Learning the tangent space of dynamical instabilities from data

Abstract

For a large class of dynamical systems, the optimally time-dependent (OTD) modes, a set of deformable orthonormal tangent vectors that track directions of instabilities along any trajectory, are known to depend "pointwise" on the state of the system on the attractor, and not on the history of the trajectory. We leverage the power of neural networks to learn this "pointwise" mapping from phase space to OTD space directly from data. The result of the learning process is a cartography of directions associated with strongest instabilities in phase space. Implications for data-driven prediction and control of dynamical instabilities are discussed.The optimally time-dependent (OTD) modes are a set of deformable orthonormal tangent vectors that track directions of instabilities along any trajectory of a dynamical system. Traditionally, these modes are computed by a time-marching approach that involves solving multiple initial-boundary-value problems concurrently with the state equations. However, for a large class of dynamical systems, the OTD modes are known to depend "pointwise" on the state of the system on the attractor, and not on the history of the trajectory. So we propose a neural-network algorithm to learn this "pointwise" mapping from phase space to OTD space directly from data. The neural network produces a cartography of directions of strongest instability in phase space, as well as accurate estimates for the leading Lyapunov exponents.

Transient linear stability of pulsating Poiseuille flow using optimally time-dependent modes

“…Although the alignment is exponentially fast, the exact time it takes in practice is highly dependent on the system as well as the initial conditions. Once converged, however, the OTD modes depend only on the trajectory pointwise and are memoryless (Blanchard & Sapsis 2019 b ).…”

Oblique projection for scalable rank-adaptive reduced-order modelling of nonlinear stochastic partial differential equations with time-dependent bases

“…they produce identical low-rank matrices [11,12], and their differences lie only in their numerical performance. TDB ROMs have also been used in other fields and applications including dynamical systems [13,14], combustion [7,15], linear sensitivity analysis [6], dynamical instabilities [16–18], deep learning [19] and singular vale decomposition (SVD) estimation for matrices that vary smoothly with a parameter [20].…”

Data-driven resolvent analysis

“…The unprecedented availability of high-fidelity numerical simulations and experimental measurements has led to incredible growth of research in data-driven modelling of dynamical systems during the past decade (Schmid 2010; Williams, Kevrekidis & Rowley 2015; Brunton, Proctor & Kutz 2016; Rudy et al. 2017; Loiseau & Brunton 2018; Blanchard & Sapsis 2019; Brunton & Kutz 2019; Duraisamy, Iaccarino & Xiao 2019; Raissi, Perdikaris & Karniadakis 2019; Qian et al. 2020; Raissi, Yazdani & Karniadakis 2020; Li et al.…”

“…The unprecedented availability of high-fidelity numerical simulations and experimental measurements has led to incredible growth of research in data-driven modelling of dynamical systems during the past decade (Schmid 2010;Williams, Kevrekidis & Rowley 2015;Rudy et al 2017;Loiseau & Brunton 2018;Blanchard & Sapsis 2019;Brunton & Kutz 2019;Duraisamy, Iaccarino & Xiao 2019;Raissi, Perdikaris & Karniadakis 2019;Qian et al 2020;Raissi, Yazdani & Karniadakis 2020;Li et al 2021). In fluid dynamics, this has led to the development and application of machine learning algorithms to extract dominant coherent structures from flow data (Taira et al 2017;Brenner, Eldredge & Freund 2019;Taira et al 2019;Brunton, Noack & Koumoutsakos 2020).…”