pyforce-1.0.0: Python Framework for data-driven model Order Reduction of multi-physiCs problEms

Antonio Cammi; Carolina Introini; Stefano Riva; Yantao Luo

arxiv: 2605.18082 · v1 · pith:YST52O4Znew · submitted 2026-05-18 · 💻 cs.LG

pyforce-1.0.0: Python Framework for data-driven model Order Reduction of multi-physiCs problEms

Stefano Riva , Yantao Luo , Carolina Introini , Antonio Cammi This is my paper

Pith reviewed 2026-05-20 13:18 UTC · model grok-4.3

classification 💻 cs.LG

keywords data-driven reduced order modelingmulti-physics problemsnuclear engineeringPython packagepyvistaVTK formatmodel order reductionROSE project

0 comments

The pith

pyforce 1.0.0 reimplements data-driven reduced order modeling for multi-physics nuclear problems using pyvista and numpy arrays.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents pyforce as a Python package that implements data-driven reduced order modeling techniques for multi-physics problems, focused on nuclear engineering applications. It forms part of the ROSE project, which develops mathematical algorithms to reduce model complexity, locate optimal sensors, and incorporate real measurements into physical system knowledge. Version 1.0.0 rewrites the package to use pyvista for mesh import, integral computation, and result visualization, while storing functions as numpy arrays. This design enables compatibility with any external solver that can export results in VTK format, unlike the prior dolfinx-based version.

Core claim

pyforce 1.0.0 is a Python package that implements data-driven reduced order modeling for multi-physics problems by using pyvista as the backend for mesh handling and visualization together with numpy arrays for data storage, thereby allowing the same algorithms to process output from any solver that exports in VTK format.

What carries the argument

The pyforce package, which employs pyvista for mesh importing, integral evaluation, and visualization while storing solution fields as numpy arrays.

If this is right

Users can apply the same reduced-order techniques to simulation results generated by any external code that writes VTK files.
Integration of real sensor data into nuclear reactor models becomes simpler through the package's measurement-assimilation tools.
Optimal sensor placement searches can be performed directly on VTK meshes without requiring a specific finite-element backend.
Model complexity reduction for coupled multi-physics systems is now accessible inside standard Python workflows.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The VTK-centric design may encourage coupling of reduced-order models with existing machine-learning pipelines that already read VTK data.
Broader use outside nuclear engineering could follow if the package is tested on other multi-physics domains that produce VTK output.
Future extensions might add direct support for time-dependent or nonlinear reduced bases while retaining the numpy/pyvista core.

Load-bearing premise

That the data-driven reduced order modeling algorithms remain effective and numerically stable when reimplemented with pyvista and numpy arrays on VTK data exported from arbitrary external solvers.

What would settle it

A side-by-side comparison on the same multi-physics test case showing that pyforce produces unstable or inaccurate reduced models from VTK data while the original dolfinx implementation succeeds.

Figures

Figures reproduced from arXiv: 2605.18082 by Antonio Cammi, Carolina Introini, Stefano Riva, Yantao Luo.

**Figure 2.** Figure 2: Scheme on how the classes in the offline subpackage are structured and how they interact with each other. 3.3 online subpackage Given the basis functions and the sensors placed in the offline phase, the objective becomes the reconstruction of the state of the system given a set of local measurements of some characteristic fields or the characteristic unseen parameter of the state. This latter approach coin… view at source ↗

**Figure 3.** Figure 3: Scheme on how the classes in the online subpackage for non-intrusive reduced order modelling techniques are structured and how they interact with each other. Online measures saved in arrays structures (TR-)GEIM online from pyforce.online.geim GEIM basis and sensors: from pyforce.offline.geim PBDW online from pyforce.online.pbdw EIM online from pyforce.online.eim EIM basis and sensors: from pyforce.offline.… view at source ↗

**Figure 4.** Figure 4: Scheme on how the classes in the online subpackage for direct (a)and indirect (b) reconstruction techniques are structured and how they interact with each other. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Some of the snapshots generated from the toy function for different values of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: Singular values of the snapshot matrix. Projection and Reconstruction Finally, a test snapshot is projected onto the reduced space and reconstructed: 1 test_index = 0 # Index of the test snapshot to project 2 test_snapshot = test_snaps [ test_index ] 3 reduced_coeffs = svd . project ( test_snapshot ) 4 reconstructed_snapshot = svd . reconstruct ( reduced_coeffs ) 8 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

**Figure 7.** Figure 7: Comparison between the original (first row) and reconstructed (second row) test snapshot. [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

read the original abstract

pyforce is a Python package implementing Data-Driven Reduced Order Modelling techniques for applications to multi-physics problems, mainly set in the Nuclear Engineering world. The package is part of the ROSE (Reduced Order modelling with data-driven techniques for multi-phySics problEms): mathematical algorithms aimed at reducing the complexity of multi-physics models (for nuclear reactors applications), at searching for optimal sensor positions and at integrating real measures to improve the knowledge on the physical systems. With respect to the previous original implementation based on dolfinx package (v0.6.0), version 1.0.0 of pyforce has been completely re-written using pyvista as backend for mesh importing, computing integrals, and visualisation of results; in addition, functions are stored as numpy arrays, improving the ease of use of the package. This choice allows to use pyforce with any software solver able to export results in VTK format.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a software release note for an updated pyforce package that swaps in pyvista and numpy to work with any VTK-exporting solver.

