Block-Structured Operator Inference

• Block-structured Operator Inference is a data-driven technique for constructing reduced-order models that preserve distinct multiphysics interactions.
• It uses separate POD bases for fluid and structural states and enforces block-wise regularization to enhance stability and dimensional reduction.
• Numerical results on aeroelastic benchmarks demonstrate up to 20% computational speedup and improved robustness in capturing complex dynamics.

Block-structured Operator Inference is a methodology for learning structured reduced-order models (ROMs) of multiphysics systems, such as coupled fluid-structure problems, using nonintrusive data-driven operator inference. The approach explicitly preserves and imposes the physical interconnection structure by enforcing a block-wise ansatz for the dynamical equations, permitting tailored regularization, dimensional reduction, and preservation of key system properties such as stability. This methodology addresses limitations of monolithic operator inference by enabling physics-informed block partitioning, leading to improved computational efficiency and robustness.

1. Formulation of the Block-structured Multiphysics System

Consider a general multiphysics model with states partitioned into structural and fluid degrees of freedom:

• Structural state: $q_{\rm s}(t)\in\mathbb R^{n_s}$
• Fluid state: $q_{\rm f}(t)\in\mathbb R^{n_f}$
• Full state: $q(t) = \begin{bmatrix}q_{\rm s}(t)\q_{\rm f}(t)\end{bmatrix} \in \mathbb{R}^{n_s+n_f}$

The governing ODEs, allowing quadratic nonlinearity and up to quadratic cross-coupling, are $$\begin{aligned} \dot{q}<em>{\rm s} &amp;= c</em>{\rm s} + A_{\rm s}q_{\rm s} + H_{\rm s}(q_{\rm s}\otimes q_{\rm s}) + F_{\rm s}(q_{\rm s}, q_{\rm f}) \ \dot{q}<em>{\rm f} &amp;= c</em>{\rm f} + A_{\rm f}q_{\rm f} + H_{\rm f}(q_{\rm f}\otimes q_{\rm f}) + F_{\rm f}(q_{\rm s}, q_{\rm f}) \end{aligned}</p> <p>$$where$$</p> <p>\begin{aligned} F_{\rm s}(q_{\rm s}, q_{\rm f}) &amp;= E_{\rm s} q_{\rm f} + L_{\rm s}(q_{\rm s}\otimes q_{\rm f}) + G_{\rm s}(q_{\rm f}\otimes q_{\rm f}) \ F_{\rm f}(q_{\rm s}, q_{\rm f}) &amp;= E_{\rm f} q_{\rm s} + L_{\rm f}(q_{\rm s}\otimes q_{\rm f}) + G_{\rm f}(q_{\rm s}\otimes q_{\rm s}) \end{aligned}</p> <p>$$Aggregated, the full system equation is$$</p> <p>\begin{bmatrix} \dot q_{\rm s} \ \dot q_{\rm f}</p> <h1 class='paper-heading' id='end-bmatrix'>\end{bmatrix}</h1> <p>\underbrace{ \begin{bmatrix} c_{\rm s} \ c_{\rm f} \end{bmatrix}}<em>{c} + \underbrace{ \begin{bmatrix} A</em>{\rm s} &amp; E_{\rm s} \ E_{\rm f} &amp; A_{\rm f} \end{bmatrix}}<em>{A} \begin{bmatrix} q</em>{\rm s} \ q_{\rm f} \end{bmatrix} + \underbrace{ \begin{bmatrix} H_{\rm s} &amp; L_{\rm s} &amp; G_{\rm s} \ G_{\rm f} &amp; L_{\rm f} &amp; H_{\rm f} \end{bmatrix} }<em>{H} \begin{bmatrix} q</em>{\rm s}\otimes q_{\rm s} \ q_{\rm s}\otimes q_{\rm f} \ q_{\rm f}\otimes q_{\rm f} \end{bmatrix}$$The block structure reflects and preserves inter-physics interactions, admitting clear separation of self-dynamics and cross-coupling, as compared to the “monolithic” vectorized form$\dot q(t)=c+A\,q(t)+H(q(t)\otimes q(t))$.

