GasNiTROM: Model Reduction via Non-Intrusive Optimization of Oblique Projection Operators and Guaranteed-Stable Latent-Space Dynamics

1. Problem Statement

The paper addresses the challenges in creating Non-Intrusive Reduced-Order Models (NI-ROMs) for high-dimensional nonlinear systems, particularly those exhibiting:

• Large transient growth: Systems where energy amplifies significantly before decaying.
• High sensitivity: Systems where low-energy or rapidly decaying states critically influence high-energy dynamics.
• Instability: Existing NI-ROMs often suffer from finite-time or infinite-time blow-up when operating far from equilibrium.

Limitations of Current Methods:

• Orthogonal Projections (e.g., POD-Galerkin, Operator Inference): These methods project data onto the most energetic modes. While computationally efficient, they often truncate "dynamically important" low-energy states, leading to poor forecasting accuracy and instability in transient regimes.
• Stability vs. Accuracy Trade-off: Recent methods (e.g., Lyapunov-stable Operator Inference) enforce global stability but still rely on orthogonal projections, sacrificing accuracy. Conversely, methods that optimize oblique projections for accuracy (e.g., NiTROM) often lack stability guarantees, leading to model blow-up.
• Intrusiveness: Many accurate methods (e.g., Balanced Truncation, LSPG) require access to the full-order solver's Jacobian and adjoint, making them inapplicable to "black-box" solvers or data-only scenarios.

2. Methodology: GasNiTROM

The authors propose GasNiTROM (Globally-asymptotically-stable Non-intrusive Trajectory-based optimization of Reduced-Order Models), a framework that simultaneously optimizes for accuracy (via oblique projections) and stability (via Lyapunov-constrained dynamics).

A. Mathematical Formulation

The reduced-order model (ROM) is defined by an encoder $\mathbf{\Psi}^\top$, a decoder $\mathbf{\Phi}(\mathbf{\Psi}^\top\mathbf{\Phi})^{-1}$, and latent-space dynamics $\mathbf{f}_r$. The goal is to minimize the forecasting error between the ROM output and the Full-Order Model (FOM) data.

1. Oblique Projections (Accuracy):
   Unlike standard methods that use orthogonal projections ($\mathbf{\Phi} = \mathbf{\Psi}$), GasNiTROM treats the encoder $\mathbf{\Psi}$ and decoder $\mathbf{\Phi}$ as independent optimization variables. This defines an oblique projection operator $\mathbb{P} = \mathbf{\Phi}(\mathbf{\Psi}^\top\mathbf{\Phi})^{-1}\mathbf{\Psi}^\top$, allowing the model to capture sensitivity mechanisms between low- and high-energy states.

   • Manifold Optimization: The optimization is performed over a product manifold:
     • Grassmann Manifold ($\mathcal{G}_{n,r}$): Parameterizes the decoder subspace (invariant to rotation/scaling).
     • Stiefel Manifold ($St_{n,r}$): Parameterizes the encoder (ensuring full column rank).

2. Guaranteed-Stable Dynamics (Stability):
   The latent dynamics are modeled as quadratic polynomials: $\dot{\mathbf{z}} = \mathbf{A}\mathbf{z} + \mathbf{H}:\mathbf{z}\mathbf{z}^\top + \mathbf{B}\mathbf{u}$.
   To guarantee Global Asymptotic Stability (GAS), the authors enforce a Lyapunov-based parameterization:

   • Linear Term ($\mathbf{A}$): Parameterized to ensure the linear part is Hurwitz (stable).
   • Quadratic Term ($\mathbf{H}$): Parameterized to be energy-conserving (skew-symmetric structure relative to a metric).
   • Parameterization Trick: To avoid rank-deficiency issues common in previous works (where $\tilde{\mathbf{Q}} = \mathbf{Q}\mathbf{Q}^\top$), the authors propose a new parameterization for the Lyapunov matrix: $\tilde{\mathbf{Q}} = \mathbf{Q}^{-1}\mathbf{Q}^{-\top}$. This guarantees positive definiteness by construction as long as $\mathbf{Q}$ is full rank.

3. Optimization and Gradients:
   The problem is solved via gradient-based optimization on the product manifold.

   • Closed-Form Gradients: A key contribution is the derivation of the gradient of the objective function using adjoint-based backpropagation in the latent space. This eliminates the need for expensive automatic differentiation (AD) of the entire time-integration loop, significantly reducing computational cost.
   • Training Strategy: The authors employ a coordinate-descent approach (alternating between optimizing tensors and projection operators) and progressively extend the time horizon to improve robustness.

3. Key Contributions

1. Unified Framework: GasNiTROM is the first non-intrusive framework to simultaneously enforce global asymptotic stability and optimize oblique projection operators.
2. Novel Parameterization: Introduction of the $\mathbf{Q}^{-1}\mathbf{Q}^{-\top}$ parameterization for the Lyapunov matrix, ensuring strict positive definiteness and avoiding the rank-deficiency pitfalls of previous stable Operator Inference methods.
3. Efficient Gradient Computation: Derivation of closed-form gradients via adjoint methods, bypassing the computational overhead of automatic differentiation for the training process.
4. Manifold Optimization: Rigorous formulation of the optimization problem over Grassmann and Stiefel manifolds to handle the geometric constraints of the projection operators.

4. Results

The method was validated on two distinct test cases:

Case 1: 3D Toy ODE System

• Setup: A system with strong coupling between a rapidly decaying state ($x_3$) and slowly decaying states ($x_1, x_2$).
• Findings:
  • Stability: While standard NiTROM and Operator Inference models became unstable during temporal extrapolation, GasNiTROM remained stable.
  • Accuracy: GasNiTROM significantly outperformed orthogonal projection methods (POD-Galerkin, GasOpInf) in predicting transient peaks. It successfully captured the "non-normal" dynamics where low-energy states drive high-energy growth.

Case 2: 2D Lid-Driven Cavity Flow ($Re=8300$)

• Setup: Incompressible Navier-Stokes equations with large transient growth due to non-normality.
• Findings:
  • Transient Growth: GasNiTROM accurately predicted the energy amplification during the transient phase, whereas orthogonal projection models (OpInf, POD-Galerkin) failed to capture the growth magnitude and phase.
  • Robustness: Under high-amplitude sinusoidal forcing, standard NiTROM models blew up in finite time. GasNiTROM remained stable and provided accurate forecasts, even when operating far outside the training regime.
  • Error Reduction: GasNiTROM demonstrated up to two orders of magnitude improvement in forecasting accuracy compared to state-of-the-art orthogonal projection methods.

5. Significance

GasNiTROM represents a significant advancement in data-driven model reduction for complex physical systems. By resolving the historical trade-off between stability and accuracy, it enables the creation of reliable "black-box" surrogates for systems that are prone to instability and large transients (e.g., hypersonic flows, combustion, fluid-structure interaction). The ability to guarantee stability without sacrificing the ability to capture critical sensitivity mechanisms makes this approach highly valuable for control design, uncertainty quantification, and real-time simulation of high-dimensional nonlinear systems.