Trajectory-Aware ROM Preconditioners

• Trajectory-aware ROM preconditioners are reduced-order modeling techniques that adapt the preconditioner, basis, and even time coordinates to the system's actual trajectory.
• They integrate online data, probabilistic formulations, and adaptive error indicators to significantly improve convergence, reduce residual errors, and enhance computational speed.
• While offering marked efficiency gains, these methods require careful management of computational budgets and robust safeguard mechanisms to handle complex, high-dimensional trajectories.

Trajectory-aware ROM preconditioners are reduced-order-model constructions in which the preconditioner, reduced basis, latent dynamics, or even the time coordinate is adapted to the trajectory actually realized by a dynamical system, an optimization process, or an iterative solver. In the literature considered here, the phrase spans strict linear-algebraic preconditioning of reduced equations, ROM-enhanced preconditioners for transport and parallel-in-time solvers, and broader conditioning strategies that use trajectory information to reshape residual norms, posterior trajectory distributions, or stretched-time coordinates (Lindsay et al., 2022, Peng, 2024, Liu et al., 2020, Standridge et al., 17 Mar 2026). The unifying idea is that static offline surrogates are often inefficient or inaccurate when the relevant low-dimensional structure is concentrated along an online path rather than across the entire admissible state or parameter domain.

1. Scope and taxonomy

The defining feature of trajectory awareness is not a single algorithmic pattern but the choice of which trajectory is treated as informative. Depending on the application, that trajectory may be a state trajectory of a parametric dynamical system, an optimization path in parameter space, a nonlinear residual trajectory, a Krylov or source-iteration correction trajectory, or a reparameterized time path used to regularize stiff dynamics. This broad usage is visible across Bayesian ROM construction for unknown trajectories (Héas et al., 2017), progressive ROM updates along optimization trajectories (Zahr et al., 2014), residual-preconditioned projection methods (Lindsay et al., 2022), and ROM-enhanced preconditioners for radiative transfer and all-at-once time discretizations (Peng, 2024, Tang et al., 5 Sep 2025, Liu et al., 2020).

Classical offline-online ROM pipelines assume that a fixed reduced basis, trained on representative snapshots, remains accurate throughout deployment. Several of the cited works challenge that assumption directly. For nonlinear PDE-constrained optimization, the solution manifold can be highly curved, so a global reduced-order basis may become unmanageably large and waste most of its information away from the actual optimization path (Zahr et al., 2014). For static non-intrusive surrogates, forecasting beyond the training manifold can lead to drift or destabilization, motivating adaptive updates of both subspace and reduced dynamics (Hedayat et al., 11 Feb 2026). For transport problems, a ROM trained on offline correction data may lose effectiveness once the online preconditioned residual trajectory departs from the offline one (Tang et al., 5 Sep 2025).

Trajectory source	Trajectory-aware mechanism	Representative papers
Unknown system realizations with incomplete observations	Bayesian data-enhanced surrogate for ROM learning	(Héas et al., 2017)
Optimization iterates in parameter space	Progressive basis enrichment and nonlinear trust region	(Zahr et al., 2014)
Nonlinear residual trajectory	Preconditioning inside LSPG residual minimization	(Lindsay et al., 2022)
Solver correction and residual trajectories	ROM-projected kinetic corrections, online greedy updates, switching preconditioners	(Peng, 2024, Tang et al., 5 Sep 2025, Liu et al., 2020, Liu et al., 20 Mar 2026)
Stretched-time trajectory for stiff dynamics	Optimization-based time reparameterization	(Standridge et al., 17 Mar 2026)

This taxonomy also clarifies a recurrent ambiguity: some papers use “preconditioner” in the conventional iterative-solver sense, while others use it more broadly for any transformation that improves the conditioning of ROM construction, ROM learning, or reduced prediction.