2. Construction of Reduced Bases

Snapshots are collected separately for the structural and fluid states: $$\begin{aligned} Q_{\rm s} &amp;= {q_{\rm s}(t_i)}<em>{i=1}^{k}, \ Q</em>{\rm f} &amp;= {q_{\rm f}(t_i)}_{i=1}^{k} \end{aligned}</p> <p>$$Each snapshot matrix undergoes a thin singular value decomposition (SVD) to produce separate Proper Orthogonal Decomposition (POD) bases:$$</p> <p>\begin{aligned} Q_{\rm s} &amp;= V_{\rm s} \Sigma_{\rm s} W_{\rm s}^T \qquad (V_{\rm s} \in \mathbb{R}^{n_s \times r_s}) \ Q_{\rm f} &amp;= V_{\rm f} \Sigma_{\rm f} W_{\rm f}^T \qquad (V_{\rm f} \in \mathbb{R}^{n_f \times r_f}) \end{aligned}</p> <p>$$The reduced subspace for the overall system is formed as a block-diagonal projection:$$</p> <p>V_r = \mathrm{diag}(V_{\rm s}, V_{\rm f}) = \begin{bmatrix} V_{\rm s} &amp; 0 \ 0 &amp; V_{\rm f} \end{bmatrix} \in\mathbb R^{{(n_s+n_f)\times(r_s+r_f)}}$$

This construction aligns with the system’s physical partitioning and enables structured reduction.

3. Block-structured Reduced-order Model Ansatz

The reduced state vector is $$\hat q = \begin{bmatrix}\hat q_{\rm s} \ \hat q_{\rm f}\end{bmatrix}$$ with $q \approx V_r \hat q$. The reduced-order model (ROM) mimics the block structure of the original system: $$\begin{bmatrix} \dot{\hat q}<em>{\rm s} \ \dot{\hat q}</em>{\rm f}</p> <h1 class='paper-heading' id='end-bmatrix'>\end{bmatrix}</h1> <p>\begin{bmatrix} \hat c_{\rm s} \ \hat c_{\rm f} \end{bmatrix} + \begin{bmatrix} \hat A_{\rm s} &amp; \hat E_{\rm s} \ \hat E_{\rm f} &amp; \hat A_{\rm f} \end{bmatrix} \begin{bmatrix} \hat q_{\rm s} \ \hat q_{\rm f} \end{bmatrix} + \begin{bmatrix} \hat H_{\rm s} &amp; \hat L_{\rm s} &amp; \hat G_{\rm s} \ \hat G_{\rm f} &amp; \hat L_{\rm f} &amp; \hat H_{\rm f} \end{bmatrix} \begin{bmatrix} \hat q_{\rm s}\otimes\hat q_{\rm s} \ \hat q_{\rm s}\otimes\hat q_{\rm f} \ \hat q_{\rm f}\otimes\hat q_{\rm f} \end{bmatrix}$$In many aeroelastic contexts, domain expertise dictates the imposition of sparsity, e.g., setting$\hat H_{\rm s}=0$,$\hat L_{\rm s}=0$,$\hat G_{\rm s}=0$,$\hat G_{\rm f}=0$,$\hat L_{\rm f}=0$, yielding a block-sparse ROM. Imposing known physics in this manner enforces physical constraints exactly and precludes spurious cross-terms due to noise or overfitting.

4. Operator Inference as Regularized Block-wise Regression

Operator Inference proceeds using reduced snapshots $\widehat Q_{\rm s} = V_{\rm s}^T\,Q_{\rm s}$, $\widehat Q_{\rm f} = V_{\rm f}^T\,Q_{\rm f}$ and their time derivatives (e.g., via finite differences). For each physics block $p\in\{\rm s, f\}$, a separate regularized linear least-squares problem is formulated:

$$\min_{\hat O_{\rm p}}\, \|D_{\rm p} \hat O_{\rm p}^T - R_{\rm p}^T\|_F^2 + \mathcal{R}_{\rm p}(\hat O_{\rm p})$$

with block-specific data matrices ($D_{\rm s}$, $D_{\rm f}$), operators, and regularizers.