read the letter

The main takeaway is that this manuscript describes version 1.0.0 of pyforce, a Python package for data-driven reduced-order modeling on multi-physics problems, mostly in nuclear engineering. The authors rewrote the code to replace the dolfinx backend with pyvista for mesh import, integral computation, and visualization, while storing everything as numpy arrays. This lets the tool accept output from any external solver that writes VTK files instead of being tied to one specific finite-element library. The underlying algorithms still come from the earlier ROSE project work on model reduction, sensor placement, and data assimilation. No new mathematical results or algorithms appear here. The rewrite improves flexibility and ease of use for practitioners who already have solvers producing standard VTK output. That is a practical engineering improvement worth noting for people who need to apply existing ROM techniques without rewriting their simulation stack. The main limitation is the lack of any fresh benchmarks or side-by-side tests. The text does not show whether the new backend changes accuracy, stability, or runtime on the same problems, so readers must take the claim that behavior is preserved on trust. There are also no quantitative performance numbers or validation cases against the prior version. This paper is aimed at nuclear engineers and applied mathematicians who work with reduced-order models and want a Python tool they can drop into an existing workflow. Someone already using the ROSE methods or looking for an open implementation that is not locked to dolfinx would get direct value from the release note. It is not a broad theoretical contribution, but documenting a usable software update in a specialized field can still be worthwhile. I would send it for peer review so that users in the target community can check the implementation details and flag any practical issues before wider adoption.

Referee Report

1 major / 2 minor

Summary. The manuscript describes pyforce-1.0.0, a Python package implementing data-driven reduced order modelling techniques for multi-physics problems, primarily in nuclear engineering applications. It forms part of the ROSE project and details a complete rewrite of version 1.0.0 that replaces the dolfinx backend with pyvista for mesh importing, integral computations, and result visualization, while storing functions as numpy arrays to improve usability and enable compatibility with any external solver that exports VTK format.

Significance. If the reimplementation faithfully preserves the numerical behavior of the original ROSE algorithms, the package could lower barriers to applying data-driven ROM methods in multi-physics settings by decoupling the framework from a specific finite-element library. This flexibility may be useful for nuclear-engineering workflows that rely on heterogeneous solvers. The manuscript itself, however, contains no benchmarks, timing comparisons, or validation against the prior dolfinx version, so the practical significance remains difficult to quantify from the text alone.

major comments (1)

Abstract: the central claim that the pyvista-based rewrite 'allows to use pyforce with any software solver able to export results in VTK format' is presented without any concrete integration example, mesh-handling test, or numerical verification that the new backend reproduces the original integral and projection operators; this assumption is load-bearing for the stated advantage of the 1.0.0 release.

minor comments (2)

The manuscript would benefit from a short 'Usage' or 'Examples' section that demonstrates loading a VTK file from an external solver and performing at least one reduced-order operation, even if only for illustration.
Consider clarifying the precise data-driven ROM algorithms retained from the ROSE project (e.g., specific POD or DMD variants) and any modifications introduced by the numpy/pyvista storage model.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation of minor revision. We address the single major comment point by point below, clarifying the design rationale while agreeing to strengthen the presentation with additional material.

read point-by-point responses

Referee: Abstract: the central claim that the pyvista-based rewrite 'allows to use pyforce with any software solver able to export results in VTK format' is presented without any concrete integration example, mesh-handling test, or numerical verification that the new backend reproduces the original integral and projection operators; this assumption is load-bearing for the stated advantage of the 1.0.0 release.

Authors: We acknowledge that the abstract states the interoperability benefit without an accompanying example or explicit verification in the current text. The reimplementation stores all field data as NumPy arrays and delegates mesh import, integral evaluation, and visualization to PyVista, which natively supports the VTK file format; this architectural choice decouples the reduced-order modeling algorithms from any particular finite-element library and thereby permits direct ingestion of results from any external solver that writes VTK files. Nevertheless, we agree that a concrete demonstration would make the claim more robust. In the revised manuscript we will add a short integration example (reading a VTK file produced by an independent solver, performing the same projection and integral operations, and comparing the numerical results against the prior dolfinx implementation) together with a brief mesh-handling test to confirm that the new backend reproduces the original operators to machine precision. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript is a software release note for the pyforce v1.0.0 package. It describes a re-implementation of data-driven reduced-order modeling techniques from the ROSE project using pyvista and numpy, with no mathematical derivations, equations, fitted parameters, predictions, or uniqueness theorems present. No load-bearing self-citations or self-definitional steps appear; references to prior ROSE work serve only as context for the package's purpose rather than as justification for any internal claim that reduces to the citation itself. The central statements concern software functionality and compatibility with VTK-exporting solvers, which are independent of any circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a software package description rather than a theoretical paper; no free parameters, axioms, or invented entities are introduced or required for the central claim.

