6 research outputs found
Multiscale velocity correlations in turbulence and Burgers turbulence: Fusion rules, Markov processes in scale, and multifractal predictions
We compare different approaches towards an effective description of
multi-scale velocity field correlations in turbulence. Predictions made by the
operator product expansion, the so-called fusion rules, are placed in
juxtaposition to an approach that interprets the turbulent energy cascade in
terms of a Markov process of velocity increments in scale. We explicitly show
that the fusion rules are a direct consequence of the Markov property provided
that the structure functions exhibit scaling in the inertial range.
Furthermore, the limit case of joint velocity gradient and velocity increment
statistics is discussed and put into the context of the notion of dissipative
anomaly. We generalize a prediction made by the multifractal (MF) approach
derived in [Phys. Rev. Lett. 80, 3244 (1998)] to correlations among inertial
range velocity increment and velocity gradients of any order. We show that for
the case of squared velocity gradients such a relation can be derived from
"first principles" in the case of Burgers equation. Our results are benchmarked
by intensive direct numerical simulations of Burgers turbulence.Comment: 18 pages, 6 figure
Reconstruction, forecasting, and stability of chaotic dynamics from partial data
The forecasting and computation of the stability of chaotic systems from
partial observations are tasks for which traditional equation-based methods may
not be suitable. In this computational paper, we propose data-driven methods to
(i) infer the dynamics of unobserved (hidden) chaotic variables (full-state
reconstruction); (ii) time forecast the evolution of the full state; and (iii)
infer the stability properties of the full state. The tasks are performed with
long short-term memory (LSTM) networks, which are trained with observations
(data) limited to only part of the state: (i) the low-to-high resolution LSTM
(LH-LSTM), which takes partial observations as training input, and requires
access to the full system state when computing the loss; and (ii) the
physics-informed LSTM (PI-LSTM), which is designed to combine partial
observations with the integral formulation of the dynamical system's evolution
equations. First, we derive the Jacobian of the LSTMs. Second, we analyse a
chaotic partial differential equation, the Kuramoto-Sivashinsky (KS), and the
Lorenz-96 system. We show that the proposed networks can forecast the hidden
variables, both time-accurately and statistically. The Lyapunov exponents and
covariant Lyapunov vectors, which characterize the stability of the chaotic
attractors, are correctly inferred from partial observations. Third, the
PI-LSTM outperforms the LH-LSTM by successfully reconstructing the hidden
chaotic dynamics when the input dimension is smaller or similar to the
Kaplan-Yorke dimension of the attractor. This work opens new opportunities for
reconstructing the full state, inferring hidden variables, and computing the
stability of chaotic systems from partial data
Instanton based importance sampling for rare events in stochastic PDEs
We present a new method for sampling rare and large fluctuations in a
non-equilibrium system governed by a stochastic partial differential equation
(SPDE) with additive forcing. To this end, we deploy the so-called instanton
formalism that corresponds to a saddle-point approximation of the action in the
path integral formulation of the underlying SPDE. The crucial step in our
approach is the formulation of an alternative SPDE that incorporates knowledge
of the instanton solution such that we are able to constrain the dynamical
evolutions around extreme flow configurations only. Finally, a reweighting
procedure based on the Girsanov theorem is applied to recover the full
distribution function of the original system. The entire procedure is
demonstrated on the example of the one-dimensional Burgers equation.
Furthermore, we compare our method to conventional direct numerical simulations
as well as to Hybrid Monte Carlo methods. It will be shown that the
instanton-based sampling method outperforms both approaches and allows for an
accurate quantification of the whole probability density function of velocity
gradients from the core to the very far tails.Comment: 8 pages, 4 figure
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Stability analysis of chaotic systems from data.
UNLABELLED: The prediction of the temporal dynamics of chaotic systems is challenging because infinitesimal perturbations grow exponentially. The analysis of the dynamics of infinitesimal perturbations is the subject of stability analysis. In stability analysis, we linearize the equations of the dynamical system around a reference point and compute the properties of the tangent space (i.e. the Jacobian). The main goal of this paper is to propose a method that infers the Jacobian, thus, the stability properties, from observables (data). First, we propose the echo state network (ESN) with the Recycle validation as a tool to accurately infer the chaotic dynamics from data. Second, we mathematically derive the Jacobian of the echo state network, which provides the evolution of infinitesimal perturbations. Third, we analyse the stability properties of the Jacobian inferred from the ESN and compare them with the benchmark results obtained by linearizing the equations. The ESN correctly infers the nonlinear solution and its tangent space with negligible numerical errors. In detail, we compute from data only (i) the long-term statistics of the chaotic state; (ii) the covariant Lyapunov vectors; (iii) the Lyapunov spectrum; (iv) the finite-time Lyapunov exponents; (v) and the angles between the stable, neutral, and unstable splittings of the tangent space (the degree of hyperbolicity of the attractor). This work opens up new opportunities for the computation of stability properties of nonlinear systems from data, instead of equations. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s11071-023-08285-1
A Hybrid Monte Carlo algorithm for sampling rare events in space-time histories of stochastic fields
We introduce a variant of the Hybrid Monte Carlo (HMC) algorithm to address large deviation statistics in stochastic hydrodynamics. Based on the path integral approach to stochastic (partial) differential equations, our HMC algorithm samples space-time histories of the dynamical degrees of freedom under the influence of random noise. First, we validate and benchmark the HMC algorithm by reproducing multi-scale properties of the one-dimensional Burgers equation driven by Gaussian and white-in-time noise. Second, we show how to implement an importance sampling protocol to significantly enhance, by order-of-magnitudes, the probability to sample extreme and rare events, making it possible for the first time to estimate moments of field variables of extremely high order (up to 30 and more). By employing reweighting techniques, we map the biased configurations back to the original probability measure in order to probe their statistical importance. Finally, we show that by biasing the system towards very intense negative gradients, the HMC algorithm is able to explore the statistical fluctuations around instanton configurations. Our results will also be interesting and relevant in lattice gauge theory since they provide a new insight on reweighting techniques