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Data-driven Closures & Assimilation for Stiff Multiscale Random Dynamics

Tyler E. Maltba, Hongli Zhao, D. Adrian Maldonado

TL;DR

This work tackles uncertainty propagation in high-dimensional, stiff multiscale RODEs driven by colored noise by deriving exact reduced-order PDF dynamics for low-dimensional QoIs (RoPDFs). Unclosed terms in the RoPDFs are represented as regression functions of the state, learned from sparse data, and a defect term is learned through data assimilation to compensate model misspecification. Two assimilation strategies are explored: nudging, which gradually relaxes the RoPDF toward observations, and deep neural networks, which learn a defect-driven, physics-informed observer. The framework is demonstrated on a stiff linear system and a 47-dimensional power-grid cascade model, achieving accurate density estimates with orders-of-magnitude speedups over direct Monte Carlo simulations, and revealing practical pathways for uncertainty quantification in complex multiscale systems.

Abstract

We introduce a data-driven and physics-informed framework for propagating uncertainty in stiff, multiscale random ordinary differential equations (RODEs) driven by correlated (colored) noise. Unlike systems subjected to Gaussian white noise, a deterministic equation for the joint probability density function (PDF) of RODE state variables does not exist in closed form. Moreover, such an equation would require as many phase-space variables as there are states in the RODE system. To alleviate this curse of dimensionality, we instead derive exact, albeit unclosed, reduced-order PDF (RoPDF) equations for low-dimensional observables/quantities of interest. The unclosed terms take the form of state-dependent conditional expectations, which are directly estimated from data at sparse observation times. However, for systems exhibiting stiff, multiscale dynamics, data sparsity introduces regression discrepancies that compound during RoPDF evolution. This is overcome by introducing a kinetic-like defect term to the RoPDF equation, which is learned by assimilating in sparse, low-fidelity RoPDF estimates. Two assimilation methods are considered, namely nudging and deep neural networks, which are successfully tested against Monte Carlo simulations.

Data-driven Closures & Assimilation for Stiff Multiscale Random Dynamics

TL;DR

This work tackles uncertainty propagation in high-dimensional, stiff multiscale RODEs driven by colored noise by deriving exact reduced-order PDF dynamics for low-dimensional QoIs (RoPDFs). Unclosed terms in the RoPDFs are represented as regression functions of the state, learned from sparse data, and a defect term is learned through data assimilation to compensate model misspecification. Two assimilation strategies are explored: nudging, which gradually relaxes the RoPDF toward observations, and deep neural networks, which learn a defect-driven, physics-informed observer. The framework is demonstrated on a stiff linear system and a 47-dimensional power-grid cascade model, achieving accurate density estimates with orders-of-magnitude speedups over direct Monte Carlo simulations, and revealing practical pathways for uncertainty quantification in complex multiscale systems.

Abstract

We introduce a data-driven and physics-informed framework for propagating uncertainty in stiff, multiscale random ordinary differential equations (RODEs) driven by correlated (colored) noise. Unlike systems subjected to Gaussian white noise, a deterministic equation for the joint probability density function (PDF) of RODE state variables does not exist in closed form. Moreover, such an equation would require as many phase-space variables as there are states in the RODE system. To alleviate this curse of dimensionality, we instead derive exact, albeit unclosed, reduced-order PDF (RoPDF) equations for low-dimensional observables/quantities of interest. The unclosed terms take the form of state-dependent conditional expectations, which are directly estimated from data at sparse observation times. However, for systems exhibiting stiff, multiscale dynamics, data sparsity introduces regression discrepancies that compound during RoPDF evolution. This is overcome by introducing a kinetic-like defect term to the RoPDF equation, which is learned by assimilating in sparse, low-fidelity RoPDF estimates. Two assimilation methods are considered, namely nudging and deep neural networks, which are successfully tested against Monte Carlo simulations.
Paper Structure (18 sections, 1 theorem, 39 equations, 8 figures)

This paper contains 18 sections, 1 theorem, 39 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Reduced-order Method of Distributions
  3. Data Assimilation
  4. Nudging
  5. Deep Neural Networks
  6. Numerics
  7. Assimilation Training
  8. Computational Complexity
  9. Experiments
  10. Stiff Linear System
  11. RoPDF Equation & Numerics
  12. Error Analysis
  13. Power System Cascade Outages
  14. RoPDF Equation & Numerics
  15. Regression
  16. ...and 3 more sections

Key Result

Theorem 2.2

\newlabelthm0 $\Pi_{x_k}(X_k,t;{\omega})$ almost surely obeys, in the sense of distributions, the linear conservation law

Figures (8)

  • Figure 1: (Left) Learned $\hat{{\mathcal{R}}}$ from $N_\text{MC}^\text{tr} = 5\times 10^2$ MC realizations of \ref{['eq:linrode']} at sparse times $\mathbf T_\nu$ with $\nu=2\times 10^2$. (Middle) The learned advection coefficient of \ref{['eq:linpdf']}. (Right) Evolution of the homogeneous solution $f_{x_1}^\text{h}$ to \ref{['eq:linpdf']}.
  • Figure 2: (Left) Temporal evolution of $f_{x_1}^\text{h}$$L_1$ error against the yardstick $f_\text{MC}$ for $\nu\in\{1,2,5\}\times 10^2$ and $N_\text{MC}^\text{tr} = 5\times 10^2$. (Middle) Snapshot of $f_\text{MC}$, $f_{x_1}^\text{h}$, $\hat{f}_{x_1}^\text{NR}$, and $\hat{f}_{x_1}^\text{DNN}$ for $\nu=2\times 10^2$ at time $t=7.2$. (Right) Defects $\hat{f}_{x_1}^\text{d}$ corresponding to the middle plot for DNN and NR observers. The latter is computed ex post facto as $\hat{f}_{x_1}^\text{NR} - f_{x_1}^\text{h}$.
  • Figure 3: Evolution of $L_1$ error for $\nu\in\{1,2,5\}\times 10^2$ and $N_\text{MC}^\text{tr} = 5\times 10^2$. (Left) $\hat{f}_{x_1}^\text{NR}$ against the assimilated $f_\text{MC}^{\text{tr},\nu}$. Green ticks on the $t$-axis represent short assimilation periods, i.e., when $\lambda_\nu(t)>0$. (Middle) $\hat{f}_{x_1}^\text{NR}$ against the yardstick $f_\text{MC}$. (Right) $\hat{f}_{x_1}^\text{DNN}$ against $f_\text{MC}$.
  • Figure 4: Convergence rates, on a log-log scale, for the spatiotemporal $L_1$ error of the observations $f_\text{MC}^{\text{tr},\nu}$ (left), $\hat{f}_{x_1}^\text{NR}$ (middle), and $\hat{f}_{x_1}^\text{DNN}$ (right) for $\nu=\{1,2,5\}\times10^2$ as $N_\text{MC}^\text{tr}$ increases.
  • Figure 5: $N_\text{MC}^\text{tr} = 2\times 10^3$ MC realizations (blue dots) of $(\gamma_{12}, \tilde{h}_{12})$ at time $t=0.1$. (Left) GLLR estimate of $\hat{{\mathcal{R}}}$ using $10$-fold CV for bandwidth selection. (Middle) Data transformed to standard Gaussian variates and corresponding GLLR fit with simple plug-in bandwidth. (Right) Fit from the middle plot transformed back to the original scale to obtain a more accurate estimate of $\hat{{\mathcal{R}}}$.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Remark 2.1
  • Theorem 2.2
  • Proof 1
  • Remark 3.1
  • Remark 3.2
  • Remark 5.1