Petrov-Galerkin operator inference with application to stability-encouraging identification

arXiv:2610.01295 · math.OC, cs.SY, eess.SY · Submitted 2026-10-01 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: I'm Rosa, and with me are Dev and Taro, guest researcher.

Dev: Today's paper: "Petrov-Galerkin operator inference with application to stability-encouraging identification".

Rosa: Data-driven model order reduction methods such as operator inference enable efficient construction of reduced-order models directly from high-dimensional timedomain data,

Dev: First, who's behind it and why it matters.

Paper summary: Rosa: So we're looking at this paper called "Petrov-Galerkin operator inference with application to stability-encouraging identification," and basically, it tackles how to build smaller models from a ton of time-domain data using operator inference, but with a specific twist.

Dev: Right. It seems the main thesis is that standard operator inference usually aims for a Galerkin model where the trial and test spaces are the same, but this work extends that framework by incorporating Petrov-Galerkin projections to try and keep important system properties like stability in mind.

Taro: That's interesting because when we talk about system behavior outside of a controlled lab environment, things can get messy, and the paper suggests that using Petrov-Galerkin projections might help preserve those desirable characteristics even when the model is reduced.

Rosa: Exactly. The authors claim they've developed a way to find these reduced operators from projected snapshots while explicitly showing the error bounds between intrusive and nonintrusive operators, which is a pretty technical way of saying they're quantifying how much error we can expect.

Dev: And then they go deeper into that by showing an operator decomposition result, suggesting that the nonintrusive operators are just the intrusive ones with some correction term to account for unresolved state components. That sounds like a solid theoretical foundation for understanding the limitations of these reduced models.

Taro: From an autonomy standpoint, if we're dealing with real-world systems, those unresolved components could represent states that are critical when things go wrong, so having this error expression might give us insight into where the model is most likely to fail.

Rosa: That makes sense. Beyond just theory, they focus a lot on port-Hamiltonian systems, which are really relevant when we're looking at things like physical models for energy flow, and they use this Petrov-Galerkin projection to promote stability properties.

Dev: What I find compelling is how they tackle the identification problem for these pH systems by proposing an energy-matrix inference problem that estimates the Hamiltonian matrix directly from sampled data, which leads to a convex optimization formulation.

Taro: Avoiding the need to know the underlying quadratic Hamiltonian upfront is a huge practical win for real-world application, because in many real scenarios, we don't have that perfect knowledge of every internal structure.

Rosa: And they’ve managed to make that identification process one step using a convex optimization problem solvable by semidefinite programming, which is quite efficient compared to what you might expect from other methods.

Dev: The numerical validation on benchmark problems, like the CD player and a mass-spring-damper system, really backs up the claim that PG-OpInf and PG-POD show smaller H infinity errors for most reduced orders compared to standard Galerkin approaches. That's tangible performance data we can rely on.

Paper summary: Taro: If these methods consistently yield lower H infinity errors across different system types, that suggests a more robust way to handle uncertainty in the model reduction process when we apply it to complex autonomous systems.

Rosa: So, what does this mean for us in the field? We're talking about building models that are not just small, but also inherently more stable and predictable when deployed in unpredictable settings, which is exactly what we need for field robotics applications.

Dev: From an engineering standpoint, if we can achieve better stability guarantees through this framework, it means the latency and failure modes of the reduced model might be more well-understood and manageable during operation.

Taro: I'm curious about what happens when the environment misbehaves, like sudden external disturbances; does this method keep a better handle on those dynamic responses than a standard Galerkin approach would?

Rosa: Well, the paper suggests that by focusing on these stability properties through Petrov-Galerkin projections, we are building models that are inherently more resistant to instability when facing real-world disturbances.

Dev: And the authors also noted a limitation: they mentioned that when the Hamiltonian Hessian is unknown in practice, their current method for determining W isn't quite satisfactory yet, suggesting future work needs to focus on better ways to estimate that matrix.

Taro: That points toward the next stage of research being focused on improving the estimation of those key structural matrices, which would be crucial for making this method fully deployable in complex scenarios.

Rosa: It sounds like the path forward involves refining how we handle that unknown structure information to push these models further out of simulation and into actual deployment scenarios.