2. Probabilistic and optimization-path formulations

A probabilistic formulation appears in "Reduced Modeling of Unknown Trajectories" (Héas et al., 2017). The setup assumes that the true trajectory distribution $p_X$ is unknown, but two information sources are available: a surrogate prior density $p$ on trajectories and incomplete observations $y$ linked to trajectories $x$ by a known conditional distribution $p(y|x)$. The central construction is the data-enhanced surrogate

$\hat{p}(x) := \int p(x|y) p_Y(y)\,dy,$

with posterior

$$p(x|y) = \frac{p(y|x) p(x)}{\int p(y|x') p(x')\,dx'}.$$

The paper states the key property

$$\mathrm{KL}(p_X, \hat{p}) \leq \mathrm{KL}(p_X, p),$$

so complementing the prior with the collected data provably enhances knowledge of the distribution of system trajectories (Héas et al., 2017). Standard ROM techniques such as Proper Orthogonal Decomposition and Dynamic Mode Decomposition are then recast as optimizations over weighted posterior samples obtained by Sequential Monte Carlo. In the paper’s terminology, this probabilistic construction can be viewed as an effective preconditioner for ROM construction, because it reshapes the surrogate objective with data-informed uncertainty rather than point estimates alone.

A different but related trajectory-aware strategy is the progressive ROM framework for PDE-constrained optimization in "Progressive construction of a parametric reduced-order model for PDE-constrained optimization" (Zahr et al., 2014). The method breaks the traditional offline-online framework by defining a sequence of reduced optimization problems whose constraining ROM is retrained from High-Dimensional Model samples taken at previously encountered solutions. The reduced optimization problems include a nonlinear trust region based on the HDM residual norm,

$\frac{1}{2} \| \mathbf{R}(\mathbf{\bar{w} + \mathbf{V}\mathbf{y}, \boldsymbol{\mu}) \|_2^2 \leq \epsilon,$

which keeps the optimization trajectory in a region where the ROM is accurate (Zahr et al., 2014). State and sensitivity snapshots are incorporated into the reduced-order basis, low-rank SVD updates are used for efficient enrichment, and the reference vector is chosen as the closest available HDM solution. Applied to subsonic aerodynamic shape optimization, the framework reduces the number of queries to the HDM by a factor of 4-5, with errors in the optimal solution far less than $0.1\%$ (Zahr et al., 2014). A plausible implication is that trajectory awareness here acts as localization: the reduced model is specialized to the path actually traversed rather than to a global parameter domain.

3. Residual-norm shaping and ideal projection in reduced equations

A strict preconditioning formulation is developed in "Preconditioned Least-Squares Petrov-Galerkin Reduced Order Models" (Lindsay et al., 2022). Standard LSPG solves

$p$0

whereas the preconditioned formulation directly embeds a nonsingular matrix $p$1 in the residual minimization: $p$2 This changes the norm defining the residual minimization, can improve the residual-based stability constant, and can scale components of the residual for problems with disparate physical scales (Lindsay et al., 2022).

The paper’s central theoretical claim is that an ideal preconditioned LSPG ROM, obtained with the inverse of the Jacobian of the corresponding Full Order Model, emulates projection of the FOM solution increment onto the reduced basis, which is described as a lower bound on the ROM solution error for a given reduced basis (Lindsay et al., 2022). Since the preconditioner may depend on the current Jacobian, this is trajectory-aware in the local nonlinear sense: the reduced minimization is adapted to the instantaneous structure of the FOM residual. Practical Jacobi, Gauss-Seidel, and ILU preconditioners are computed using the Ifpack library within Trilinos and tested in the Albany HPC code. Numerically, the method yields improvements of 2–7 orders of magnitude in ROM solution error, drops reduced Jacobian condition numbers by up to 10 orders of magnitude, and reduces the number of nonlinear iterations and wall time by up to $p$3 (Lindsay et al., 2022).

This formulation is significant because it distinguishes two meanings of ROM preconditioning. One meaning is external: preconditioning the linear systems that arise while solving a ROM. The other, used here, is intrinsic: preconditioning the reduced optimization principle itself. The latter more directly controls approximation error, stability, and reduced conditioning.

4. Solver-trajectory ROM preconditioners for transport and all-at-once systems

The most literal use of the phrase “trajectory-aware ROM preconditioner” appears in parametric radiative transfer and related iterative solvers. In "A Flexible GMRES Solver with Reduced Order Model Enhanced Synthetic Acceleration Preconditioenr for Parametric Radiative Transfer Equation" (Peng, 2024), the ideal kinetic correction equation for the discrete RTE is projected onto a ROM space: $p$4 The resulting ROMSAD preconditioner is nonlinear and iteration-dependent because it may switch after a fixed number of early iterations from ROM-based correction to Diffusion Synthetic Acceleration, which motivates the use of FGMRES (Peng, 2024). The paper introduces a greedy offline construction based on the largest residual of the projected correction, and reports that FGMRES with the ROMSAD preconditioner is more efficient than GMRES with the right DSA preconditioner, can achieve up to $p$5 speedup in some examples, and benefits from offline training cost reductions sometimes by a factor of 4 or more (Peng, 2024).

