Source-linked AI summary
High-Fidelity Digital Twin Data Models by Randomized Dynamic Mode Decomposition and Deep Learning with Applications in Fluid Dynamics
Diana A. Bistrian
TL;DR
The paper addresses the difficulty of modeling non-intrusive fluid-dynamics data and identifying faithful reduced-order models for rapidly changing shock-wave phenomena. It combines randomized dynamic mode decomposition with neural-network nonlinear autoregressive estimators, producing digital twin data models that match source dynamics with reduced complexity and computational costs.
Problem
Modeling discrete non-intrusive fluid-dynamics data is difficult, while POD applicability to complex systems is limited mainly by truncation errors.
Method
The framework combines adaptive randomized dynamic mode decomposition for offline reduced-basis construction with neural-network nonlinear autoregressive estimators for online digital twin identification.
Results
The digital twin data models produce outputs consistent with original source data while enabling high-accuracy dynamics mapping with reduced CPU-time and hardware costs across three shock-wave phenomena.
Takeaways & Limitations
The proposed non-intrusive framework provides reduced-complexity digital twin data models for accurately reproducing rapidly changing dynamics.
Takeaways & Limitations
POD applicability to complex systems is limited mainly by errors associated with truncating the POD modes.
Abstract
from arXiv · showhide
The purpose of this paper is the identification of high-fidelity digital twin data models from numerical code outputs by non-intrusive techniques (i.e., not requiring Galerkin projection of the governing equations onto the reduced modes basis). In this paper the author defines the concept of the digital twin data model (DTM) as a model of reduced complexity that has the main feature of mirroring the original process behavior. The significant advantage of a DTM is to reproduce the dynamics with high accuracy and reduced costs in CPU time and hardware for settings difficult to explore because of the complexity of the dynamics over time. This paper introduces a new framework for creating efficient digital twin data models by combining two state-of-the-art tools: randomized dynamic mode decomposition and deep learning artificial intelligence. It is shown that the outputs are consistent with the original source data with the advantage of reduced complexity. The DTMs are investigated in the numerical simulation of three shock wave phenomena with increasing complexity. The author performs a thorough assessment of the performance of the new digital twin data models in terms of numerical accuracy and computational efficiency.
1 Introduction
The paper addresses reduced-complexity modeling of complex, non-intrusive fluid data and introduces DTMs that mirror process behavior with high accuracy and lower computational cost.
- 1 Introduction: POD and DMD are widely used modal-decomposition methods because they connect flow dynamics with spatial patterns.DMD additionally links spatial structures to individual frequencies and growth or decay rates.
- 1 Introduction: POD applicability to complex systems is limited by errors associated with truncating its modes.
- 1 Introduction: Non-intrusive fluid-data modeling is difficult when discrete data lack an associated mathematical model.
- 1 Introduction: DTMs are reduced-complexity models intended to mirror original process behavior while reducing CPU-time and hardware costs.The framework targets rapidly changing dynamics and timescales difficult to explore directly.
- 1 Introduction: The framework combines randomized dynamic mode decomposition with deep-learning artificial intelligence to identify efficient DTMs from non-intrusive data.
2 Shock Wave Phenomena: Full-order Model of Nonlinear Viscous Burgers Equation
The paper uses the nonlinear viscous Burgers equation as a full-order test problem for three shock-wave regimes whose oscillatory complexity increases with Reynolds number.
- 2 Shock Wave Phenomena: Full-order Model of Nonlinear Viscous Burgers Equation: The model uses viscosity ν = 1/Re and a discontinuous initial condition, producing a shock-wave phenomenon.
- 2 Shock Wave Phenomena: Full-order Model of Nonlinear Viscous Burgers Equation: The initial-value problem is solved by finite-difference discretization in conservative form followed by a parabolic integration scheme.
- 2 Shock Wave Phenomena: Full-order Model of Nonlinear Viscous Burgers Equation: Training data contain 301 regularly spaced time snapshots with Δt = 0.01 and 101 spatial measurements per snapshot.
- 2 Shock Wave Phenomena: Full-order Model of Nonlinear Viscous Burgers Equation: Three Burgers-equation shock-wave experiments use Re = 10^2, Re = 10^3, and Re = 10^4, with increasing Reynolds number producing more oscillations and complexity.The exact solutions are shown for the three Reynolds-number cases.
- 2 Shock Wave Phenomena: Full-order Model of Nonlinear Viscous Burgers Equation: The study seeks a low-complexity reduced-order model that faithfully approximates the nonlinear viscous Burgers equation and forms a digital twin data model.
3 Reduced order modeling based on Dynamic Mode Decomposition
The DMD reduction constructs a propagator approximation from time-shifted snapshots, decomposes dynamics into modes with temporal characteristics, and retains leading modes for ROM reconstruction.
- 3 Reduced order modeling based on Dynamic Mode Decomposition: DMD forms time-shifted snapshot matrices and approximates a propagator mapping each state snapshot to the next.The snapshot sequence is modeled through relations such as u_i = A u_{i−1}.
- 3 Reduced order modeling based on Dynamic Mode Decomposition: SVD-based DMD is used as an alternative to the Moore–Penrose approach, particularly when the snapshot matrix is rank deficient.The SVD separates proper orthogonal modes, singular values, and the conjugate-transpose factor.
- 3 Reduced order modeling based on Dynamic Mode Decomposition: Reducing the residual improves convergence of the companion-matrix eigenvalues and eigenvectors toward Koopman-operator quantities.
- 3 Reduced order modeling based on Dynamic Mode Decomposition: DMD modes represent snapshots through spatial modes weighted by amplitudes and complex temporal factors, with eigenvalues encoding growth rates and frequencies.
