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1. Data-Driven Model Reduction for Stochastic Burgers Equations

   https://doi.org/10.3390/e22121360  

   Lu, Fei ( December 2020 , Entropy)

   null (Ed.)

   We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variable’s trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model’s stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step, where the K-mode Galerkin system’s mean Courant–Friedrichs–Lewy (CFL) number agrees with that of the full model. 

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2. On the stability of shocks in isothermal black hole accretion discs

   https://doi.org/10.1093/mnras/stac731  

   Hester, Eric W. ; Vasil, Geoffrey M. ; Wechselberger, Martin ( March 2022 , Monthly Notices of the Royal Astronomical Society)

   Most black holes possess accretion discs. Models of such discs inform observations and constrain the properties of the black holes and their surrounding medium. Here, we study shocks in a thin isothermal black hole accretion flow. Modelling infinitesimal viscosity allows the use of multiple-scales matched asymptotic methods. We thus derive the first explicit calculations of isothermal shock stability. We find that the inner shock is always unstable, and the outer shock is always stable. The growth/decay rates of perturbations depend only on an effective potential and the incoming–outgoing flow difference at the shock location. We give a prescription of accretion regimes in terms of angular momentum and black hole radius. Accounting for viscous angular momentum dissipation implies unstable outer shocks in much of parameter space, even for realistic viscous Reynolds numbers of the order ≈1020.

    

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3. State Space Modeling of Viscous Unsteady Aerodynamic Loads

   https://doi.org/10.2514/6.2021-1830  

   Taha, Haitham E. ; Rezaei, Amir S. ( January 2021 , AIAA SciTech)

   null (Ed.)

   In this paper, we started by summarizing our recently developed viscous unsteady theory based on coupling potential flow with the triple deck boundary layer theory. This approach provides a viscous extension of potential flow unsteady aerodynamics. As such, a Reynolds- number-dependence could be determined. We then developed a finite-state approximation of such a theory, presenting it in a state space model. This novel nonlinear state space model of the viscous unsteady aerodynamic loads is expected to serve aerodynamicists better than the classical Theodorsen’s model, as it captures viscous effects (i.e., Reynolds number dependence) as well as nonlinearity and additional lag in the lift dynamics; and allows simulation of arbitrary time-varying airfoil motions (not necessarily harmonic). Moreover, being in a state space form makes it quite convenient for simulation and coupling with structural dynamics to perform aeroelasticity, flight dynamics analysis, and control design. We then proceeded to develop a linearization of such a model, which enables analytical results. So, we derived an analytical representation of the viscous lift frequency response function, which is an explicit function of, not only frequency, but also Reynolds number. We also developed a state space model of the linearized response. We finally simulated the nonlinear and linear models to a non- harmonic, small-amplitude pitching maneuver at 100 , 000 Reynolds number and compared the resulting lift and pitching moment with potential flow, in reference to relatively higher fidelity computations of the Unsteady Reynolds-Averaged Navier-Stokes equations. 

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4. Variational multiscale reinforcement learning for discovering reduced order closure models of nonlinear spatiotemporal transport systems

   https://doi.org/10.1038/s41598-022-22598-y  

   San, Omer ; Pawar, Suraj ; Rasheed, Adil ( October 2022 , Scientific Reports)

   A central challenge in the computational modeling and simulation of a multitude of science applications is to achieve robust and accurate closures for their coarse-grained representations due to underlying highly nonlinear multiscale interactions. These closure models are common in many nonlinear spatiotemporal systems to account for losses due to reduced order representations, including many transport phenomena in fluids. Previous data-driven closure modeling efforts have mostly focused on supervised learning approaches using high fidelity simulation data. On the other hand, reinforcement learning (RL) is a powerful yet relatively uncharted method in spatiotemporally extended systems. In this study, we put forth a modular dynamic closure modeling and discovery framework to stabilize the Galerkin projection based reduced order models that may arise in many nonlinear spatiotemporal dynamical systems with quadratic nonlinearity. However, a key element in creating a robust RL agent is to introduce a feasible reward function, which can be constituted of any difference metrics between the RL model and high fidelity simulation data. First, we introduce a multi-modal RL to discover mode-dependant closure policies that utilize the high fidelity data in rewarding our RL agent. We then formulate a variational multiscale RL (VMRL) approach to discover closure models without requiring access to the high fidelity data in designing the reward function. Specifically, our chief innovation is to leverage variational multiscale formalism to quantify the difference between modal interactions in Galerkin systems. Our results in simulating the viscous Burgers equation indicate that the proposed VMRL method leads to robust and accurate closure parameterizations, and it may potentially be used to discover scale-aware closure models for complex dynamical systems.

    

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5. Contraction for large perturbations of traveling waves in a hyperbolic–parabolic system arising from a chemotaxis model

   https://doi.org/10.1142/S0218202520500104  

   Choi, Kyudong ; Kang, Moon-Jin ; Kwon, Young-Sam ; Vasseur, Alexis F. ( February 2020 , Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   We consider a hyperbolic–parabolic system arising from a chemotaxis model in tumor angiogenesis, which is described by a Keller–Segel equation with singular sensitivity. It is known to allow viscous shocks (so-called traveling waves). We introduce a relative entropy of the system, which can capture how close a solution at a given time is to a given shock wave in almost [Formula: see text]-sense. When the shock strength is small enough, we show the functional is non-increasing in time for any large initial perturbation. The contraction property holds independently of the strength of the diffusion. 

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