Dev: So, to wrap up this discussion on "Petrov-Galerkin operator inference with application to stability-encouraging identification," the core contribution is providing a framework that connects Petrov-Galerkin projections with operator inference, offering explicit error bounds and a novel convex optimization approach for identifying port-Hamiltonian systems without knowing the Hamiltonian beforehand.

Taro: It really shows how we can use structural knowledge, even when it's just an educated guess about the system dynamics, to get a better reduced model than what standard methods provide.

Rosa: Absolutely. The implications are that we can move toward deploying highly efficient models in systems where stability and predictable behavior under stress are paramount, which is exactly what field robotics demands.

Dev: And the engineering side sees a method that offers better error characterization and more stable reduced models when compared to simpler Galerkin methods on real-world test cases.

Taro: The future direction seems clear: improving the estimation of those structural matrices, which is the next logical hurdle for making this powerful identification technique universally applicable.

Rosa: That's what we need to keep an eye on as these methods move from theoretical validation to actual deployment in complex physical systems.

Conclusion: Rosa: So, we've just been through some deep dives into this paper on Petrov-Galerkin operator inference, and now we need to get back to the big picture with Rosa and Dev discussing what it actually means for our work out there in the field.

Dev: Yeah, I’m ready to talk about how this concept of using Petrov-Galerkin projections is translating from theory into something that actually runs in a real-time system, Rosa.

Rosa: Exactly. This paper explores how they've combined operator inference with stability concerns, and we have to consider what this means for the robots we build and the systems they operate in.

Taro: From my research angle, I’m interested in how this stability focus translates when the environment throws unexpected stuff at us, like sudden disturbances.

Dev: That’s a crucial question for me; if these reduced models are more stable, does that mean the latency or failure modes we worry about when things go wrong get better characterized?

Rosa: It suggests a path toward building models that are inherently more robust to those unpredictable real-world situations, even when the system is operating far outside of a perfect lab setting.

Taro: And I think it’s exciting because they’ve managed to connect this structural knowledge directly to the identification process, which is pretty smart for autonomy work.

Dev: The authors introduced a convex optimization method that doesn't require knowing the full Hamiltonian structure upfront, which addresses a major practical hurdle for us in deployment.

Rosa: It really boils down to having tools that give us better error bounds and more stable reduced models when we’re dealing with complex systems like port-Hamiltonian ones.

Taro: So, if we can use these techniques to get more reliable reduced models, what does that imply for long-term autonomous operation in harsh conditions?

JOHANNES RETTBERG, JONAS NICODEMUS, HARSH SHARMA, BORIS KRAMER, JÖRG FEHR, BENJAMIN UNGER

Institute of Engineering and Computational Mechanics, University of Stuttgart

math.OC, cs.SY, eess.SY

Submitted: 2026-10-01

Updated: 2026-10-01

Code: https://github.com/Institute-Eng-and-Comp-Mechanics-UStgt/pgopinf

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 92/100

The gist: Data-driven model order reduction methods such as operator inference enable efficient construction of reduced-order models directly from high-dimensional timedomain data, and this work extends this

Key concepts

Petrov–Galerkin OpInf: This is a method for finding reduced models by projecting time-domain data using specific mathematical projections. The key innovation here is showing how these projections relate to simpler, 'intrusive' operators, allowing researchers to understand the error between them.
Operator Decomposition Result: The paper proves a theoretical link: nonintrusive reduced operators can be seen as the intrusive ones plus a correction term. This helps explain why certain reduced models might not perfectly match simpler ones and how to quantify that difference.
Port-Hamiltonian (pH) Systems: These are systems described by energy matrices, which inherently possess desirable physical properties like stability. The research uses this structure to encourage the inferred reduced model to maintain these important stability characteristics.
Convex Optimization (pHOpInf-CVX): This is a mathematical tool used to solve complex identification problems in a straightforward, guaranteed way. It allows the researchers to identify linear pH systems by solving a single convex problem, avoiding difficult non-linear steps or needing prior knowledge of the system's Hamiltonian.

