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10 joss submission review reviewer 2 software paper
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19 changes: 11 additions & 8 deletions joss-paper/paper.bib
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}


@misc{RMP_2024,
title={Multi-Physics Model Bias Correction with Data-Driven Reduced Order Modelling Techniques: Application to Nuclear Case Studies},
author={Stefano Riva and Carolina Introini and Antonio Cammi},
year={2024},
eprint={2401.07300},
archivePrefix={arXiv},
primaryClass={math.NA},
doi = {10.48550/arXiv.2401.07300}
@article{RMP_2024,
title = {Multi-physics model bias correction with data-driven reduced order techniques: Application to nuclear case studies},
journal = {Applied Mathematical Modelling},
volume = {135},
pages = {243-268},
year = {2024},
issn = {0307-904X},
doi = {https://doi.org/10.1016/j.apm.2024.06.040},
url = {https://www.sciencedirect.com/science/article/pii/S0307904X24003196},
author = {Stefano Riva and Carolina Introini and Antonio Cammi},
keywords = {Reduced order modelling, Data driven, Nuclear reactors, Multi-physics, Model correction},
}

@misc{BarattaEtal2023,
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46 changes: 40 additions & 6 deletions joss-paper/paper.md
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---

# Summary
Innovative reactor technologies in the framework of Generation IV are usually characterised by harsher and more hostile environments compared to standard nuclear systems, for instance, due to the liquid nature of the fuel or the adoption of liquid salt and molten as coolant. This framework poses more challenges in the monitoring of the system itself; since placing sensors inside the reactor itself is a nearly impossible task, it is crucial to study innovative methods able to combine together different sources of information, namely mathematical models and real data (i.e., measurements) in a quick, reliable and efficient way. These methods fall into the Data-Driven Reduced Order Modelling framework, they can be very useful to learn the missing physics or the dynamics of the problem, in particular, they can be adapted to generate surrogate models able to map the out-core measurements of a simple field (e.g., neutron flux and temperature) to the dynamics of un-observable complex fields (precursors concentration and velocity).
*pyforce* (Python Framework for data-driven model Order Reduction of multi-physiCs problEms) is a Python library implementing Data-Driven Reduced Order Modelling (DDROM) techniques [@RMP_2024] for applications to multi-physics problems, mainly in the nuclear engineering world. These techniques have been implemented upon the dolfinx package [@BarattaEtal2023] (currently v0.6.0), part of the FEniCSx project, to handle mesh generation, integral calculation and functions storage. The package is part of the ROSE (Reduced Order modelling with data-driven techniques for multi-phySics problEms) framework, which is one of the main research topics investigated at the [ERMETE-Lab](https://github.com/ERMETE-Lab#reduced-order-modelling-with-data-driven-techniques-for-multi-physics-problems-rose-): in particular, the focus of the research activities is on mathematical algorithms aimed at reducing the complexity of multi-physics models with a focus on nuclear reactor applications, searching for optimal sensor positions and integrating experimental data to improve the knowledge on the physical systems.

# Statement of need
*pyforce* is a Python library (Python Framework for data-driven model Order Reduction of multi-physiCs problEms) implementing Data-Driven Reduced Order Modelling (DDROM) techniques [@RMP_2024] for applications to multi-physics problems, mainly for the Nuclear Engineering world. These techniques have been implemented upon the dolfinx package [@BarattaEtal2023] (currently v0.6.0), part of the FEniCSx project, to handle mesh generation, integral calculation and functions storage. The package is part of the ROSE (Reduced Order modelling with data-driven techniques for multi-phySics problEms) framework, which investigates mathematical algorithms aimed at reducing the complexity of multi-physics models with a focus on nuclear reactor applications, at searching for optimal sensor positions and at integrating experimental data to improve the knowledge on the physical systems.
Innovative reactor technologies in the framework of Generation IV are usually characterised by harsher and more hostile environments than standard nuclear systems, for instance, due to the liquid nature of the fuel or the adoption of liquid salt and molten as coolant. This framework poses more challenges in the monitoring of the system itself; since placing sensors inside the reactor itself is a nearly impossible task, it is crucial to study innovative methods able to combine different sources of information, namely mathematical models and measurements data (i.e., local evaluations of quantities of interest) in a quick, reliable and efficient way. These methods fall into the Data-Driven Reduced Order Modelling framework, they can be very useful to learn the missing physics or the dynamics of the problem, in particular, they can be adapted to generate surrogate models able to map the out-core measurements of a simple field (e.g., neutron flux and temperature) to the dynamics of non-observable complex fields (precursors concentration and velocity).

