Hierarchical deep learning for data-driven identification of reduced-order models of nonlinear dynamical systems

Identifying reduced-order models (ROMs) of nonlinear dynamical systems is difficult, especially when the system equation is unknown with only measurement data available. In such a case, not only a reduced subspace but also the associated dynamics need to be identified from data only, leading to a challenging data-driven ROM problem. In this study, we present a hierarchical deep learning approach to identify ROM from measurement only; it simultaneously identifies the nonlinear normal modal (NNM) subspace with a hierarchical order and the associated nonlinear modal dynamics. We conduct study to validate such an approach on both unforced and forced nonlinear dynamical systems, and find that the identified hierarchical NNMs-spanned subspace enables an efficient and effective dimensional truncation to achieve optimally lowest-dimensional ROM. We discuss in detail its performance and applicability

Data-driven identification of nonlinear normal modes via physics-integrated deep learning

Identifying the characteristic coordinates or modes of nonlinear dynamical systems is critical for understanding, analysis, and reduced-order modeling of the underlying complex dynamics. While normal modal transformation exactly characterizes any linear systems, there exists no such a general mathematical framework for nonlinear dynamical systems. Nonlinear normal modes (NNMs) are natural generalization of the normal modal transformation for nonlinear systems; however, existing research for identifying NNMs has relied on theoretical derivation or numerical computation from the closed-form equation of the system, which is usually unknown. In this work, we present a new data-driven framework based on physics-integrated deep learning for nonlinear modal identification of unknown nonlinear dynamical systems from the system response data only. Leveraging the universal modeling capacity and learning flexibility of deep neural networks, we first represent the forward and inverse nonlinear modal transformations through the physically interpretable deep encoder–decoder architecture, generalizing the modal superposition to nonlinear dynamics. Furthermore, to guarantee correct nonlinear modal identification, the proposed deep learning architecture integrates prior physics knowledge of the defined NNMs by embedding a unique dynamics-coder with physics-based constraints, including generalized modal properties, dynamics evolution, and future-state prediction. We test the proposed method by a series of study on the conservative and non-conservative Duffing systems with cubic nonlinearity and observe that the proposed data-driven framework is able to identify NNMs with invariant manifolds, energy-dependent nonlinear modal spectrum, and future-state prediction for unknown nonlinear dynamical systems from response data only; these identification results are found consistent with those from theoretically derived or numerically computed from closed-form equations. We also discuss its implementations and limitations for nonlinear modal identification of dynamical systems

A deep generative framework for data-driven surrogate modeling and visualization of parameterized nonlinear dynamical systems

Nonlinear dynamical systems in applications such as design and control typically depend on a set of variable parameters that represent system geometry, boundary conditions, material properties, etc. Such a parameterized dynamical system requires a parameterized model (e.g., a parameterized differential equation) to describe. On the one hand, to discover the wide variety of the parameter-dependent dynamical behaviors, repeated simulations with the parameterized model are often required over a large range of parameter values, leading to significant computational burdens especially when the system is complex (strongly nonlinear and/or high-dimensional) and the high-fidelity model is inefficient to simulate. Thus, seeking surrogate models that mimic the behaviors of high-fidelity parameterized models while being efficient to simulate is critically needed. On the other hand, the governing equations of the parameterized nonlinear dynamical system (e.g., an aerodynamic system with a physical model (full-scale or scaled in the laboratory) for optimization or design tasks) may be unknown or partially unknown due to insufficient physics knowledge, leading to an inverse problem where we need to identify the models from measurement data only. Accordingly, this work presents a novel deep generative framework for data-driven surrogate modeling/identification of parameterized nonlinear dynamical systems from data only. Specifically, the presented framework learns the direct mapping from simulation parameters to visualization images of dynamical systems by leveraging deep generative convolutional neural networks, yielding two advantages: (i) the surrogate simulation is efficient because the calculation of transient dynamics over time is circumvented; (ii) the surrogate output retains characterizing ability and flexibility as the visualization image is customizable and supports any visualization scheme for revealing and representing high-level dynamics feature (e.g., Poincaré map). We study and demonstrate the framework on Lorenz system, forced pendulum system, and forced Duffing system. We present and discuss the prediction performance of the obtained surrogate models. It is observed that the obtained model has promising performance on capturing the sensitive parameter dependence of the nonlinear dynamical behaviors even when the bifurcation occurs. We also discuss in detail the limitation of this work and potential future work

Data-Driven Nonlinear Modal Analysis: A Deep Learning Approach

We present a data-driven method based on deep learning for identifying nonlinear normal modes of unknown nonlinear dynamical systems using response data only. We leverage the modeling capacity of deep neural networks to identify the forward and inverse nonlinear modal transformations and the associated modal dynamics evolution. We test the method on Duffing systems with cubic nonlinearity and observe that the identified NNMs with invariant manifolds from response data agree with those analytical or numerical ones using closed-form equations