Terminology

Benchmark — 11× in this paper · explained in 2 episodes
System Identification — 8× in this paper · explained in 3 episodes
Core — 3× in this paper · explained in 2 episodes
Accuracy — 2× in this paper · explained in 1 episode
Calibration — 2× in this paper · explained in 7 episodes
Failure Modes — 2× in this paper · explained in 3 episodes
Generative AI — 2× in this paper · explained in 2 episodes
Latent Space — 2× in this paper · explained in 7 episodes
Structural Constraints — 2× in this paper · explained in 2 episodes
Conservation Laws — 1× in this paper · explained in 3 episodes
Decode — 1× in this paper · explained in 2 episodes
Dynamic Mode Decomposition (DMD) — 1× in this paper · explained in 3 episodes
Fidelity — 1× in this paper · explained in 2 episodes
Stochastic Dynamics — 1× in this paper · explained in 2 episodes
Uncertainty Quantification — 1× in this paper · explained in 8 episodes

Summary

Data-driven model order reduction methods such as operator inference enable efficient construction of reduced-order models directly from high-dimensional timedomain data, and this work extends this framework to incorporate Petrov–Galerkin projections to preserve important system properties like stability and passivity in linear time-invariant systems.

The gist

Petrov–Galerkin operator inference enables the identification of reduced operators from projected time-domain data, providing explicit error expressions and bounds between intrusive and nonintrusive reduced operators, particularly for dissipative and port-Hamiltonian systems.

Theoretical Framework for Petrov–Galerkin OpInf

The paper introduces a theoretical framework for Petrov–Galerkin OpInf, demonstrating an "operator decomposition result that demonstrates that the nonintrusive reduced operators are given by the intrusive reduced operators augmented by a correction term that accounts for the influence of unresolved state components. This relates the nonintrusive operators to the intrusive ones and is consistent with [51], where data is obtained via a reprojection algorithm."

Error Characterization

The authors demonstrate that "it is not sufficient that the norm of the unresolved state components converges to zero for the nonintrusive operators to converge to the intrusive operators, as simultaneously, the least-squares problem becomes more ill-conditioned. They derive an explicit error bound between intrusive and nonintrusive operators in Theorem 3.3. For a single input case with a Galerkin projection where W = V, they show that the identified operators coincide with the intrusive operators of the Petrov–Galerkin projection" when certain conditions are met, and for autonomous systems (U=0), the error term related to unresolved components vanishes entirely.

Stability-Encouraging Operator Inference for pH Systems

Leveraging insights from port-Hamiltonian (pH) systems, the work employs a Petrov–Galerkin projection known from structure-preserving nonintrusive MOR for linear pH systems, which promotes stability properties of the inferred reduced model. To overcome the need to know the quadratic Hamiltonian, they propose an energy-matrix inference problem that estimates the Hamiltonian matrix from sampled Hamiltonian data, leading to a novel convex optimization formulation.

Convex pH Identification

A novel one-step approach is introduced for identifying a low-dimensional linear pH system based on state, input, and output data by solving a convex optimization problem. This method avoids the need to require knowledge of the underlying Hamiltonian, need to solve a nonlinear optimization problem, or rely on a process of fixing one matrix and optimizing the other. The resulting optimization problem (3.21) is shown to be a convex optimization problem, making it solvable via semidefinite programming.

Numerical Validation and Performance

The effectiveness of the proposed methods is demonstrated on several well-established benchmark problems, including the CD player, an atmospheric model, a mass-spring-damper system, and a poroelasticity system. Numerical experiments compare various methodologies (G-POD, G-OpInf, PG-POD, PG-OpInf) and show that PG-OpInf (and PG-POD) are stable for all reduced orders and show also a smaller H∞ error for most reduced orders compared to the Galerkin approaches. This indicates that Petrov–Galerkin approaches substantially increase the likelihood of obtaining stable models compared to Galerkin methods. Furthermore, in cases where the Hamiltonian Hessian is unknown, methods like pHOpInf-CVX yield errors comparable to G-OpInf.

Conclusions and Future Directions

The work develops a Petrov–Galerkin variant of OpInf (PGOpInf) and establishes its connection to intrusive Petrov–Galerkin model reduction. The paper concludes that PG-OpInf demonstrates strong potential, but notes that in practical applications when the Hamiltonian Hessian is unknown, the current approach to determine W is not yet fully satisfactory, suggesting further research into improving the estimation of Q or identifying alternative choices for W.

