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Data-assisted reduced-order modeling of extreme events in complex dynamical systems
Zhong Yi Wan, Pantelis R. Vlachas, Petros Koumoutsakos, Themistoklis P. Sapsis
TL;DR
High-dimensional dynamical systems with rare extreme events are difficult to reduce because truncated modes can matter, while data-driven methods struggle where observations are sparse. The paper combines an imperfect projected model with an LSTM trained on reduced-space model–data mismatch, and reports important advantages over either approach alone, especially for extreme-event regions.
Problem
High-dimensional attractors, rare transient events, and sparse data make both classical equation-based reduction and purely data-driven modeling difficult for extreme-event dynamics.
Method
The framework combines an imperfect projected physical model with data streams and uses an LSTM recurrent neural network to model truncated or unaccounted-for dynamics.
Results
The hybrid strategy has important advantages over purely data-driven methods and reduced-order models alone in prototype systems exhibiting extreme events.
Takeaways & Limitations
The imperfect model supplies dynamical information while the recurrent network captures nonlinear dynamics, supporting reduced-order prediction in systems with high dimensionality and transient events.
Takeaways & Limitations
The reduction can fail when omitted dimensions, though small on average, play important dynamical roles such as buffering energy transfer between modes.
Abstract
from arXiv · showhide
Dynamical systems with high intrinsic dimensionality are often characterized by extreme events having the form of rare transitions several standard deviations away from the mean. For such systems, order-reduction methods through projection of the governing equations have limited applicability due to the large intrinsic dimensionality of the underlying attractor but also the complexity of the transient events. An alternative approach is data-driven techniques that aim to quantify the dynamics of specific modes utilizing data-streams. Several of these approaches have improved performance by expanding the state representation using delayed coordinates. However, such strategies are limited in regions of the phase space where there is a small amount of data available, as is the case for extreme events. In this work, we develop a blended framework that integrates an imperfect model, obtained from projecting equations into a subspace that still contains crucial dynamical information, with data-streams through a recurrent neural network (RNN) architecture. In particular, we employ the long-short-term memory (LSTM), to model portions of the dynamics which cannot be accounted by the equations. The RNN is trained by analyzing the mismatch between the imperfect model and the data-streams, projected in the reduced-order space. In this way, the data-driven model improves the imperfect model in regions where data is available, while for locations where data is sparse the imperfect model still provides a baseline for the prediction of the system dynamics. We assess the developed framework on two challenging prototype systems exhibiting extreme events and show that the blended approach has improved performance compared with methods that use either data streams or the imperfect model alone. The improvement is more significant in regions associated with extreme events, where data is sparse.
Introduction
Extreme events challenge both equation-based reduction and purely data-driven modeling in high-dimensional systems. The paper proposes blending an imperfect physical model with data streams through an RNN to improve reduced-order descriptions of complex, transient dynamics.
- Extreme events occur across turbulent, geophysical, optical, water-wave, and mechanical systems.
- Classical Galerkin reduction struggles because truncated degrees of freedom can remain essential in systems with high intrinsic dimensionality and rare transient events.
- The proposed hybrid reduced-order model combines an imperfect physical model with available data streams for complex systems with multiscale attractors and strongly transient nonlinear dynamics.
- An LSTM-based RNN represents truncated degrees of freedom while retaining instability information from the imperfect model and learning nonlinear attractor dynamics.The approach uses recurrent memory to represent dimensions omitted from the reduced model.
- The framework builds on earlier blended approaches but targets prediction of complex systems with high dimensionality and strongly transient dynamics.
- Prototype examples show important advantages over purely data-driven methods and reduced-order models used alone.
Materials and methods
The method combines reduced-order equations with an LSTM that learns complementary dynamics from delayed reduced-space data. It addresses truncation and imperfect parametrization while retaining the equation-based model as a structured baseline.
- Reduced-order formulation: The state is decomposed into retained coordinates ξ and truncated coordinates η using orthonormal subspaces Y and Z.The offset b is typically the attractor mean state, and the reduced dimension m is intended to be much smaller than d.
- Reduced-order formulation: Flat Galerkin reduction sets η = 0 when its average magnitude is much smaller than ξ, yielding the reduced dynamics Fξ(ξ).This approximation produces an m-dimensional system but can omit dynamically important truncated modes.
