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Non-Intrusive Balancing for Data-Driven Model Reduction with Oblique Projections

This paper introduces a non-intrusive framework that computes balancing-based oblique projections from forward simulation data to learn reduced-order models via Operator Inference, significantly outperforming traditional orthogonal projection methods in accuracy for systems with large transient energy growth.

Original authors: Alberto Padovan

Published 2026-10-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Alberto Padovan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of engineering and physics, scientists often face a dilemma when trying to predict how complex systems will behave. Whether it is the air flowing over an airplane wing or the movement of fluids inside a pipe, the underlying mathematics can be so vast and detailed that running a full simulation takes days or even weeks on the most powerful supercomputers. To make these problems manageable, researchers use a technique called model reduction. The goal is to create a much simpler, faster version of the system that still captures the essential behavior. For decades, the standard way to do this has been to look at a system's history, find the patterns that carry the most energy, and build a simplified model based on those dominant patterns. This approach works well for many steady, predictable situations. However, it often fails when a system is jolted by a sudden disturbance. In these moments, a system might experience a massive, temporary surge in energy before settling down. The traditional method, which focuses only on the most energetic parts, tends to discard the subtle, low-energy states that are actually responsible for that sudden surge. As a result, the simplified model might look correct on paper but fail spectacularly when asked to predict what happens after a shock.

This is the specific challenge addressed by a new study from Alberto Padovan at the New Jersey Institute of Technology. The researcher set out to fix a blind spot in how engineers create these simplified models for complex, non-linear systems. The problem is that the most accurate methods for handling these sudden energy surges usually require access to the internal "blueprint" of the computer code running the simulation. Most real-world engineering software is a "black box," meaning its internal workings are hidden or proprietary, so engineers cannot use the most precise tools available. Padovan developed a new, non-intrusive method that bypasses the need for this hidden information. Instead of trying to reverse-engineer the code, the new method learns directly from the output of standard forward simulations. It uses a clever mathematical trick to identify which parts of the system are most sensitive to disturbances and which parts are most visible to sensors, even without seeing the inner mechanics. By balancing these two perspectives, the method constructs a simplified model that knows exactly where to look for trouble.

The core of this new approach involves a process called balancing. In simple terms, this means finding a way to compress the system's state that respects both how easily the system can be pushed by an external force and how easily its changes can be seen by an observer. Traditional methods often ignore this balance, focusing only on the energy levels. Padovan's technique, however, computes a special kind of projection that aligns the simplified model with the system's actual sensitivity. This is done by running a series of standard simulations where the system is given small pushes, or impulses, and then carefully analyzing the resulting data. The researcher introduced a "checkpointing" strategy, where snapshots of the system's state are taken at specific moments during these simulations. These snapshots are then reused as starting points for new, shorter simulations to gather information about how the system responds to being observed. This allows the method to build a complete picture of the system's behavior without ever needing to open the black box or run expensive, extra simulations.

The results of this new method were tested on several increasingly difficult scenarios, ranging from mathematical equations describing fluid waves to complex simulations of air flowing over wings. In one test, the researchers simulated air flowing over a two-dimensional airfoil, a shape similar to an airplane wing, with a state space containing nearly 700,000 variables. In an even more demanding test, they modeled a three-dimensional wing with approximately 80 million degrees of freedom. In every case, the new balanced models were compared against the standard models that rely on the traditional energy-based approach. The difference was stark. When the systems were hit with sudden disturbances or forced to oscillate in ways they had never seen before, the standard models often failed completely, predicting errors that were orders of magnitude too large or developing wild, unrealistic oscillations. In contrast, the new balanced models tracked the true behavior with high precision. They successfully captured the massive, temporary spikes in energy that the old models missed, reducing forecasting errors by up to two orders of magnitude.

The success of this method lies in its ability to recognize that the most important parts of a system are not always the ones with the most energy. In the airfoil example, the standard model focused on the large swirls of air far behind the wing, where the energy was highest. The new balanced model, however, identified that the most critical dynamics were happening right at the surface of the wing and in the immediate wake, where the flow is most sensitive to changes. The trial and test modes—the mathematical building blocks of the simplified model—were found to be spatially separated, with one set concentrated near the point where the system is pushed and the other near where it is measured. This separation is a hallmark of complex, non-normal dynamics, and it is something that traditional orthogonal projections simply cannot represent. By allowing the model to use this oblique, or slanted, projection, the method preserves the delicate mechanisms that drive transient growth.

The study demonstrates that it is possible to build highly accurate, reduced-order models for massive, complex systems without needing access to the source code or the ability to calculate complex adjoint information. The computational cost of the new method is comparable to that of the standard techniques, requiring only a similar amount of time and storage as a standard analysis of the training data. This makes the approach practical for real-world engineering applications where speed and accuracy are paramount. The work suggests that for systems prone to large, temporary energy surges, the old rule of "keep the most energetic parts" is insufficient. Instead, a balanced view that accounts for both sensitivity and observability is required to truly understand and predict the behavior of complex physical systems. The findings offer a new path forward for engineers working with black-box solvers, providing a way to create reliable, fast models that can handle the unpredictable nature of real-world disturbances.

Technical Summary: Non-Intrusive Balancing for Data-Driven Model Reduction with Oblique Projections

Problem Statement
Non-intrusive reduced-order modeling (ROM) is critical for controlling, optimizing, and forecasting high-dimensional physical systems simulated with black- or gray-box solvers where source code access is restricted. However, existing non-intrusive methods typically rely on orthogonal projections onto manifolds identified by minimizing static reconstruction losses (e.g., Proper Orthogonal Decomposition, POD). This approach fails for systems exhibiting large-amplitude transient energy growth, as it truncates low-energy states that are dynamically critical for forecasting. While intrusive methods like Balanced Truncation (BT) address this by using adjoint information to define dynamics-aware oblique projections, such adjoint capabilities are often unavailable in proprietary solvers. The central challenge addressed is how to identify dynamically appropriate coordinates and oblique projections for nonlinear systems without access to adjoint information or full-order linearization.