For the structural part: $$\begin{aligned} D_{\rm s} &amp;= [\mathbf{1}<em>k^{T\,|\,\widehat} Q</em>{\rm s}^{T\,|\,\widehat} Q_{\rm f}^T] \ \hat O_{\rm s} &amp;= [\hat c_{\rm s}, \hat A_{\rm s}, \hat E_{\rm s}] \ R_{\rm s} &amp;= \dot{\widehat Q}_{\rm s} \end{aligned}</p> <p>$$For the fluid part:$$</p> <p>\begin{aligned} D_{\rm f} &amp;= [\mathbf{1}<em>k^{T\,|\,\widehat} Q</em>{\rm s}^{T\,|\,\widehat} Q_{\rm f}^{T\,|\,(\widehat} Q_{\rm f} \otimes \widehat Q_{\rm f})^T] \ \hat O_{\rm f} &amp;= [\hat c_{\rm f}, \hat E_{\rm f}, \hat A_{\rm f}, \hat H_{\rm f}] \ R_{\rm f} &amp;= \dot{\widehat Q}_{\rm f} \end{aligned}</p> <p>$$Tikhonov regularization is applied with independent hyperparameters per block:$$</p> <p>\begin{aligned} \mathcal{R}<em>{\rm s} &amp;= \gamma</em>{\rm s}^{\rm lin}(|\hat c_{\rm s}|<em>F² + |\hat A</em>{\rm s}|<em>F² + |\hat E</em>{\rm s}|<em>F²⁾ \ \mathcal{R}</em>{\rm f} &amp;= \gamma_{\rm f}^{\rm lin}(|\hat c_{\rm f}|<em>F² + |\hat E</em>{\rm f}|<em>F² + |\hat A</em>{\rm f}|<em>F²⁾⁺ \gamma</em>{\rm f}^{\rm quad}|\hat H_{\rm f}|_F² \end{aligned}$$ The regularization parameters are selected by grid search using a “bounded growth” constraint to promote stability, as in prior work (cf. McQuarrie et al.).

The block separation substantially reduces the dimensionality of the individual regression problems and admits block-specific regularization strategies, unlike monolithic regression.

5. Preservation of Physics-informed Structure and System Properties

Enforcing a block structure confers multiple benefits:

• It preserves the interpretable separation of multi-physics components, maintaining original partitioning of self-dynamics and coupling.
• Linear subsystems, such as structural mechanics, remain strictly linear in the ROM, preserving second-order Hamiltonian or modal structure where present.
• Block-wise regularization mitigates overfitting; each physics block is regularized independently, yielding greater robustness and improved numerical stability relative to a monolithic approach.
• Directly zeroing or constraining blocks (e.g., setting known-inactive couplings to zero) prevents spurious or nonphysical terms due to data noise or limited training data. A plausible implication is that these features facilitate transferability and generalization, as structure is maintained by design rather than learned anew for each problem instance.

6. Numerical Demonstration on Coupled Aeroelastic Systems

Block-structured Operator Inference has been numerically demonstrated on the AGARD 445.6 wing benchmark, a canonical aeroelastic test involving:

• Full-order simulation: FUN3D RANS CFD (∼18 million DOF) coupled to a four-mode structural FE model.
• Training on nine flow regimes (Mach 0.901, 0.957, 1.141; $q_\infty =$ 50, 70, 90 psf) spanning subsonic, supersonic, and near-flutter conditions.
• POD dimensions: $r_{\rm s}=4$ (structure), $r_{\rm f}\in\{8,12\}$ (fluid), yielding $r=12$ or $16$.

Empirical findings include:

• Both monolithic and block-structured ROMs achieve lift-coefficient errors (relative RMSE) of $O(10^{-3}$–$10^{-2})$ during extrapolative prediction intervals.
• Block-structured ROMs match in-sample accuracy and exhibit enhanced robustness for out-of-sample initial conditions; monolithic ROMs frequently diverge or mispredict post-perturbation.
• Median per-timestep online CPU time is reduced from approximately $0.23$ s (monolithic) to $0.18$ s (block-structured): ∼21% speedup. The average speedup across all cases is ≈20%.
• Operator parameter count for $r_s=4$ is reduced by up to 40% at $r_f=8$ and by ∼20% at $r_f=12$; this reduction translates directly into per-step computational savings.
• Block-structured ROMs successfully generalize to new initial conditions without retraining, a property not observed for the monolithic method.

These results confirm the practical advantages of the block-structured approach: lower online cost, fewer learned parameters, and enhanced stability—while retaining the accuracy of monolithic Operator Inference, even for challenging supersonic and flutter-proximate regimes.

7. Summary, Significance, and Implications

Block-structured Operator Inference provides a principled, nonintrusive methodology for data-driven model reduction in multiphysics systems, rigorously preserving system partitioning and physical constraints. By allowing both physics-informed block ansatzes and independently tailored regularization, it yields robust, stable ROMs with substantial computational savings (∼20% online speedup), lower operator dimensionality, and resilient generalization capabilities. Explicitly enforced zero blocks ensure that known physics constraints are respected by construction. This suggests that block-structured Operator Inference is broadly applicable to complex coupled multiphysics problems where physical fidelity, stability, and efficiency are critical.