reduced order modeling framework that reconstructs high-dimensional fields from sparse sensor histories by combining a temporal LSTM encoder with a shallow latent-to-state decoder. Grounded in a separation-of-variables philosophy and supported by compressive training via POD or spherical harmonics, SHRED-ROM can handle parametric and chaotic dynamics without requiring encoder inversions, with online reconstruction $\hat{{\bf u}}_k^{\boldsymbol{\mu}}={\boldsymbol{\Psi}}{\bf f}_X({\bf f}_T({\bf s}_{k-L}^{\boldsymbol{\mu}},...,{\bf s}_k^{\boldsymbol{\mu}}))$. Across five challenging tests—from SWE on a sphere to GoPro videos and complex fluid-structure flows—it achieves accurate reconstructions for unseen parameters and sensor configurations with modest training and hardware demands, and demonstrates robustness to sensor placement and parametric variation. The approach offers fast, real-time capable reconstructions and parameter estimation, with potential extensions to forecasting and uncertainty quantification.

Abstract

Reduced Order Modeling is of paramount importance for efficiently inferring high-dimensional spatio-temporal fields in parametric contexts, enabling computationally tractable parametric analyses, uncertainty quantification and control. However, conventional dimensionality reduction techniques are typically limited to known and constant parameters, inefficient for nonlinear and chaotic dynamics, and uninformed to the actual system behavior. In this work, we propose sensor-driven SHallow REcurrent Decoder networks for Reduced Order Modeling (SHRED-ROM). Specifically, we consider the composition of a long short-term memory network, which encodes the temporal dynamics of limited sensor data in multiple scenarios, and a shallow decoder, which reconstructs the corresponding high-dimensional states. SHRED-ROM is a robust decoding-only strategy that circumvents the numerically unstable approximation of an inverse which is required by encoding-decoding schemes. To enhance computational efficiency and memory usage, the full-order state snapshots are reduced by, e.g., proper orthogonal decomposition, allowing for compressive training of the networks with minimal hyperparameter tuning. Through applications on chaotic and nonlinear fluid dynamics, we show that SHRED-ROM (i) accurately reconstructs the state dynamics for new parameter values starting from limited fixed or mobile sensors, independently on sensor placement, (ii) can cope with both physical, geometrical and time-dependent parametric dependencies, while being agnostic to their actual values, (iii) can accurately estimate unknown parameters, and (iv) can deal with different data sources, such as high-fidelity simulations, coupled fields and videos.

Figures (7)

• Figure 1: Graphical summary of the SHallow REcurrent Decoder-based Reduced Order Model (SHRED-ROM). Sparse sensor values ${\bf s}_{k}^{\boldsymbol{\mu}_i}$ over time windows of length $L$ in multiple scenarios are encoded through a long short-term memory (LSTM), while a shallow decoder network (SDN) projects the resulting latent representation in the high-dimensional state space. Full-order state snapshots ${\bf u}_k^{\boldsymbol{\mu}_i}$ are reduced by proper orthogonal decomposition (POD), allowing for compressive training at the POD level. After training, in the online phase, it is possible to reconstruct high-dimensional state trajectories $\hat{{\bf u}}_k^{\boldsymbol{\mu}}$ for new parameters $\boldsymbol{\mu}$.
• Figure 2: Shallow Water Equations. Ground truth (first row), SHRED-ROM reconstructions with POD-based compressive training (second row) and SHRED-ROM reconstructions with spherical harmonics-based compressive training (third row). The following test cases are considered: height reconstruction from $3$ fixed height sensors (first column); velocity reconstruction from $3$ fixed height sensors (second column); velocity reconstruction from the coordinates of $1$ mobile sensor at $t=11$ hours (third column) and $t=957$ hours (fourth column). The sparse sensors exploited by SHRED-ROM are depicted with magenta dots.
• Figure 3: GoPro Physics. Original videos (first row), preprocessed data (second row) and SHRED-ROM reconstructions from $3$ fixed pixels in the cylinder wake (first column). The following test cases are considered: symmetric shedding reconstruction of the $400$th (first column) and the $800$th frame (second column); alternating symmetric shedding reconstruction of the $269$th (third column) and the $1002$nd frame (fourth column). The sparse pixels exploited by SHRED-ROM are depicted with yellow dots.
• Figure 4: Kuramoto-Sivashinsky. (a) Ground truth (first row) and SHRED-ROM reconstructions (second row). The following test cases are considered: state reconstruction from $2$ fixed sensors with $\boldsymbol{\mu} = [1.27, 3.71]^{\top}$ (first column), $\boldsymbol{\mu} = [1.18, 4.34]^{\top}$ (second column) and $\boldsymbol{\mu} = [1.33, 1.35]^{\top}$ (third column). (b) Relative reconstruction errors committed by SHRED-ROM on test data for different lag values (first column) and different number of sensors (second column), while considering different sensor placements.
• Figure 5: Fluidic Pinball. (a) Ground truth (first row) and SHRED-ROM reconstructions (second row). The following test cases are considered: state reconstruction from $3$ fixed sensors with $\boldsymbol{\mu} = [-1.38, -4.56, -2.54]^{\top}$ at $t=1$ seconds (first column) and $t=3$ seconds (second column); state reconstruction from the coordinates of $1$ mobile sensor with $\boldsymbol{\mu} = [-1.42, 4.65, 2.68]^{\top}$ at $t=1$ seconds (third column) and $t=3$ seconds (fourth column). (b) Relative reconstruction errors committed by SHRED-ROM on test data for different lag values (first column) and different number of sensors (second column), while considering different sensor placements. The sparse sensors exploited by SHRED-ROM are depicted with magenta dots.

Theorems & Definitions (1)

• Example 1