- 3 Reduced order modeling based on Dynamic Mode Decomposition: The reduced model retains selected leading modes rather than all computed modes, because weak-contribution modes are unnecessary for the targeted representation.The number of retained modes is denoted NDMD and is selected algorithmically rather than simply taking the first modes.
4 Offline stage: Randomized Dynamic Mode Decomposition
The offline stage uses randomized singular value decomposition within DMD to identify a low-dimensional dynamical representation while reducing computational cost. It selects modes through an optimization-based rank determination and targets accurate reconstruction without additional mode-selection procedures.
- Randomized DMD: Randomized DMD reduces the problem dimension and avoids computationally expensive SVD while retaining a reduced-order subspace of Ritz values.The randomized procedure aims to improve reduced-order linear-model accuracy and produces a subspace whose dimension matches the randomized SVD rank.
- Rank selection: The optimal ROM rank is determined by minimizing the reconstruction error over the temporal interval using a constrained optimization problem.The error is defined with a time average and the L2 norm, and the selected rank minimizes the low-rank model error.
- Model construction: Unlike POD, DMD generally produces nonorthogonal modes, so DMD-based ROMs may require closure or regularization in projection-based applications.The paper proposes a randomized DMD variant augmented with deep learning as an alternative framework for the reduced-order model.
- Randomized DMD: The randomized DMD pipeline forms a rank-k randomized SVD, solves for reduced dynamics and eigenvalues, computes dynamic modes and amplitudes, and reconstructs the DMD model.The workflow uses randomized sampling, orthonormalization, projection, low-dimensional SVD, eigenvalue decomposition, and mode-amplitude calculation.
- k-RSVD: The k-RSVD procedure uses a random test matrix, constructs and orthonormalizes a sampling matrix, projects snapshots into a smaller space, and computes an economy-size SVD.The procedure then computes right singular vectors from the reduced decomposition.
5 Online stage: Fast Digital Twin Data Model Identification Using Deep Learning Nonlinear Autoregressive Estimators
The online stage identifies a reduced-order digital twin model by learning the temporal coefficients of the DMD-ROM with nonlinear autoregressive estimators. NLARX models use delayed inputs and outputs, with a cascade-forward neural network serving as the nonlinear estimator.
- DTM identification: The online DTM represents the reduced dynamics through DMD modes and estimated temporal coefficients obtained from a nonlinear estimator.The preceding algorithm supplies leading dynamic modes and discrete temporal coefficients, which the online model approximates.
- NLARX formulation: NLARX predicts each current output from delayed outputs and delayed inputs through a nonlinear function plus modeling error.The parameters include the numbers of past outputs and inputs and the pure input delay.
- NLARX formulation: The NLARX structure feeds previous network outputs back into the input layer and lets users choose input and output delays for representing system dynamics.Training modifies the parameter vector over the complete trajectory to minimize the objective in Eq. (33).
- Neural estimator: The nonlinear estimator is implemented as a cascade-forward neural network with 10 hidden layer sizes.Figure 3 presents this network as the nonlinear estimator used for the NLARX models.
6 Numerical Results: Computational Efficiency of the Algorithm
The randomized DMD–NLARX framework identifies low-rank digital twin models for three increasingly complex Burgers shock-wave cases, with low errors, strong correlations, and low computational cost.
- Model-order selection: Randomized DMD identifies an optimal low-rank subspace and selects the most influential dynamic modes through global optimization of the target rank.Genetic-algorithm/sequential-quadratic-programming and simulated-annealing approaches required similar computational effort.
- Model-order selection: Target-rank evaluation compares relative error and correlation versus retained modes for Re = 10^2, 10^3, and 10^4, selecting 15, 20, and 20 leading modes.The three cases correspond to Figures 4–6 and the randomized-DMD results are summarized using 300 snapshots in Table 1.
- Temporal modeling: DMD coefficients are supplied as inputs to NLARX estimators to estimate the reduced model’s temporal coefficients over the entire time window.The estimator configurations for the three cases are reported in Tables 2–4.
- Overall results: Low errors and very good correlation coefficients across all three test cases confirm the digital twin models’ computational efficiency.Solutions are illustrated in Figures 10–12, while the error and correlation values are reported in Tables 2–4.
- Computational efficiency: Offline CPU time stays below 2 seconds, while online estimation requires 8–17 seconds and increases for very high Reynolds number.The higher online cost at Re = 104 is attributed to the NLARX estimator’s temporal-behavior estimation.
7 Conclusions
The paper identifies high-fidelity digital twin data models non-intrusively by combining adaptive randomized dynamic mode decomposition with neural-network nonlinear autoregressive estimators. Across three shock-wave experiments with Reynolds numbers from 10^2 to 10^4, the models reproduce the dynamics accurately while reducing computational costs.
- The approach identifies high-fidelity digital twin models from numerical outputs using non-intrusive techniques that do not require Galerkin projection onto the reduced-mode basis.
- The framework combines adaptive randomized dynamic mode decomposition for the offline reduced basis with neural-network nonlinear autoregressive estimators for online temporal-coefficient estimation.
- The models were evaluated on three shock-wave phenomena of increasing complexity, with Reynolds numbers ranging from 10^2 to 10^4.
- The proposed digital twin data models map the dynamics with high accuracy and reduced CPU-time and hardware costs, including for rapidly changing timescales.
- Online estimation of the temporal coefficients produced fast and accurate digital twin identification, supported by the computational-efficiency assessment and qualitative analysis across the three experiments.