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doi: 10.3934/dcdss.2021096
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## Effective Mori-Zwanzig equation for the reduced-order modeling of stochastic systems

 1 Department of Applied Mathematics, University of California, Merced, Merced (CA) 95343, USA 2 Department of Computational Mathematics, Michigan State University, East Lansing (MI) 48824, USA

* Corresponding author

Received  February 2021 Revised  June 2021 Early access August 2021

Built upon the hypoelliptic analysis of the effective Mori-Zwanzig (EMZ) equation for observables of stochastic dynamical systems, we show that the obtained semigroup estimates for the EMZ equation can be used to derive prior estimates of the observable statistics for systems in the equilibrium and nonequilibrium state. In addition, we introduce both first-principle and data-driven methods to approximate the EMZ memory kernel and prove the convergence of the data-driven parametrization schemes using the regularity estimate of the memory kernel. The analysis results are validated numerically via the Monte-Carlo simulation of the Langevin dynamics for a Fermi-Pasta-Ulam chain model. With the same example, we also show the effectiveness of the proposed memory kernel approximation methods.

Citation: Yuanran Zhu, Huan Lei. Effective Mori-Zwanzig equation for the reduced-order modeling of stochastic systems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021096
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##### References:
Sample path of the tagged oscillator momentum $p_{50}(t)$. We display the result for the stochastic FPU system (52) with weak ($\theta = 0.1$) and strong nonlinearity ($\theta = 1$) at high ($\beta = 1$) and low ($\beta = 20$) temperature
Temporal auto-correlation function of the tagged oscillator momentum $p_j(t)$ for weakly nonlinear FPU system at different temperature $T\propto 1/\beta$. We compare results we obtained by calculating the EMZ memory from first principles using $14$-th order Faber polynomials with results from MC simulation ($10^6$ sample paths). In the subplots, we display $|C(t)/C(0)|$ and the exponentially decaying upper bound $ce^{-\alpha t}$ with an estimated decaying rate $\alpha$
">Figure 3.  Approximated EMZ memory kernel corresponding to the tagged particle momentum correlation function $C(t)$. The subplots display $|K(t)/K(0)|$ and the exponentially decaying upper bound $c_{{\mathcal{Q}}}e^{-\alpha_{{\mathcal{Q}}}t}$ with an estimated decaying rate $\alpha_{{\mathcal{Q}}}$. Other setting is same as Figure 2
Temporal auto-correlation function of the tagged oscillator momentum $p_j(t)$ for strongly nonlinear FPU system at different temperature $T\propto 1/\beta$. The MC simulation results ($10^6$ sample paths) of the correlation function are compared with the one obtained by the data-driven memory kernel using Faber series (20th order) and the standard Laguerre polynomials (20th order). In the subplots, we display $|C(t)/C(0)|$ and the exponentially decaying upper bound $ce^{-\alpha t}$ with an estimated decaying rate $\alpha$
Comparison of the dynamics of the particle momentum $p_{50}(t)$ generated by the MC simulation and the ROM (55). The displayed results are for a stochastic FPU system with strong nonlinearity ($\theta = 1$) at high temperature $\beta = 1$ (first row) and low temperature $\beta = 20$ (second row). In the first column, we compare the simulated sample paths. The time autocorrelation functions $C(t)/C(0)$ (second column) are obtained by averaging a cluster of the sample trajectories. The third column compares the stationary distribution of the stochastic process $\rho_{p_{50}}$ which are obtained via kernel density estimations
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