Symplectic Induced-Order Modeling for Hamiltonian Systems

• Symplectic induced-order modeling is a reduced-order technique that preserves key geometric invariants in Hamiltonian systems, ensuring stability and energy conservation.
• It employs greedy basis generation, structure-preserving integrators, and nonintrusive operator inference to construct accurate and efficient reduced models for high-dimensional dynamics.
• Applications in nonlinear wave equations, elasticity, and quantum mechanics highlight its critical role in maintaining symplectic structure for long-term simulation fidelity.

Symplectic induced-order modeling (ROM) encompasses a collection of advanced methodologies for reduced-order modeling of Hamiltonian and multi-symplectic systems, prioritizing the exact preservation of underlying geometric structures—primarily the symplectic form and energy invariants—that are critical for ensuring stability and long-term fidelity. Recent advances in symplectic ROM span projection-based greedy algorithms, structure-preserving decomposition techniques, randomized matrix factorizations, manifold-based model reduction using neural networks, and non-intrusive operator inference. These approaches are widely used in parametric PDEs, nonlinear wave equations, elasticity, and computational physics where accurate reduced models are required for large-scale, high-dimensional systems.

1. Symplectic Projection Principles and Structure Preservation

Canonical Hamiltonian systems are described by the evolution equation

$\dot{z} = J_{2n} \nabla_z H(z)$

where $J_{2n}$ is the canonical symplectic matrix and $H(z)$ is the Hamiltonian. The essential requirement for symplectic ROM is that the reduction transformation (basis matrix $A \in \mathbb{R}^{2n \times 2k}$) satisfies

$A^T J_{2n} A = J_{2k}$

ensuring that the reduced variables $y$ (with $z \approx A y$) inherit the symplectic structure. The symplectic inverse—$A^+ = J_{2k}^T A^T J_{2n}$—acts as a projection operator onto this subspace. Projecting the dynamics yields a reduced system

$\dot{y} = J_{2k} \nabla_y H(Ay)$

preserving invariants, e.g., the Hamiltonian error

$\Delta H(t) = |H(z(t)) - H(AA^+z(t))|$

remains constant, supporting energy conservation and stability (Afkham et al., 2017). This principle extends to multi-symplectic PDEs, where symmetry conditions (involving both temporal and spatial symplectic matrices $K$, $L$) must be preserved in projection (Uzunca et al., 2022).

2. Greedy Symplectic Basis Generation and Error Indicators

A principal innovation is the use of greedy algorithms for symplectic basis construction. At each iteration, the parameter space is searched for the configuration $\omega^{k+1}$ maximizing the Hamiltonian error (or projection error), generating snapshots of the full system on which the largest error sample is selected. Basis enrichment proceeds via a symplectic Gram-Schmidt process, adding vector pairs (e.g., $v$ and $J_{2n}v$) to maintain symplecticity. The method converges exponentially under Kolmogorov $n$-width decay assumptions; error indicators such as

$\sigma_{2k}(s) = \| s - P_{2k}(s) \|$

guide the selection (Afkham et al., 2017). Similar greedy and randomized approaches can be generalized using non-orthonormal bases (PSD SVD-like), selecting column pairs optimized on weighted symplectic singular values, which achieves lower projection error and enhanced flexibility (Buchfink et al., 2019, Herkert et al., 2023).

3. Structure-Preserving Time Integration and Hyper-Reduction

Energy and symplectic preservation must be enforced at both continuous and discrete levels. Structure-preserving integrators, such as the average vector field (AVF) method and the implicit midpoint rule, exactly conserve polynomial invariants and Hamiltonians for the underlying system (Karasözen et al., 2019, Uzunca et al., 2022). For nonlinear problems, hyper-reduction techniques (e.g., Discrete Empirical Interpolation Method, DEIM) are combined with symplectic basis construction to accelerate computations while maintaining approximate invariance. POD and tensorial-POD approaches exploit the quadratic structure where possible, and an a priori error bound for DEIM ensures control over energy drift:

$$\| \delta_t \epsilon_h^k \| \leq \sum_{j=1}^N \Delta x \| (P^T\Phi)^{-1} \| \| (I-\Phi\Phi^T) \| \| \delta_t S_j(\alpha^k) \|$$

(Uzunca et al., 2022).

