Data-driven resolvent analysis (2010.02181v1)

Abstract: Resolvent analysis identifies the most responsive forcings and most receptive states of a dynamical system, in an input--output sense, based on its governing equations. Interest in the method has continued to grow during the past decade due to its potential to reveal structures in turbulent flows, to guide sensor/actuator placement, and for flow control applications. However, resolvent analysis requires access to high-fidelity numerical solvers to produce the linearized dynamics operator. In this work, we develop a purely data-driven algorithm to perform resolvent analysis to obtain the leading forcing and response modes, without recourse to the governing equations, but instead based on snapshots of the transient evolution of linearly stable flows. The formulation of our method follows from two established facts: $1)$ dynamic mode decomposition can approximate eigenvalues and eigenvectors of the underlying operator governing the evolution of a system from measurement data, and $2)$ a projection of the resolvent operator onto an invariant subspace can be built from this learned eigendecomposition. We demonstrate the method on numerical data of the linearized complex Ginzburg--Landau equation and of three-dimensional transitional channel flow, and discuss data requirements. The ability to perform resolvent analysis in a completely equation-free and adjoint-free manner will play a significant role in lowering the barrier of entry to resolvent research and applications.

Data-driven Resolvent Analysis in Dynamical Systems

The paper introduces a novel methodology for conducting resolvent analysis using a purely data-driven approach. This technique is significant in its departure from traditional methods that require access to high-fidelity numerical solvers and the governing equations of the dynamical systems under paper. In particular, the authors demonstrate how dynamic mode decomposition (DMD) can be exploited to approximate the eigendecomposition of the underlying linear operator using data snapshots of transient system evolution.

Methodological Advancement

Resolvent analysis has been a tool of growing interest due to its application in uncovering crucial structures within turbulent flows, optimizing sensor and actuator placement, and guiding flow control strategies. Traditionally, resolvent analysis required the linearized dynamics operator, which is typically derived from the governing equations of the system. However, this paper circumvents these requirements by relying solely on the data-driven reconstruction of the system dynamics. The authors achieve this through DMD, a data-driven technique, to approximate the eigenvalues and eigenvectors. Once these elements are obtained, the resolvent operator is projected onto the span of the learned eigenvectors to identify the system's most responsive inputs and outputs.

Examples and Results

To illustrate the efficacy and applicability of their approach, the authors apply their method to two fundamental examples: the linearized complex Ginzburg--Landau equation and a three-dimensional transitional channel flow. In both cases, they demonstrate that their data-driven method can achieve results that align closely with those obtained from conventional operator-based approaches. Importantly, they highlight the importance of the data's richness, noting that the design of initial disturbances in simulations plays a crucial role in enabling the discovery of meaningful resolvent modes.

In the context of the complex Ginzburg--Landau equation, the authors establish that sufficient trajectory data allows for the accurate extraction of forcing and response modes, thus validating their method's effectiveness. Similarly, with the transitional channel flow, different simulation setups revealed insights into how localized actuations could be used to probe specific flow dynamics, affirming the method's practical utility in complex multi-dimensional systems.

Implications and Future Directions

The implications of this work for practical and theoretical advancements in fluid dynamics and beyond are substantial. By eliminating the need for explicit access to the governing equations and adjoint simulations, this data-driven resolvent analysis method reduces barriers to entry for researchers and practitioners, offering potential applications with experimental setups. Additionally, the reliance on data snapshots suggests that similar methods could be extended to real-world systems where high-fidelity simulations are infeasible.

The authors' insights pave the way for further exploration in several directions. One significant avenue is the refinement of strategies for disturbance design in simulations to ensure a maximally informative data output, enhancing the discovery potential of the data-driven resolvent modes. Furthermore, extending the applicability of this method to turbulent flow scenarios presents a challenge that warrants further investigation, especially in disentangling linear and nonlinear flow contributions.

Conclusion

This paper's contribution lies in advancing the resolvent analysis framework towards a fully data-driven model, which holds promise for broader application and accessibility. Through rigorous examples and methodical validation, the authors have introduced a significant extension to the toolset available for analyzing dynamical systems, opening up opportunities for both academic research and industrial application in fluid dynamics and potentially other fields reliant on dynamical systems analysis.