| name | equation-free-digital-twins |
| description | Equation-free digital twin framework using Koopman operator theory and Hankel-matrix embeddings for real-time structural state reconstruction without physical models. Use when: (1) building digital twins for complex engineering structures, (2) virtual sensing from partial observations, (3) Koopman-based system identification, (4) real-time monitoring of nonlinear structural dynamics.
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Equation-Free Digital Twins via Koopman-Hankel Framework
Core Methodology (arXiv:2605.00950)
Build digital twins for high-dimensional engineering structures using Koopman operator theory,
Hankel-matrix embeddings, and dynamic mode decomposition (DMD) — no mass/stiffness matrices required.
Key Concepts
Koopman Operator Theory
- Lifts nonlinear dynamics into a linear invariant subspace
- Enables linear analysis tools on inherently nonlinear systems
- Infinite-dimensional operator approximated via finite-dimensional projection
Hankel-Matrix Embedding
- Constructs trajectory-based state-space from input-output data
- Captures system dynamics through time-delayed observations
- Rank-optimized to separate physical modes from noise/harmonics
Virtual Sensing
- Reconstruct unmeasured states from limited sensor data
- Rolling-horizon strategy for real-time estimation
- Achieves R-squared > 0.95 at 1 Hz, > 0.99 at higher rates
Workflow
Step 1: Collect Operational Data
Gather time-series from available sensors. No input measurements needed (input-blind).
Step 2: Build Hankel Matrix
Construct trajectory-based Hankel matrix from time-delayed sensor observations.
Use past/future windows to capture system dynamics.
Step 3: Koopman-Hankel Decomposition
- SVD of Hankel matrix: H = U * S * V^T
- Truncate to dominant modes via rank optimization (gap statistic)
- Extract Koopman eigenvalues and modes
- Separate structural resonances from deterministic harmonics (e.g., 3P rotor)
Step 4: Virtual Sensing (Rolling-Horizon)
Project partial observations into Koopman subspace, evolve using Koopman dynamics,
map back to physical coordinates for full-state reconstruction.
Step 5: Predictability Analysis
Estimate Lyapunov time from Koopman eigenvalues to define predictability horizon.
Example: ~1.0 s for floating offshore wind turbine.
Validation Metrics
- R-squared at 1 Hz: > 0.95
- R-squared at higher rates: > 0.99
- Mode separation: reliable with rank optimization
- Predictability horizon: ~1.0 s Lyapunov time
Applications
- Floating offshore wind turbine monitoring
- Bridge and building structural health monitoring
- Aerospace structure vibration analysis
- Any system with partial observability + nonlinear dynamics
Pitfalls
- Rank selection critical: too low loses dynamics, too high overfits noise
- 3P rotor harmonics can mask structural resonances without proper separation
- Requires sufficient data length for reliable Hankel construction
- Predictability horizon limits forecasting range
Reference
arXiv:2605.00950 — Abaei, BahooToroody, Polojarvi, Remes (2026)