4. Extending to Manifolds and Nonlinear Symplectic Embeddings

Linear-subspace ROMs are insufficient for systems with slowly decaying Kolmogorov $n$-widths. Symplectic Induced-Order Modeling advances to nonlinear trial manifolds using symplectic manifold Galerkin (SMG) projection. The solution is reconstructed as

$x(t; \mu) \approx \text{ref}(\mu) + d(x_r(t; \mu))$

where $d\colon \mathbb{R}^{2n} \to \mathbb{R}^{2N}$ is a (weakly) symplectic map, often realized as a neural network decoder optimized with a symplecticity penalty term:

$$\text{sympl\_loss}(\theta) = \frac{1}{(2n)^2|X|} \sum_{x \in X} \|\nabla d(e(x; \theta))^T J_{2N} \nabla d(e(x; \theta)) - J_{2n}\|_F^2$$

The reduced model is then generated by enforcing that the projection of residual dynamics onto the tangent space vanishes, recovering reduced Hamiltonian evolution. Analytical results guarantee energy preservation and Lyapunov stability (Buchfink et al., 2021).

5. Data-Driven, Nonintrusive Symplectic Operator Learning

Recent frameworks provide physics-preserving, data-driven inference of reduced models in settings where explicit discretization operators are inaccessible. Nonintrusive operator inference uses snapshot data to learn structure-preserving reduced operators ($\hat{D}_q$, $\hat{D}_p$, etc.). Symmetry constraints are imposed so that the learned model

$$\dot{\hat{y}} = J_{2r} \nabla_{\hat{y}} \hat{H}(\hat{y})$$

preserves the canonical structure (Sharma et al., 2021). For multi-symplectic PDEs, operator learning is formulated as least-squares minimization subject to skew-symmetry:

$$\min_{D_x = -D_x^T} \| K_r \dot{\hat{Z}} + L_r D_x \hat{Z} - \hat{F} \|_F$$

enforcing energy and multi-symplectic conservation laws. This “grey box” approach is particularly effective for black-box solvers and generalizes robustly outside the training interval (Yıldız et al., 16 Sep 2024).

6. Neural Network-Based Symplectic Modeling

End-to-end symplectic ROMs governed by neural networks have been instantiated using HénonNets—deep architectures composed of invertible, exact symplectic layers:

$$H(V, \eta): [x; y] \mapsto [y + \eta; x + \nabla V(y)]$$

with stacked layers yielding universal approximation of symplectic diffeomorphisms. The encoder maps from high-dimensional to low-dimensional latent symplectic manifolds, while the latent-space flow map is also symplectic (via HénonNet). A truncated symplectic subspace and its canonical inclusion (with optional linear G-reflector corrections) are employed for latent manifold assembly:

$$f_\text{enc}(x) = \tau \circ G_\text{full} \circ \mathcal{A}_\text{full}(x),\qquad f_\text{dec}(y) = \mathcal{A}_\text{full}^{-1} \circ G_\text{full}^{-1} \circ \iota(y)$$

These neural architectures guarantee exact preservation of the symplectic structure at all stages (Chen et al., 16 Aug 2025).

7. Applications and Numerical Validation

Symplectic induced-order ROMs have been validated on a broad range of problems including the parametric wave equation, nonlinear Schrödinger equation, shallow water equations, linear elasticity, the Korteweg–de Vries equation, and the Zakharov–Kuznetsov equation (Afkham et al., 2017, Buchfink et al., 2019, Karasözen et al., 2019, Uzunca et al., 2022, Yıldız et al., 16 Sep 2024, Chen et al., 16 Aug 2025). In all cases, symplectic ROMs demonstrate:

• Near-constant Hamiltonian behavior over long integrations
• Dramatically reduced projection error (for non-orthonormal bases and manifold-based reductions)
• Exponential convergence rates under greedy or decomposition-based algorithms
• Robust generalization capabilities, particularly in data-driven frameworks with operator inference
• Substantial computational speedups in multi-query and real-time scenarios, when randomized matrix factorizations or hyper-reduction are employed

The methods ensure the preservation of key invariants (energy, mass, momentum, enstrophy, circulation) crucial for long-term stability and fidelity—directly extending modeling capabilities for high-dimensional, parametric, and strongly nonlinear Hamiltonian systems.

8. Implications, Practical Contexts, and Future Directions

Symplectic induced-order modeling provides a rigorously validated pathway for reduced-order modeling that does not compromise geometric integrity. Core implications include:

• Accurate, structure-preserving model reduction in parametric and nonlinear Hamiltonian systems
• Applicability in black-box, large-scale industrial codes due to nonintrusive data-driven frameworks
• Theoretical guarantees (e.g., a-posteriori error bounds, Lyapunov stability, energy invariance)
• Fusion of geometric numerical integration, randomized linear algebra, tensor decomposition, and deep learning to address high dimensionality and nonlinearity

A plausible implication is the expansion of symplectic ROM frameworks in molecular dynamics, plasma physics, quantum mechanics, and real-time simulation environments, where long-term accurate and efficient reduced models are essential. Anticipated future research directions include extending manifold-based reduction methods to more complex nonlinear systems, integrating deeper symplectic neural architectures, and systematic paper of error propagation and stability in both intrusive and nonintrusive ROMs.