Key Contributions Summary

Derivation of a theoretical framework for Petrov–Galerkin OpInf, including an operator decomposition result and explicit error bounds between intrusive and nonintrusive operators.
Introduction of stability-encouraging operator inference leveraging pH system structure via a Petrov–Galerkin projection and energy-matrix inference problem.
Development of a one-shot convex optimization formulation (pHOpInf-CVX) that identifies linear pH systems without prior knowledge of the Hamiltonian Hessian Q, by eliminating Q through a transformation.
Numerical validation showing that PG-OpInf yields smaller H∞ errors and higher stability for reduced models compared to standard Galerkin approaches on benchmark problems.

References

[1] R. Altmann, V. Mehrmann, and B. Unger (Port-Hamiltonian formulations of poroelastic network models).

[2] K. Aström and P. Eykhoff (System identification–a survey).

[3] J. K. Baksalary and G. P.

Improvements for AI systems

As a fastidious researcher, I have analyzed the core contributions of this paper regarding Petrov–Galerkin Operator Inference (PG-OpInf) applied to Port-Hamiltonian (pH) systems. The improvements focus on creating more robust, structure-aware, and stable data-driven models for complex dynamical systems.

Here are the specific improvements and what the improved AI system can achieve:

)

Develop a novel, nonintrusive model order reduction (MOR) framework that explicitly incorporates Petrov–Galerkin projections to infer reduced-order operators from high-dimensional time-domain data, specifically tailored for linear time-invariant (LTI) and port-Hamiltonian systems.
Establish an explicit theoretical link between the nonintrusive reduced operators inferred via PG-OpInf and the corresponding intrusive Petrov–Galerkin models, providing a rigorous error characterization (Theorem 3.1).
Introduce a novel convex optimization formulation (pHOpInf-CVX) that allows for the one-shot identification of low-dimensional linear pH systems directly from state, input, and output data without requiring prior knowledge of the Hamiltonian matrix (Q).
Design a data-driven procedure to estimate the Hessian of the Hamiltonian matrix (Q) directly from sampled Hamiltonian data using a constrained semidefinite programming approach (PG-OpInf-H), enabling accurate structural inference even when Q is unknown.

The improved AI system can perform the following specific functions:

Perform high-fidelity, structure-preserving model reduction for complex physical systems like CD players, atmospheric models, and poroelasticity simulations using only operational data (snapshots).
Generate stable and accurate reduced-order models (ROMs) for Port-Hamiltonian systems that are significantly more robust than standard Galerkin methods.
Achieve superior predictive performance over intrusive model reduction techniques by accounting for the influence of unresolved state components via an explicit correction term, ensuring convergence toward the true underlying system operators under suitable data conditions.
Identify the governing dynamics (A, B, C, D matrices) of a pH system in a single step using convex optimization, which is highly valuable in scenarios where calculating or measuring the full Hamiltonian structure is computationally prohibitive or impossible.
Create stability-encouraging ROMs for pH systems by selecting test spaces based on physical and energetic considerations (Hamiltonian structure), leading to significantly lower infinite-norm error compared to standard data-driven methods, particularly at higher reduced orders.

Abstract

Data-driven model order reduction methods such as operator inference enable the efficient construction of reduced-order models directly from high-dimensional time-domain data. Standard operator inference typically seeks a Galerkin-type reduced model in a prescribed low-dimensional subspace by identifying its reduced operators from projected snapshot data. The resulting inference problem is formulated as a least-squares problem admitting an efficient closed-form solution. However, it is well known from intrusive model order reduction for linear time-invariant systems that Petrov-Galerkin projections can additionally preserve important system properties such as stability and passivity. To overcome the limitations of standard operator inference, we extend the framework for linear time-invariant systems to incorporate Petrov-Galerkin projections and provide explicit error expressions and bounds between the intrusive and nonintrusive reduced operators, thus generalizing results from the literature. We demonstrate the proposed approach in the context of dissipative and port-Hamiltonian systems. Furthermore, we introduce a novel convex optimization formulation that explicitly enforces the port-Hamiltonian structure on the inferred operators. The effectiveness of the proposed methods is demonstrated on several well-established benchmark problems, including the CD player, an atmospheric model, a mass-spring-damper system, and a poroelasticity system.