Efficient Data-Driven Modeling of Nonlinear Dynamical Systems via Metalearning

Data-driven modeling of nonlinear dynamical systems is essential because of the need for a trade-off among complexity, efficiency, and reliability in analytical or numerical studies as well as the difficulty of deriving fully physics-based models. An important limitation is commonly seen in existing works: modeling of a new dynamical system typically starts from scratch, requiring a large amount of data and intensive computation, though some prior experience or knowledge is available from a previously collected database of similar but different systems. However, on the one hand, the data amount for the new dynamical system is often limited, especially for real-world dynamical systems. On the other hand, the computational resource is also limited and a data-driven modeling task is usually computationally expensive, especially for large-scale systems. To improve data efficiency and computational efficiency in data-driven modeling of nonlinear systems, we present an enhanced data-driven modeling approach by incorporating metalearning into a physics-integrated deep learning framework. The core idea is to learn the metaknowledge about how to model a new system from a previously collected database of similar but different systems. Then this metaknowledge is leveraged to enable efficient modeling of a new system with limited data. For validations we conducted numerical experiments on three sets of fundamental nonlinear systems, including Duffing oscillators, nonlinear pendulums, and van der Pol oscillators. We performed both interpolation and extrapolation modelings to investigate the generalization ability of the presented approach. Furthermore, we conducted a quantitative analysis on data efficiency, addressing two critical issues: how few data are sufficient for the new system modeling and how much prior experience (previously collected database of similar but different systems) is needed for the metalearning. The results show that the presented approach improves both data efficiency and computational efficiency, compared with the conventional data-driven modeling approach (without leverage of the prior database) and the pretraining-based approach (simply using the prior database but without metalearning idea). We also discuss the limitations of this work and potential future study

Super-sensitivity incoherent optical methods for full-field displacement measurements

The sensitivity of incoherent optical methods using video cameras (e.g., optical flow and digital image correlation) for full-field displacement measurements, defined by the minimum measurable displacements, is essentially limited by the finite bit depth of the digital camera due to the quantization with round-off error. Quantitatively, the theoretical sensitivity limit is determined by the bit depth B as δp = 1/(2B − 1) [pixel] which corresponds to a displacement causing an intensity change of one gray level. Fortunately, the random noise in the imaging system may be leveraged to perform a natural dithering to overcome the quantization, rendering the possibility of breaking the sensitivity limit. In this work we study such a theoretical sensitivity limit and present a spatiotemporal pixel-averaging method with dithering to achieve super-sensitivity. The numerical simulation results indicate that super-sensitivity can be achieved and is quantitatively determined by the total pixel number N for averaging and the noise level σn as δp∗ ∝ (σn/√N)δp

A recurrent neural network framework with an adaptive training strategy for long-time predictive modeling of nonlinear dynamical systems

Long-time prediction of future states has been challenging in data-driven modeling of nonlinear dynamical systems as the prediction error accumulates over the prediction horizon. One of the potential reasons is the lack of robustness for the data-driven model. In this study we present a recurrent neural network (RNN) framework with an adaptive training strategy to model nonlinear dynamical systems from data for long-time prediction of future states. Specifically, we exploit the recurrence of network to improve the model robustness by explicitly incorporating the multi-step prediction with error accumulation into model training. Furthermore, we introduce an adaptive training strategy, where the prediction horizon gradually increases from a small value to facilitate the RNN training. We demonstrate the proposed approach on a family of Duffing oscillators, including autonomous and non-autonomous systems with various attractors, and discuss its advantages and limitations

Efficient regional seismic risk assessment via deep generative learning of surrogate models

Efficient regional seismic risk assessment including ground motion prediction and damage risk estimation is needed for emergency response planning. However, a conventional regional assessment suffers from low data- and time- efficiency as it generally involves a large number of locations and infrastructure systems that have specific soil conditions, and geometric, material, and structural properties, requiring access to large data and massive individual calculations with complicated procedures. To achieve efficient regional seismic risk assessment, this work presents a deep generative learning framework to construct input–output surrogate models of regional seismic risk by learning the underlying complex relation between earthquake source parameters and regional seismic risk involving many locations and structures from data. The learned deep surrogate models directly output the ground motion intensity map and the risk map of a region given earthquake source parameters, circumventing massive individual calculations and data access to individual locations and structures. The presented framework is validated on the bridge network risk assessment using simulated scenario earthquakes of the San Francisco Bay Area. We observe that the obtained deep surrogate models perform well without the need of data access to locations and structures and are time-efficient. We also discuss the applicability and limitations of the presented framework

Data-Driven Modeling of Parameterized Nonlinear Dynamical Systems with a Dynamics-Embedded Conditional Generative Adversarial Network

Nonlinear dynamical systems in applications such as design and control generally depend on a set of variable parameters that represent system geometry, boundary conditions, material properties, etc. Modeling of such parameterized nonlinear systems from first principles is often challenging due to insufficient knowledge of the underlying physics (e.g., damping), especially when the physics-associated parameters are considered to be variable. In this study, we present a dynamics-embedded conditional generative adversarial network (Dyn-cGAN) for data-driven modeling and identification of parameterized nonlinear dynamical systems, capturing transient dynamics conditioned on the system parameters. Specifically, a dynamics block is embedded in a modified conditional generative adversarial network, thereby identifying temporal dynamics and its dependence on the system parameters, simultaneously. The data-driven Dyn-cGAN model is learned to perform long-term prediction of the dynamical response of a parameterized nonlinear dynamical system (equivalently a family of nonlinear systems with different parameter values), given any initial conditions and system parameter values. The capability of the presented Dyn-cGAN is evaluated by numerical studies on a variety of parameterized nonlinear dynamical systems including pendulums, Duffing, and Lorenz systems, considering various combinations of initial conditions and system (physical) parameters as inputs and different ranges of nonlinear dynamical behaviors including chaotic. It is observed that the presented data-driven framework is reasonably effective for predictive modeling and identification of parameterized nonlinear dynamical systems. Further analysis also indicates that its prediction accuracy degrades gracefully as the complexity of the nonlinear system increases, such as strongly nonlinear systems and systems with multiple parameters changing. The limitations of this work and potential future work are also discussed