The trajectory-aware extension is made explicit in "Synthetic Acceleration Preconditioners for Parametric Radiative Transfer Equations based on Trajectory-Aware Reduced Order Models" (Tang et al., 5 Sep 2025). That paper identifies a specific failure mode of ROMSAD: the underlying ROM ignores the preconditioner-dependence of the residual trajectory, producing a mismatch between offline and online residual trajectories (Tang et al., 5 Sep 2025). The proposed remedy is to iteratively construct a sequence of ROMs, one for each early iteration, so that each basis is generated from residuals produced by the preceding ROM-informed preconditioner. For early iterations $p$6, the online correction takes the form

$p$7

Numerical tests show superior efficiency over DSA and substantial gains in both efficiency and robustness over ROMSAD; for a parametric lattice problem, trajectory-aware ROM preconditioners achieve rapid convergence within only $p$8-$p$9 iterations online (Tang et al., 5 Sep 2025). The conceptual point is precise: the object being reduced is not merely the solution manifold, but the residual-correction trajectory induced by the preconditioned iteration itself.

A related online paradigm appears in "A ROM-accelerated parallel-in-time preconditioner for solving all-at-once systems from evolutionary PDEs" (Liu et al., 2020). There, Step-(b) of the ParaDIAG preconditioner requires the solution of many complex-shifted spatial systems,

$y$0

which are replaced by Galerkin ROM solves

$y$1

Because the ROM must be built online at each iteration, the basis is generated greedily from the current residual trajectory using a residual-based error indicator

$y$2

The paper introduces heuristic accelerations such as recursive reduced-matrix updates, random subsampling, and coarse residual evaluation, and reports more than an order-of-magnitude speed-up over multigrid-based ParaDIAG preconditioning, with 8x–20x speed-ups in the reported numerics (Liu et al., 2020). The same paper also notes a limitation that recurs throughout the area: when the trajectory complexity increases, especially in convection-dominated problems, the reduced basis dimension grows and the efficiency gain narrows.

An offline-free time-dependent transport variant is given in "On-the-Fly ROM-Based Acceleration of SI-DSA for Implicit Time Marching of the Radiative Transfer Equation" (Liu et al., 20 Mar 2026). ROMs are built on the fly from streaming solution data using dynamical mode decomposition, incremental low-rank singular value decomposition, and error indicators. They enhance both initial guesses and preconditioners during time marching, and the reported overall speedup over classical SI-DSA is $y$3 to $y$4, with only marginal overhead for ROM construction and updates (Liu et al., 20 Mar 2026). This is trajectory-aware in the temporal sense: the reduced model exploits low-rank structure across successive time steps rather than across a fixed offline ensemble.

5. Time-reparameterized and adaptive learning-compatible formulations

Trajectory awareness also appears as a coordinate transformation rather than as a basis update or residual-space preconditioner. "Trajectory-Optimized Time Reparameterization for Learning-Compatible Reduced-Order Modeling of Stiff Dynamical Systems" (Standridge et al., 17 Mar 2026) treats time reparameterization as a stiffness-mitigation mechanism for neural ODE ROMs. A strictly increasing time map $y$5 yields the stretched-time dynamics

$y$6

and the proposed trajectory-optimized time reparameterization chooses a traversal-speed profile in arc-length coordinates to penalize acceleration in stretched time (Standridge et al., 17 Mar 2026). The paper states that this produces reparameterized trajectories that are better conditioned and easier to learn than existing time-reparameterization methods, and reports loss reductions of one to two orders of magnitude compared to benchmark algorithms (Standridge et al., 17 Mar 2026). In this setting, the time map itself functions as a preconditioner for reduced learning dynamics.

