Nan Jiang

Max Gunzburger, Nan Jiang, Feifei Xu

We study a turbulence closure model in which the fractional Laplacian $(-Δ)^α$ of the velocity field represents the turbulence diffusivity. We investigate the energy spectrum of the model by applying Pao's energy transfer theory. For the case $α=1/3$, the corresponding power law of the energy spectrum in the inertial range has a correction exponent on the regular Kolmogorov -5/3 scaling exponent. For this case, this model represents Richardson's particle pair-distance superdiffusion of a fully developed homogeneous turbulent flow as well as Lévy jumps that lead to the superdiffusion. For other values of $α$, the power law of the energy spectrum is consistent with the regular Kolmogorov -5/3 scaling exponent. We also propose and study a modular time-stepping algorithm in semi-discretized form. The algorithm is minimally intrusive to a given legacy code for solving Navier-Stokes equations by decoupling the local part and nonlocal part of the equations for the unknowns. We prove the algorithm is unconditionally stable and unconditionally, first-order convergent. We also derive error estimates for full discretizations of the model which, in addition to the time stepping algorithm, involves a finite element spatial discretization and a domain truncation approximation to the range of the fractional Laplacian.

Nan Jiang, William Layton

Standard eddy viscosity models, while robust, cannot represent backscatter and have severe difficulties with complex turbulence not at statistical equilibrium. This report gives a new derivation of eddy viscosity models from an equation for the evolution of variance in a turbulent flow. The new derivation also shows how to correct eddy viscosity models. The report proves the corrected models preserve important features of the true Reynolds stresses. It gives algorithms for their discretization including a minimally invasive modular step to adapt an eddy viscosity code to the extended models. A numerical test is given with the usual and over diffusive Smagorinsky model. The correction (scaled by $10^{-8}$ ) does successfully exhibit intermittent backscatter.

Nan Jiang, Michael Schneier

Studying the propagation of uncertainties in a nonlinear dynamical system usually involves generating a set of samples in the stochastic parameter space and then repeated simulations with different sampled parameters. The main difficulty faced in the process is the excessive computational cost. In this paper, we present an efficient, partitioned ensemble algorithm to determine multiple realizations of a reduced Magnetohydrodynamics (MHD) system, which models MHD flows at low magnetic Reynolds number. The algorithm decouples the fully coupled problem into two smaller sub-physics problems, which reduces the size of the linear systems that to be solved and allows the use of optimized codes for each sub-physics problem. Moreover, the resulting coefficient matrices are the same for all realizations at each time step, which allows faster computation of all realizations and significant savings in computational cost. We prove this algorithm is first order accurate and long time stable under a time step condition. Numerical examples are provided to verify the theoretical results and demonstrate the efficiency of the algorithm.

Nan Jiang, Jinzhao Li, Yexiang Xue

In reinforcement learning, Reverse Experience Replay (RER) is a recently proposed algorithm that attains better sample complexity than the classic experience replay method. RER requires the learning algorithm to update the parameters through consecutive state-action-reward tuples in reverse order. However, the most recent theoretical analysis only holds for a minimal learning rate and short consecutive steps, which converge slower than those large learning rate algorithms without RER. In view of this theoretical and empirical gap, we provide a tighter analysis that mitigates the limitation on the learning rate and the length of consecutive steps. Furthermore, we show theoretically that RER converges with a larger learning rate and a longer sequence.

Nan Jiang, Ziming Wu, De-Chuan Zhan et al.

Chain-of-Thought (CoT) reasoning has significantly advanced Large Language Models (LLMs) in solving complex tasks. However, its autoregressive paradigm leads to significant computational overhead, hindering its deployment in latency-sensitive applications. To address this, we propose \textbf{DART} (\textbf{D}istilling \textbf{A}utoregressive \textbf{R}easoning to Silent \textbf{T}hought), a self-distillation framework that enables LLMs to replace autoregressive CoT with non-autoregressive Silent Thought (ST). Specifically, DART introduces two training pathways: the CoT pathway for traditional reasoning and the ST pathway for generating answers directly from a few ST tokens. The ST pathway utilizes a lightweight Reasoning Evolvement Module (REM) to align its hidden states with the CoT pathway, enabling the ST tokens to evolve into informative embeddings. During inference, only the ST pathway is activated, leveraging evolving ST tokens to deliver the answer directly. Extensive experimental results demonstrate that DART offers significant performance gains compared with existing non-autoregressive baselines without extra inference latency, serving as a feasible alternative for efficient reasoning.

Max Gunzburger, Nan Jiang, Zhu Wang

We consider settings for which one needs to perform multiple flow simulations based on the Navier-Stokes equations, each having different values for the physical parameters and/or different initial condition data, boundary conditions data, and/or forcing functions. For such settings, we propose a second-order time accurate ensemble-based method that to simulate the whole set of solutions, requires, at each time step, the solution of only a single linear system with multiple right-hand-side vectors. Rigorous analyses are given proving the conditional stability and error estimates for the proposed algorithm. Numerical experiments are provided that illustrate the analyses.

Max Gunzburger, Nan Jiang, Zhu Wang

Many applications of computational fluid dynamics require multiple simulations of a flow under different input conditions. In this paper, a numerical algorithm is developed to efficiently determine a set of such simulations in which the individually independent members of the set are subject to different viscosity coefficients, initial conditions, and/or body forces. The proposed scheme applied to the flow ensemble leads to need to solve a single linear system with multiple right-hand sides, and thus is computationally more efficient than solving for all the simulations separately. We show that the scheme is nonlinearly and long-term stable under certain conditions on the time-step size and a parameter deviation ratio. Rigorous numerical error estimate shows the scheme is of first-order accuracy in time and optimally accurate in space. Several numerical experiments are presented to illustrate the theoretical results.