Basis Optimization Sharpens Reduced Port-Hamiltonian Power and Lattice System Models

Computational engineers simulating complex physical networks frequently confront a difficult trade-off between dynamical fidelity and strict thermodynamic stability. When constructing reduced-order models of large-scale nonlinear systems, standard projection techniques like proper orthogonal decomposition excel at compressing state vectors into low-dimensional subspaces but routinely omit derivative components critical to transient trajectories. In a study published in Results in Engineering, an engineering research team led by Hamad Alduaij, Jafarali Parol, Haoran Li, and Yang Weng demonstrated that coupling coordinate subspace selection directly with an energy-constrained operator refit resolves this operational bottleneck. By iteratively scoring projection candidates against both state reconstruction limits and held-out derivative residuals, their framework preserves physical dissipation inequalities while outperforming conventional fixed-basis inference across nonlinear power network and multi-particle lattice benchmarks.

The study centers on port-Hamiltonian systems, a mathematical framework that models physical networks by separating energy storage from skew-symmetric power-preserving interconnections, symmetric dissipation, and external ports. In nonlinear settings, such as electrical power grids experiencing disturbance-driven oscillations, standard reduced-order models enforce stability through dissipative constraints on the inferred operators. However, standard operator inference historically selects an orthonormal proper orthogonal decomposition basis based solely on state reconstruction error before fitting a dissipative operator to snapshot trajectories. Because state reconstruction and constrained fitting optimize separate quantities, derivative directions discarded during initial coordinate selection remain entirely inaccessible to any operator confined to that reduced subspace. Consequently, an energy-stable model fit within a poorly chosen subspace can fail to capture true transient dynamics.

To overcome this limitation, the authors developed an analytical formulation that quantifies admissible-model distance on the Grassmann manifold. Their mathematical derivation proves that the squared Frobenius residual between full-order derivative snapshots and predictions from an admissible reduced model splits into two orthogonal components: the derivative content lying strictly outside the chosen projection subspace, and the in-subspace fitting defect of the best dissipative operator. Zero total distance requires both derivative closure and an exact dissipative operator match within the subspace. Furthermore, the researchers proved that orthogonal rotations of the basis preserve both the minimal regularized fitting error and the lifted trajectory predictions, establishing that the subspace optimization depends strictly on the subspace itself rather than the specific coordinate orientation chosen.

Connecting algebraic subspace properties to time-domain performance, the authors proved a refit-aware state-error bound governing trajectory simulations. For systems evolving near equilibrium, the analytical bound demonstrates that the Euclidean error between the true full-order trajectory and the lifted reduced simulation is controlled by the initial projection error, the accumulated vector-field discrepancy outside the subspace, and the integrated operator-fit defect within the subspace, modulated by a Lipschitz exponential growth factor. This analytical result established that lowering state projection error alone cannot guarantee accurate time-domain simulation, formally motivating a search algorithm that balances state reconstruction retention with successive reductions in operator fitting residuals.

To generate physically meaningful search candidates, the researchers derived a local Taylor expansion of the port-Hamiltonian equations about an equilibrium state. This expansion revealed how interconnection, dissipation, and higher-order dynamic remainders generate derivative directions outside any trial subspace. By evaluating the leading singular vectors of the linearized interconnection matrix and the leading eigenvectors of the local dissipation matrix, the authors extracted candidate coordinate directions specifically linked to interconnection response, state variation, and energy dissipation. These structured physical candidate vectors were assembled into a candidate pool alongside the initial basis vectors, creating a targeted set of trial coordinates for subspace synthesis.

The optimization routine executes a columnwise replacement search over the candidate pool, guided by two strict gating criteria. First, a candidate replacement must satisfy a reconstruction feasibility constraint scaled by a positive diagonal matrix of state deviations, ensuring that average state reconstruction error on the fitting data does not degrade beyond ten to twenty percent relative to the initial seed basis. Second, the candidate basis undergoes an accelerated projected-gradient solve to obtain a trial dissipative operator, which is evaluated across independent selection folds. A trial is accepted only if it improves the mean held-out derivative residual by more than a numerical tolerance while avoiding residual deterioration across any individual validation fold. Once the columnwise search terminates, a final semidefinite program refits the operator on the complete training snapshot dataset, followed by an eigenvalue cone projection that guarantees energy stability.

