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Inverse problems for history-enriched linear model reduction

Arjun Vijaywargiya, George Biros

TL;DR

This work tackles closure error in projection-based reduced-order models for high-dimensional driven linear systems by deriving exact history-enriched dynamics from the Mori-Zwanzig formalism, yielding a Volterra integro-differential equation for the resolved state that includes a Markovian term, memory integral, and a deterministic noise term. It then formulates two inverse problems to learn the operators $R$, $K$, and $B$ from trajectory data under a stationarity assumption, and develops a greedy time-marching least-squares scheme to identify these operators efficiently. The authors establish identifiability and well-posedness results under mild conditions, showing that full-state data preserve identifiability for time-varying $A(t)$, while partial data require a time-invariant $A$ or stabilization via time-smoothing regularization. Numerical experiments on a one-dimensional reaction-diffusion-advection equation demonstrate faithful reconstruction of the history-enriched MZ operators and accurate prediction of the resolved dynamics, highlighting the practical potential of history-aware ROMs for linear systems and guiding future extensions to nonlinear settings.

Abstract

Standard projection-based model reduction for dynamical systems incurs closure error because it only accounts for instantaneous dependence on the resolved state. From the Mori-Zwanzig (MZ) perspective, projecting the full dynamics onto a low-dimensional resolved subspace induces additional noise and memory terms arising from the dynamics of the unresolved component in the orthogonal complement. The memory term makes the resolved dynamics explicitly history dependent. In this work, based on the MZ identity, we derive exact, history-enriched models for the resolved dynamics of linear driven dynamical systems and formulate inverse problems to learn model operators from discrete snapshot data via least-squares regression. We propose a greedy time-marching scheme to solve the inverse problems efficiently and analyze operator identifiability under full and partial observation data availability. For full observation data, we show that, under mild assumptions, the operators are identifiable even when the full-state dynamics are governed by a general time-varying linear operator, whereas with partial observation data the inverse problem has a unique solution only when the full-state operator is time-invariant. To address the resulting non-uniqueness in the time-varying case, we introduce a time-smoothing Tikhonov regularization. Numerical results demonstrate that the operators can be faithfully reconstructed from both full and partial observation data and that the learned history-enriched MZ models yield accurate trajectories of the resolved state.

Inverse problems for history-enriched linear model reduction

TL;DR

This work tackles closure error in projection-based reduced-order models for high-dimensional driven linear systems by deriving exact history-enriched dynamics from the Mori-Zwanzig formalism, yielding a Volterra integro-differential equation for the resolved state that includes a Markovian term, memory integral, and a deterministic noise term. It then formulates two inverse problems to learn the operators , , and from trajectory data under a stationarity assumption, and develops a greedy time-marching least-squares scheme to identify these operators efficiently. The authors establish identifiability and well-posedness results under mild conditions, showing that full-state data preserve identifiability for time-varying , while partial data require a time-invariant or stabilization via time-smoothing regularization. Numerical experiments on a one-dimensional reaction-diffusion-advection equation demonstrate faithful reconstruction of the history-enriched MZ operators and accurate prediction of the resolved dynamics, highlighting the practical potential of history-aware ROMs for linear systems and guiding future extensions to nonlinear settings.

Abstract

Standard projection-based model reduction for dynamical systems incurs closure error because it only accounts for instantaneous dependence on the resolved state. From the Mori-Zwanzig (MZ) perspective, projecting the full dynamics onto a low-dimensional resolved subspace induces additional noise and memory terms arising from the dynamics of the unresolved component in the orthogonal complement. The memory term makes the resolved dynamics explicitly history dependent. In this work, based on the MZ identity, we derive exact, history-enriched models for the resolved dynamics of linear driven dynamical systems and formulate inverse problems to learn model operators from discrete snapshot data via least-squares regression. We propose a greedy time-marching scheme to solve the inverse problems efficiently and analyze operator identifiability under full and partial observation data availability. For full observation data, we show that, under mild assumptions, the operators are identifiable even when the full-state dynamics are governed by a general time-varying linear operator, whereas with partial observation data the inverse problem has a unique solution only when the full-state operator is time-invariant. To address the resulting non-uniqueness in the time-varying case, we introduce a time-smoothing Tikhonov regularization. Numerical results demonstrate that the operators can be faithfully reconstructed from both full and partial observation data and that the learned history-enriched MZ models yield accurate trajectories of the resolved state.
Paper Structure (26 sections, 3 theorems, 72 equations, 4 figures, 8 tables)

This paper contains 26 sections, 3 theorems, 72 equations, 4 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Outline
  5. Formulation
  6. Mori-Zwanzig formalism
  7. Inverse problems
  8. Formulation
  9. Discrete formulation based on the Backward Euler rule
  10. Reconstruction algorithms
  11. Reconstruction with full observation data
  12. Reconstruction of $~R$
  13. Reconstruction of $~K$ and $~B$
  14. Reconstruction under a finite-memory approximation
  15. Reconstruction with partial data
  16. ...and 11 more sections

Key Result

Theorem 3.1

The Full Data inverse problem is well-posed under the following assumptions: Moreover, if $~G(t)$ and $\widetilde{~G}(t)$ are zero matrices, the condition numbers of least-squares operators in eq:Rn, eq:R, and eq:KB satisfy where $\lambda_{\max}(~H)$ and $\lambda_{\min}(~H)$ are the extremal eigenvalues of the Hermitian part $~H = \frac{~A + ~A^\star}{2}$ of the full-state linear operator $~A$ i

Figures (4)

  • Figure 1: (a) True and predicted trajectories for two degrees of freedom (selected as the resolved variables) for a one-dimensional advection equation with periodic boundary conditions. Predictions are shown for a data-driven Markovian model and a data-driven MZ model \ref{['eq:vide']}. (b) Absolute error in the predicted trajectories. The MZ model recovers the true trajectories, whereas the Markovian model exhibits large errors.
  • Figure 2: Unknowns to be reconstructed at the first three time steps in \ref{['eq:KB']}, \ref{['eq:partialreform']}, and \ref{['eq:nonstat2']}, respectively.
  • Figure 3: Predicted trajectories of the resolved variables $~\phi$ in all the reaction-diffusion-advection test cases for a single test initial condition. The trajectories are obtained by solving \ref{['eq:vide']}, with operators $~R$, $~K$, and $~B$ reconstructed using full observation data.
  • Figure 4: Predictions under a finite-memory approximation. Left: time evolution of the Frobenius norm $||~K||_F$ of the reconstructed full-memory kernel. Right: predicted trajectories of the first component of $\phi$ for a single test initial condition in the first six reaction-diffusion-advection test cases, with $\Delta t = 0.03125$. The operators were reconstructed using full observation data. Finite memory is enforced by truncating the kernel support, setting $K_n=0$ for all lags $n>m$, so that only contributions up to $t_m=m\Delta t$ are retained; $m=N_T$ recovers the full-memory model. In (a) and (c), trajectory accuracy degrades as the truncation becomes more severe (smaller $m$), whereas in (f) the trajectories are not predicted accurately for any truncation level. This behavior is attributable to non-decaying memory kernels, indicating a strong influence of the memory term, particularly in case (f).

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.2
  • proof
  • ...and 7 more