Methodology
The paper proposes a non-intrusive, optimization-free framework to compute balancing-based oblique projections using only forward simulation data. The methodology consists of two primary stages:

Data-Driven Balancing via Checkpointing:
- Empirical Gramians: The method adapts the empirical Gramian paradigm (Lall et al., 1999) for nonlinear systems. It computes an empirical reachability Gramian ( $W_R$ ) from impulse responses and a projected empirical observability Gramian ( $\tilde{W}_O$ ) from output trajectories.
- Projection Strategy: Recognizing that computing the full observability Gramian is intractable in high dimensions, the method restricts the observability calculation to a subspace spanned by a set of initial conditions ( $X_0$ ).
- Minimax Optimality: Theoretical analysis (Theorem 2) demonstrates that to minimize the error between the projected and full Hankel matrices, the restriction subspace should span the dominant eigenspaces of the reachability Gramian. Since the range of the reachability Gramian is well-approximated by the span of impulse responses, the method samples initial conditions for the observability calculation directly from the forward impulse response trajectories.
- Checkpointing Algorithm: To avoid running additional full-order simulations, the algorithm extracts "checkpoints" from stored impulse response snapshots. These checkpoints serve as initial conditions for new sub-trajectories, the outputs of which are used to construct the projected observability Gramian. This allows the computation of the Gramian using only data already generated for the reachability analysis.
- Balancing: The factors of $W_R$ and $\tilde{W}_O$ are balanced via Singular Value Decomposition (SVD) to yield the encoder ( $\Psi$ ) and decoder ( $\Phi$ ) matrices, defining an oblique projection operator $\mathbb{P} = \Phi(\Psi^\top\Phi)^{-1}\Psi^\top$ .
Latent-Space Dynamics Learning:
- Once the balanced projection is established, the high-dimensional state is projected onto the low-dimensional latent space.
- The latent-space dynamics are learned using Operator Inference (OpInf) or its globally-asymptotically-stable variant (GasOpInf). This involves solving a linear least-squares problem to fit polynomial dynamics to the projected data. The input term is computed via Petrov-Galerkin projection of the full-order input matrix, while the nonlinear terms are inferred from data.

Key Contributions

Non-Intrusive Oblique Projections: The paper introduces a procedure to compute balancing-based oblique projections for nonlinear systems without adjoint information, overcoming the limitations of standard POD-based orthogonal projections.
Theoretical Guarantees: The author proves that when dynamics are linear and the restriction subspace contains the range of either Gramian, the procedure recovers the classical Balanced Truncation projection. They provide error bounds quantifying the deviation from BT in terms of Hankel singular values and principal angles between subspaces.
Minimax-Optimal Sampling: The work establishes that the optimal restriction subspace for the observability Gramian in the absence of adjoints spans the dominant eigenspaces of the reachability Gramian, informing a practical sampling strategy.
Computational Efficiency: The proposed checkpointing algorithm computes the necessary Gramians using only forward simulation data, with a computational cost comparable to a standard SVD of the training data.
Scalability: The method is demonstrated on systems with up to $8 \times 10^7$ degrees of freedom, utilizing lossless compression to manage memory constraints.

Results
The method was validated on three benchmark problems of increasing complexity:

Complex Ginzburg-Landau (CGL) Equation: Tested in both stable and unstable configurations. The Balanced Operator Inference (BOpInf) models significantly outperformed POD-based OpInf models, reducing forecasting errors by up to two orders of magnitude. The BOpInf models successfully captured large transient growth and limit cycle behavior, whereas POD-based models failed to track transient amplification and developed spurious oscillations.
2D Flow Past a NACA 0012 Airfoil ( $Re=600$ ): The Balanced GasOpInf model accurately predicted perturbation energy growth and decay for impulse responses at both training and unseen forcing locations. In contrast, the POD-based GasOpInf model underestimated energy by several orders of magnitude. The model also correctly reproduced responses to sinusoidal inputs at fundamental and harmonic frequencies, which were not present in the training data.
3D Flow Past a Finite Wing ( $Re=1000$ , $n \approx 8 \times 10^7$ ): The method successfully handled a massive 3D system. The BOpInf model tracked the transient energy growth and decay of wavepackets and correctly predicted the vortical structures in the wake. The POD-based counterpart significantly underestimated the response energy and failed to capture the correct spatial structures.

In all cases, the spatial separation between the trial (decoder) and test (encoder) modes identified by the balancing procedure was observed, reflecting the non-normal nature of the dynamics—a feature orthogonal POD projections cannot capture.

Significance
The paper claims that its primary significance lies in enabling the application of balancing-based model reduction to high-dimensional, nonlinear systems simulated with black-box solvers where adjoint information is unavailable. By proving that oblique projections can be computed non-intrusively from forward data alone, the work bridges the gap between the theoretical accuracy of Balanced Truncation and the practical constraints of modern engineering solvers. The results suggest that for systems with large transient energy growth, dynamics-aware oblique projections are essential for accurate forecasting, and the proposed framework provides a computationally efficient pathway to achieve this without requiring intrusive access to the full-order model's internal structure.

Technical Summary: Non-Intrusive Balancing for Data-Driven Model Reduction with Oblique Projections

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