pith-pipeline@v0.9.0 · 5705 in / 979 out tokens · 53260 ms · 2026-05-20T13:18:59.689941+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The solution manifold U is usually infinite-dimensional... any solution u(x;µ) ≃ Σ αi(µ)·φi(x)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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2026 , publisher =

Stefano Riva and Carolina Introini and Antonio Cammi , title =. 2026 , publisher =. doi:10.21105/joss.06950 , url =

work page doi:10.21105/joss.06950 2026
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Multi-physics model bias correction with data-driven reduced order techniques: Application to nuclear case studies , journal =

Stefano Riva and Carolina Introini and Antonio Cammi , keywords =. Multi-physics model bias correction with data-driven reduced order techniques: Application to nuclear case studies , journal =. 2024 , issn =. doi:https://doi.org/10.1016/j.apm.2024.06.040 , url =

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Data-driven model order reduction for sensor positioning and indirect reconstruction with noisy data: Application to a Circulating Fuel Reactor , journal =

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2026 , publisher =

Stefano Riva and Carolina Introini and Antonio Cammi , title =. 2026 , publisher =. doi:10.21105/joss.06950 , url =

work page doi:10.21105/joss.06950 2026

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Multi-physics model bias correction with data-driven reduced order techniques: Application to nuclear case studies , journal =

Stefano Riva and Carolina Introini and Antonio Cammi , keywords =. Multi-physics model bias correction with data-driven reduced order techniques: Application to nuclear case studies , journal =. 2024 , issn =. doi:https://doi.org/10.1016/j.apm.2024.06.040 , url =

work page doi:10.1016/j.apm.2024.06.040 2024

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Data-driven model order reduction for sensor positioning and indirect reconstruction with noisy data: Application to a Circulating Fuel Reactor , journal =

Antonio Cammi and Stefano Riva and Carolina Introini and Lorenzo Loi and Enrico Padovani , keywords =. Data-driven model order reduction for sensor positioning and indirect reconstruction with noisy data: Application to a Circulating Fuel Reactor , journal =. 2024 , issn =. doi:https://doi.org/10.1016/j.nucengdes.2024.113105 , url =

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[5] [5]

Lassila, Toni and Manzoni, Andrea and Quarteroni, Alfio and Rozza, Gianluigi , year =. Model. Reduced. doi:10.1007/978-3-319-02090-7_9 , pages =

work page doi:10.1007/978-3-319-02090-7_9

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and Dean, Joseph P

Baratta, Igor A. and Dean, Joseph P. and Dokken, J. doi:10.5281/zenodo.10447666 , year =

work page doi:10.5281/zenodo.10447666

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ACM Transactions on Mathematical Software , year =

Construction of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes , author =. ACM Transactions on Mathematical Software , year =. doi:10.1145/3524456 , pages =

work page doi:10.1145/3524456

[8] [8]

Journal of Open Source Software , year =

Basix: a runtime finite element basis evaluation library , author =. Journal of Open Source Software , year =. doi:10.21105/joss.03982 , pages =

work page doi:10.21105/joss.03982

[9] [9]

2014 , volume =

Unified Form Language: A domain-specific language for weak formulations of partial differential equations , author =. 2014 , volume =

work page 2014

[10] [10]

2019 , month =

Bane Sullivan and Alexander Kaszynski , title =. 2019 , month =. doi:10.21105/joss.01450 , url =

work page doi:10.21105/joss.01450 2019

[11] [11]

Brunton, Steven L and Kutz, J Nathan , year =. Data-

work page

[12] [12]

Wiley Interdisciplinary Reviews: Climate Change, 9(5), e535, https://doi.org/10.1002/wcc.535

Data assimilation in the geosciences:. WIREs Climate Change , author =. 2018 , keywords =. doi:10.1002/wcc.535 , number =

work page doi:10.1002/wcc.535 2018

[13] [13]

Rozza, Gianluigi and Hess, Martin and Stabile, Giovanni and Tezzele, Marco and Ballarin, Francesco and Gräßle, Carmen and Hinze, Michael and Volkwein, Stefan and Chinesta, Francisco and Ladeveze, Pierre and Maday, Yvon and Patera, Anthony and Farhat Char, J , year =. Model

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2025 , pages =

Computer Physics Communications , author =. 2025 , pages =. doi:10.1016/j.cpc.2024.109359 , language =

work page doi:10.1016/j.cpc.2024.109359 2025

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Goodfellow, Ian J and Bengio, Yoshua and Courville, Aaron , year =. Deep

work page

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Weller, H G and Tabor, G and Jasak, H and Fureby, C , doi =. Comput. Phys. , keywords =

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An integrated data-driven computational pipeline with model order reduction for industrial and applied mathematics , abstract =

Tezzele, Marco and Demo, Nicola and Mola, Andrea and Rozza, Gianluigi , year =. An integrated data-driven computational pipeline with model order reduction for industrial and applied mathematics , abstract =. Novel. doi:10.1007/978-3-030-96173-2_7 , url=

work page doi:10.1007/978-3-030-96173-2_7