The techniques implemented here follow the same underlying idea expressed in \autoref{fig:darom}: in the offline (training) phase, a dimensionality reduction process retrieves a reduced coordinate system onto which the information of the mathematical model is encoded; the sensor positioning algorithm then uses this reduced set to select the optimal location of sensors according to some optimality criterion, which depends on the adopted algorithm. In the online phase, the data assimilation process begins, retrieving a novel set of reduced variables and then computing the reconstructed state through a decoding step.
The techniques implemented here follow the same underlying idea expressed in \autoref{fig:darom}. They all share the typical offline/online paradigm of ROM techniques: the former is computationally expensive and it is performed only once, whereas the latter is cheap from the computational point of view and allows to have quick and reliable evaluations of the state of the system by merging background model knowledge and real evaluations of quantities of interest [@maday_parameterized-background_2014].
During the offline (also called training) phase, a *high-fidelity* or Full Order Model (FOM), usually parameterised partial differential equations, is solved several times to obtain a collections of snapshots $\mathbf{u}_{FOM}\in\mathbb{R}^{\mathcal{N}_h}$, given $\mathcal{N}_h$ the dimension of the spatial mesh, which are dependent on some parameters $\boldsymbol{\mu}_n$; then, these snapshots are used to generate a reduced representation through a set of basis functions $\{\psi_n(\mathbf{x})\}$, in this way the degrees of freedom are decresed from $\mathcal{N}_h$ to $N$, provided that $\mathcal{N}_h>>N$. This allows to approximate any solution of the FOM as follows

\begin{equation}\label{eq:rb}
u(\mathbf{x};\boldsymbol{\mu}) \simeq \sum_{n=1}^N\alpha_n(\boldsymbol{\mu})\cdot \psi_n(\mathbf{x})
\end{equation}
with $\alpha_n(\boldsymbol{\mu})$ as the reduced coefficients, embedding the parametric dependence. Moreover, a reduced representation allows for the search of the optimal positions of sensors in the physical domain in a more efficient manner.

![General scheme of DDROM methods [@RMP_2024].\label{fig:darom}](../images/tie_frighter.pdf){ width=100% }

Up to now, the following techniques have been implemented [@DDMOR_CFR;@RMP_2024]:
All these steps are performed during the offline phase, the online phase aim consists in obtaining in a quick and reliable way a solution of the FOM for an unseen parameter $\boldsymbol{\mu}^\star$, using as input a set of measurements $\mathbf{y}\in\mathbb{R}^M$. The DDROM online takes place which produces a novel set of reduced variables, $\boldsymbol{\alpha}^\star$, and then computing an improved reconstructed state $\hat{u}_{DDROM}$ through a decoding step from the low dimensional state to the high dimensional one.

Up to now, the techniques, reported in the following tables, have been implemented [@DDMOR_CFR;@RMP_2024]: they have been split into offline and online, including how they connect with \autoref{fig:darom}.

1. Proper Orthogonal Decomposition (POD) [@rozza_model_2020] with Projection and Interpolation [@demo_complete_2019] for the Online Phase
2. Generalised Empirical Interpolation Method (GEIM) [@maday_generalized_2015], either regularised with Tikhonov [@introini_stabilization_2023] or not
<!---
1. Proper Orthogonal Decomposition (POD) [@rozza_model_2020] with Projection and Interpolation [@demo_complete_2019] for the online phase
2. Generalised Empirical Interpolation Method (GEIM) [@maday_generalized_2015], either with or without Tikhonov regulation [@introini_stabilization_2023]
3. Parameterised-Background Data-Weak (PBDW) [@maday_parameterized-background_2014]
4. an Indirect Reconstruction [@introini_non-intrusive_2023] algorithm to reconstruct non-observable fields
-->

| Offline algorithm | Basis Generation | Sensor Placement |
| --------------------------------------------------------------------------- | ---------------- | ---------------- |
| Proper Orthogonal Decomposition (POD) [@rozza_model_2020] | X | |
| SGreedy [@maday_parameterized-background_2014] | | X |
| Generalised Empirical Interpolation Method (GEIM) [@maday_generalized_2015] | X | X |


| Online algorithm | Input is parameter $\boldsymbol{\mu}$ | Input is measurement vector $\mathbf{y}$ |
| -------------------------------------------------------------------------------- | ------------------------------------- | ---------------------------------------- |
| POD Projection [@rozza_model_2020] | X | |
| POD with Interpolation (PODI) [@demo_complete_2019] | X | |
| GEIM [@maday_generalized_2015] | | X |
| Tikhonov-Regularised (TR)-GEIM [@introini_stabilization_2023] | | X |
| Parameterised-Background Data-Weak (PBDW) [@maday_parameterized-background_2014] | | X |
| Indirect Reconstruction: parameter estimation [@introini_non-intrusive_2023] | | X |


This package aims to become a valuable tool for other researchers, engineers, and data scientists working in various fields where multi-physics problems play an important role, and its scope of application is not only restricted to the Nuclear Engineering world. The package also includes tutorials showing how to use the library and its main features, ranging from snapshot generation in dolfinx, import and mapping from OpenFOAM [@weller_tensorial_1998], to the offline and online phase of each of the aforementioned DDROM algorithms. The case studies are taken from the fluid dynamics and neutronics world, being the most important physics involved in nuclear reactor physics, although the methodologies can be extended to any physics of interest.

# Authors contribution with [CRediT](https://credit.niso.org/)

- Stefano Riva: Conceptualization, Data curation, Formal analysis, Software, Visualization, Writing – original draft
- Carolina Introini: Conceptualization, Formal analysis, Software, Supervision, Writing – review & editing
- Antonio Cammi: Conceptualization, Project administration, Resources, Supervision, Writing – review & editing


# References

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