The primary experimental evaluation tested the methodology on twenty independently generated 29-state multimachine swing systems equipped with first-order turbine governors, representing ten interconnected generators under lossy line conditions. Across reduced orders of four, six, and eight, the proposed optimization was initialized from a Darboux-inspired structure-oriented basis that pairs state coordinates using the system Jacobian and observability Gramian. Across all sixty matched system-order test cases, the proposed framework outperformed conventional fixed proper orthogonal decomposition operator inference, reducing the average held-out derivative regression residual by a difference of 0.138 and lowering normalized root-mean-square trajectory error by 0.481. Every tested recovery trajectory satisfied the analytical negative Hamiltonian derivative condition and empirical energy monotonicity checks without exhibiting numerical instability.

To dissect the exact origins of performance improvements, the researchers conducted detailed mechanism controls. Comparing the proposed search against the fixed Darboux-inspired seed model revealed that the initial seed provided the bulk of the trajectory recovery accuracy, while the subsequent search delivered an average held-out residual reduction of 0.030 across nineteen of the twenty power systems. Further mechanism controls at order eight demonstrated that refit-aware selection score outperformed projection-only selection on the identical physical candidate pool, reducing residual error by 0.032 and trajectory error by 0.072. When initialized directly from a standard proper orthogonal decomposition basis, the physical search improved both metrics across all twenty systems, reducing trajectory error by 0.350 relative to static proper orthogonal decomposition and confirming that the search operates effectively even without structure-informed initialization.

To examine scalability, the researchers extended the evaluation to two deterministic high-dimensional benchmarks: a 100-state nonlinear ladder network and a 2,000-state Toda lattice model under input-driven forced conditions. On the 1,000-particle Toda lattice compressed to order twenty, a refit-aware enriched proper orthogonal decomposition family reduced state tracking error by 66.4 percent under sinusoidal excitation and by over 90 percent under constant-input transfer relative to fixed-basis inference. A direct columnwise data-pool search on the Toda lattice lowered selection residuals by 34.5 percent and cut sinusoidal state error from 0.271 to 0.126. However, the tests also showed that candidate pool composition is critical: on the Toda system, the physical pool struggled with reconstruction feasibility due to degenerate dissipation eigenspaces, whereas the data-driven candidate pool successfully identified valid update directions.

In their analysis of study scope and operational boundaries, the authors emphasized several methodological constraints. The mathematical guarantees assume autonomous or collocated input-output structures, continuously differentiable energy functions, and direct access to full-state derivative snapshots. Furthermore, when proper orthogonal decomposition captures nearly all snapshot energy at higher reduced orders, such as order sixty on the Toda lattice, classical projection models tied or marginally exceeded inferred models, indicating that coordinate optimization delivers its greatest practical benefits under aggressive model compression. The authors concluded that deploying this coordinate optimization framework in broader operational contexts will require developing robust pre-filtering techniques for noisy state measurements and investigating hyper-reduction strategies to eliminate full-dimensional state evaluations during time integration.

Subject of Research: Reduced-order modelling of nonlinear port-Hamiltonian dynamical systems using coordinate basis optimization

Article Title: Reduced-order modelling of port-Hamiltonian systems using basis optimization

Article References: Alduaij, H., Parol, J., Li, H., & Weng, Y. (2026). Reduced-order modelling of port-Hamiltonian systems using basis optimization. Results in Engineering, 32, Article 114007. https://doi.org/10.1016/j.rineng.2026.114007

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.114007

Keywords: port-Hamiltonian systems, reduced-order modeling, operator inference, basis optimization, energy stability, Lyapunov stability, proper orthogonal decomposition, power network simulation, swing-governor dynamics, Toda lattice, Grassmann manifold, semidefinite programming, nonlinear dynamics, matrix optimization, projection methods, dynamical systems, computational engineering