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Machine Learning-based quadratic closures for non-intrusive Reduced Order Models

This paper introduces a data-driven approach that enhances the accuracy of under-resolved non-intrusive Reduced Order Models by employing a Multi-Input Operators Network (MIONet) to learn a continuous, parameter-dependent quadratic correction term for neglected modes, thereby improving generalization compared to traditional least-squares methods.

Original authors: Gabriele Codega, Anna Ivagnes, Nicola Demo, Gianluigi Rozza

Published 2026-05-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Gabriele Codega, Anna Ivagnes, Nicola Demo, Gianluigi Rozza

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

Imagine you are trying to predict how water flows through a complex pipe system or a room with a moving lid. To get a perfectly accurate answer, you need a super-computer to run a massive simulation that calculates the movement of every single drop of water. This is like trying to count every grain of sand on a beach to understand the tide; it's incredibly precise, but it takes forever and costs a fortune in electricity.

The Problem: The "Lazy" Shortcut
Scientists have developed a "shortcut" called a Reduced Order Model (ROM). Instead of tracking every drop, this method takes a few snapshots of the flow, finds the main patterns (like the big swirls), and tries to guess the rest based on those patterns. It's like looking at a few photos of a dancer and trying to guess the whole dance routine.

Usually, this works great for simple dances. But when the flow gets messy, turbulent, or changes shape quickly (like water rushing around a sharp corner), the shortcut fails. It misses the "small details" because it only looks at the big, obvious patterns. In the paper, the authors call this the "under-resolved" regime—meaning the shortcut is too lazy to see the whole picture.

The Old Fix: The "One-Size-Fits-All" Patch
To fix the lazy shortcut, researchers previously tried adding a "correction term." Think of this as a patch you sew onto a torn shirt.

The Old Method (Quad-LS): They used a mathematical formula to figure out exactly how to patch the shirt. However, this patch was rigid. It was designed for a specific size and shape. If you tried to use it on a slightly different shirt (a different flow condition), the patch didn't fit well, and the shirt still looked torn. It was like trying to use a patch meant for a winter coat on a summer t-shirt.

The New Solution: The "Smart, Stretchy" Patch
The authors of this paper propose a new, smarter way to patch the shortcut. Instead of a rigid formula, they use Machine Learning (specifically, a type of AI called a Neural Network) to create a "smart patch."

They call their new models QuadNet and QuadNet-µ. Here is how they work using simple analogies:

The Smart Patch (QuadNet):
Imagine the patch isn't just a piece of cloth, but a stretchy, intelligent fabric that can change its shape depending on where on the shirt you put it.
- In the old method, the patch was calculated for specific points on the mesh (like specific stitches).
- In the new method, the AI learns a continuous rule. It understands that if the flow is turbulent at point A, the patch needs to be thick there, and if it's calm at point B, the patch can be thin. It doesn't need to be re-taught for every single point; it just "knows" the shape of the flow everywhere.
The Custom-Fit Patch (QuadNet-µ):
This is the even smarter version. Not only does the patch know where it is on the shirt, but it also knows what kind of shirt it is.
- The "µ" stands for the changing conditions (like how fast the water is flowing).
- If the water speed changes, the AI instantly adjusts the patch's shape to fit that new speed perfectly. It's like having a patch that automatically stretches or shrinks to fit the exact tension of the fabric at that moment.

What They Found
The authors tested these new "smart patches" on two classic fluid problems:

The Backward-Facing Step: Water flowing over a sudden drop in a channel (like a waterfall in a pipe).
The Lid-Driven Cavity: Water in a box where the top lid slides back and forth, dragging the water with it.

The Results:

Accuracy: The new AI patches were much better at fixing the errors than the old rigid patches. In some cases, they reduced the error by 90% compared to the basic shortcut and 80% compared to the old patch method.
Efficiency: Even though training the AI takes a little bit of time upfront, the resulting model is actually smaller (uses less memory) than the old rigid method.
Data Starvation: The new method works well even when you don't have many training examples (scarcity). The old method often broke down when data was scarce, but the AI patch could still figure out the pattern.
Partial Data: Because the AI patch is "continuous" (it understands the whole space), you don't need to show it every single point of the flow to train it. You can show it just 10% or 20% of the points, and it still learns the rest. This saves a lot of time.

In Summary
The paper introduces a way to make fast, low-quality flow simulations much more accurate by using a "smart, stretchy" AI patch. Unlike old methods that were rigid and broke when conditions changed, these new models learn the underlying rules of the flow, allowing them to adapt to new situations, work with less data, and provide highly accurate predictions without needing a supercomputer for every single calculation.

Technical Summary: Machine Learning-based Quadratic Closures for Non-intrusive Reduced Order Models

Problem Statement

Reduced Order Models (ROMs) are essential for accelerating computational fluid dynamics (CFD) simulations in applications requiring fast predictions, such as design optimization and control. While Proper Orthogonal Decomposition (POD) is a standard linear reduction technique, it often yields inaccurate approximations in under-resolved or marginally-resolved regimes where the number of retained modes is insufficient to capture complex system dynamics.

Existing data-driven approaches to correct these inaccuracies typically rely on quadratic closure models where a quadratic operator is learned via least-squares optimization (Quad-LS). However, these state-of-the-art methods suffer from two primary limitations:

Overfitting and Poor Generalization: The least-squares approach is prone to overfitting, particularly when the training manifold does not reflect the full complexity of the high-fidelity model or when the problem is ill-conditioned.
Discretization Dependence: Traditional operators are discrete, meaning they are tied to the specific mesh used for training. They cannot naturally generalize to different spatial discretizations or provide continuous field corrections.