A full-trajectory training perspective is developed in "Full trajectory optimizing operator inference for reduced-order modeling using differentiable programming" (Chakrabarti et al., 2023). The method maintains the form of the ROM-ODE obtained from Galerkin projection, but coefficients are learned directly from data using gradient descent facilitated by differentiable programming. The paper states that the method minimizes a more rigorous full trajectory error norm than a linearized calibration objective, stabilizes the ROM by displacing the eigenvalues of the linear dynamics matrix to the complex left half-plane, and adds more dissipation to the trailing POD modes, which explains its better long-term performance (Chakrabarti et al., 2023). The relevant trajectory-aware feature is the objective functional: optimization is carried out against the entire predicted trajectory rather than local one-step behavior.

The adaptive non-intrusive line of work generalizes this logic to continual updates. "Toward Adaptive Non-Intrusive Reduced-Order Models: Design and Challenges" (Hedayat et al., 11 Feb 2026) formalizes adaptive ROMs that update both latent subspace and reduced dynamics online, using Adaptive OpInf, Adaptive NiTROM, and a hybrid that initializes NiTROM with an OpInf update. The paper explicitly separates training, adaptation, and deployment regimes, introduces an online data window and adaptation window, and argues that predictive claims for ROMs must be cost-aware and transparent, with explicit reporting of online budgets and full-order model queries (Hedayat et al., 11 Feb 2026). On a transiently perturbed lid-driven cavity flow, static Galerkin, OpInf, and NiTROM models drift or destabilize beyond training, whereas Adaptive OpInf suppresses amplitude drift, Adaptive NiTROM can attain near-exact energy tracking under frequent updates, and the hybrid is the most reliable under regime changes and minimal offline data (Hedayat et al., 11 Feb 2026). A plausible interpretation is that online adaptation acts as a recurrent trajectory-aware preconditioning of both coordinates and reduced operators.

6. Performance characteristics, limitations, and common misunderstandings

Across the literature, trajectory-aware ROM preconditioners are associated with substantial but highly problem-dependent gains. Reported improvements include a factor of 4-5 fewer HDM queries in PDE-constrained optimization (Zahr et al., 2014), 2–7 orders of magnitude smaller ROM errors and up to $y$7 reductions in nonlinear iterations and wall time for preconditioned LSPG (Lindsay et al., 2022), more than an order-of-magnitude speed-up and 8x–20x gains for ParaDIAG acceleration (Liu et al., 2020), $y$8 to $y$9 speedup for on-the-fly ROM acceleration of SI-DSA (Liu et al., 20 Mar 2026), rapid convergence within only $x$0-$x$1 iterations online for trajectory-aware RTE preconditioners (Tang et al., 5 Sep 2025), and loss reductions of one to two orders of magnitude for trajectory-optimized time reparameterization in stiff ML-ROMs (Standridge et al., 17 Mar 2026).

Several misunderstandings recur. First, trajectory awareness is not synonymous with online basis enrichment alone. Bayesian posterior weighting over unknown trajectories (Héas et al., 2017), full-trajectory training objectives in operator learning (Chakrabarti et al., 2023), and optimized stretched-time coordinates (Standridge et al., 17 Mar 2026) are also trajectory-aware, even though the mechanism is probabilistic, variational, or representational rather than basis-adaptive. Second, trajectory-aware ROMs do not remove the need for safeguards. Nonlinear trust regions based on residual indicators (Zahr et al., 2014), switching from ROM-based correction to DSA (Peng, 2024), and explicit error indicators with adaptive updates (Liu et al., 20 Mar 2026) remain central to robustness. Third, “trajectory-aware” does not mean cost-free adaptivity. Online greedy updates, moving data windows, manifold optimization, and full-order queries introduce nontrivial computational budgets; this is why the adaptive non-intrusive literature emphasizes transparent reporting of training, adaptation, and deployment costs (Hedayat et al., 11 Feb 2026).

The main technical limitation is structural. These methods succeed when the relevant online trajectory, residual trajectory, or stretched-time representation is low-dimensional, smooth, or otherwise compressible. When the solution manifold is highly curved (Zahr et al., 2014), the residual trajectory changes under preconditioning (Tang et al., 5 Sep 2025), or the time evolution becomes more complex, the reduced basis can grow and efficiency gains can diminish (Liu et al., 2020). The strongest current results therefore come from methods that explicitly align the reduced representation with the trajectory induced by the actual solver, optimizer, or learned dynamics, rather than with a static offline approximation of the full problem.