- Reduced-order formulation: Truncated coordinates can remain dynamically important because they may buffer energy transfer, so neglecting them can compromise reliable forecasts.The issue arises when the reduction subspace is too large for η ≈ 0 or when Z reflects statistics without accounting for dynamics.
- Nonlinear Galerkin correction: Nonlinear Galerkin projection represents η as Φ(ξ), but Φ is well-defined only when the inertial manifold is fully parametrized by Y.Even under that condition, systematically finding Φ remains challenging.
- Data-assisted correction: The framework learns complementary dynamics ψ = G(ξ,η) with an LSTM-based model ˆG and adds it to the flat Galerkin dynamics.Delayed ξ states infer information unavailable in the imperfect reduction, while preserving the projected-equation structure.
- RNN architectures: Architecture I predicts complementary dynamics from accurate input sequences for training, whereas Architecture II integrates iterative predictions for fine-tuning and multi-step forecasting.Architecture II uses a setup stage to spin up LSTM memory, followed by a prediction stage in which errors can propagate through later inputs.
Results and discussion
The data-assisted reduced-order model combines projected equations with LSTM-based learning and improves prediction of chaotic and extreme-event dynamics in the CDV and Kolmogorov systems.
- CDV system: The five-mode POD projection retains 99.6% of CDV energy but produces a single fixed point instead of the full system’s chaotic attractor.The data-assisted model preserves the original attractor’s geometric features.
- CDV system: The data-assisted CDV model significantly outperforms both the projected and purely data-driven models across prediction lead times.It maintains low errors when the comparison methods exhibit significant errors.
- CDV system: The equation-driven component supports long-term stability, while the data-driven component improves short-term prediction accuracy and reproduces a chaotic structure absent from either component alone.The two components therefore complement one another in the reduced-order model.
- Kolmogorov flow: At lead time 0.5, the data-assisted Kolmogorov model achieves RMSE values of 0.13, 0.005, and 0.058 for modes [0, 4], [1, 0], and, respectively.These results correspond to approximately one eddy turnover time.
- Kolmogorov flow: Complementary dynamics have concentrated, bimodal-like distributions whose density falls below 10^-4 within 3 standard deviations, leaving extreme-event dynamics to the projected equations.This provides a better-conditioned target for the data-driven component.
Conclusion
The framework combines projected imperfect models with data-stream information through an LSTM-based recurrent architecture to improve reduced-order prediction of extreme transient events. Across two prototype systems, it performs especially well in sparse-data extreme-event regions while retaining the imperfect model as a prediction baseline.
- Conclusion: The framework complements projected imperfect-model dynamics with data-stream information extracted from reduced-space trajectory histories using an LNN-based recurrent strategy.The LSTM models dynamics omitted by the projection and uses delayed coordinates to improve performance.
- Conclusion: In the Charney-DeVore model, the data-driven component significantly improves short-term prediction in a high-energy reduction subspace while reproducing the original chaotic attractor.The study evaluates this behavior alongside a high-dimensional Kolmogorov-flow example.
- Conclusion: In Kolmogorov flow, data-assisted prediction is more effective near extreme events than either fully data-driven prediction or projected equations alone.The reported extreme-event region is associated with rare observations and includes improvements for wavenumbers (0, 4) and (1, 4).
- Conclusion: Near the main attractor, the purely data-driven and data-assisted approaches exhibit comparable accuracy.The advantage of the blended approach is therefore most pronounced in extreme-event regions rather than near typical states.
- Conclusion: The method improves prediction where data are available while preserving the imperfect model as a baseline where data are sparse.This design specifically targets regions associated with rare extreme events.
- Conclusion: The approach assumes that the imperfect model retains relevant dynamical information for modes associated with extreme events, which need not be the most energetic modes.The authors identify predictive control and suppression of extreme events as future applications.
Supporting information
The supporting information provides background theory and additional computations, while the study’s code is openly available and training and testing data can be requested from the authors.
- Supporting information: The supplementary notes cover recurrent neural networks, LSTM, momentum-based optimization, and additional Charney-DeVore and Kolmogorov-flow results.These materials are identified as S1 Appendix.
- Supporting information: The study’s Python source code is available online, and training and testing data are available from the authors upon request.The code repository is identified in the supporting-information passage.