Methodology

The authors propose a novel, fully data-driven, non-intrusive framework that replaces the static least-squares operator with a learnable, continuous neural operator. The core strategy involves adding a quadratic correction term, $\tau(\hat{x}; \mu)$ , to the standard POD expansion:
$u(\hat{x}, \mu) \simeq \sum_{i=1}^r a_i(\mu)\phi_i(\hat{x}) + \tau(\hat{x}; \mu)$
where $\phi_i$ are POD modes, $a_i$ are reduced coefficients, and $\tau$ is the correction term.

The paper introduces two specific neural architectures to parameterize the quadratic operator acting on the POD coefficients:

QuadNet (DeepONet-inspired): This model approximates the correction as $\tau(x; \mu) = a^T(\mu)C(\Phi_r, x)a(\mu)$ . It utilizes a Deep Operator Network (DeepONet) architecture with:
- A branch network processing the POD modes ( $\Phi_r$ ).
- A trunk network processing spatial coordinates ( $x$ ).
- A third sub-network combining outputs to produce the continuous quadratic operator.
- Key Feature: The operator is continuous in space, allowing evaluation at arbitrary collocation points independent of the training mesh.
QuadNet- $\mu$ (MIONet-inspired): This extends QuadNet by incorporating the problem parameters ( $\mu$ ) as an additional input via a Multi-Input Operator Network (MIONet) structure.
- It includes a second branch network specifically for the parameters.
- Key Feature: The operator is continuous in both space and parameter space, making the model "local" in the parameter domain and agnostic to the specific parameter instances used during training.

Training Strategy:

The models are trained to minimize a relative error loss function, which normalizes the correction term to handle varying scales of magnitude.
The authors investigate "partial data" regimes where the neural networks are trained on a subset of mesh nodes. They propose a Boltzmann-like probability distribution for sampling nodes, prioritizing regions where the correction term is large and rapidly changing, rather than uniform random sampling.

Key Contributions

Neural Operator Closures: The introduction of DeepONet and MIONet architectures to learn quadratic closure terms for non-intrusive ROMs, moving beyond traditional least-squares optimization.
Continuity and Generalization: The development of models that are continuous in both spatial coordinates and parameter space. This allows the models to be independent of the domain discretization and to generalize effectively to unseen parameter configurations.
Efficient Data Usage: Demonstration that these continuous models can be trained on reduced sets of spatial collocation points (partial data) and fewer parameter instances (scarce data) while maintaining or improving accuracy compared to linear and traditional quadratic baselines.
Parameter Efficiency: The neural models utilize fewer parameters than the tensor-based Quad-LS approach for the tested problems, despite the increased offline training cost.

Results

The methodology was validated on two standard fluid dynamics benchmarks: the backward-facing step (steady-state, varying inlet velocity) and the lid-driven cavity (time-dependent, varying Reynolds number).

Accuracy: Both QuadNet and QuadNet- $\mu$ $μ$ significantly outperformed the baseline POD-RBF (linear) and the state-of-the-art Quad-LS (quadratic least-squares) models.
- QuadNet- $\mu$ achieved the highest accuracy, reducing errors by up to 90% compared to POD-RBF and up to 80% compared to Quad-LS in specific test configurations.
- The improvement was most pronounced in the backward-facing step case, where flow behaviors varied significantly across the parameter range, highlighting the benefit of the parameter-local nature of QuadNet- $\mu$ .
Robustness in Scarce Data: In regimes with very few training snapshots (e.g., $N_\mu < 30$ ) and high POD mode counts, Quad-LS often failed or performed worse than the linear baseline due to overfitting. QuadNet- $\mu$ remained more robust, though its performance also degraded when the correction term became negligible (i.e., when the linear POD approximation was already very precise).
Partial Data: Training QuadNet- $\mu$ on only 10% to 50% of the mesh nodes yielded errors comparable to training on the full mesh, provided the sampling strategy prioritized high-correction regions. This reduced offline training time by approximately 10% in the tested setups.
Computational Cost: While the offline training time for neural models (8–25 minutes) was significantly higher than the near-instantaneous training of Quad-LS (<10 ms), the authors argue this cost is amortized in many-query industrial applications where the cost of a single full-order simulation (e.g., 3 minutes) far exceeds the training overhead.

Significance and Claims

The paper claims that the proposed "corrected-ROMs" offer a superior balance between accuracy, generalization, and data efficiency compared to existing quadratic closure strategies. By leveraging neural operators, the authors demonstrate that:

Interpretability and Non-linearity can coexist: The quadratic structure maintains a degree of physical interpretability (interaction between modes) while utilizing the non-linear approximation power of deep learning.
Discretization Independence: The continuous nature of the learned operators removes the dependency on specific mesh resolutions, a critical advantage for practical engineering applications where mesh refinement or changes are common.
Feasibility with Limited Data: The approach is particularly valuable for scenarios where high-fidelity data is scarce or expensive to generate, as the models can learn effective corrections from limited spatial and parametric samples.

The authors conclude that while the offline training cost is higher, the resulting models provide significantly more accurate predictions for unseen configurations, making them a viable and effective tool for accelerating CFD simulations in real-world applications. Future work is suggested to explore higher-order corrections and integration with more sophisticated base models like POD-NN.