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Multi-Physics Model Bias Correction with Data-Driven Reduced Order Modelling Techniques: Application to Nuclear Case Studies

Stefano Riva, Carolina Introini, Antonio Cammi

TL;DR

This work tackles bias in Multi-Physics nuclear simulations by fusing mathematical MP models with data through two data-driven ROM approaches: TR-GEIM and PBDW. It details offline-online workflows, stability considerations, and error analyses, and demonstrates the methods on the IAEA 2D PWR and TWIGL2D benchmarks under weakly coupled and linearly coupled coupling schemes. Results show that TR-GEIM and PBDW can accurately correct online state estimates using a limited number of sensors, bringing aFOM predictions in line with ground-truth FOM trajectories and field distributions. The findings support the viability of data-driven bias correction for real-time, multi-query nuclear analysis and indicate avenues for scaling to more realistic scenarios with surrogate coupling data.

Abstract

Nowadays, interest in combining mathematical knowledge about phenomena and data from the physical system is growing. Past research was devoted to developing so-called high-fidelity models, intending to make them able to catch most of the physical phenomena occurring in the system. Nevertheless, models will always be affected by uncertainties related, for example, to the parameters and inevitably limited by the underlying simplifying hypotheses on, for example, geometry and mathematical equations; thus, in a way, there exists an upper threshold of model performance. Now, research in many engineering sectors also focuses on the so-called data-driven modelling, which aims at extracting information from available data to combine it with the mathematical model. Focusing on the nuclear field, interest in this approach is also related to the Multi-Physics modelling of nuclear reactors. Due to the multiple physics involved and their mutual and complex interactions, developing accurate and stable models both from the physical and numerical point of view remains a challenging task despite the advancements in computational hardware and software, and combining the available mathematical model with data can further improve the performance and the accuracy of the former. This work investigates this aspect by applying two Data-Driven Reduced Order Modelling (DDROM) techniques, the Generalised Empirical Interpolation Method and the Parametrised-Background Data-Weak formulation, to literature benchmark nuclear case studies. The main goal of this work is to assess the possibility of using data to perform model bias correction, that is, verifying the reliability of DDROM approaches in improving the model performance and accuracy through the information provided by the data. The obtained numerical results are promising, foreseeing further investigation of the DDROM approach to nuclear industrial cases.

Multi-Physics Model Bias Correction with Data-Driven Reduced Order Modelling Techniques: Application to Nuclear Case Studies

TL;DR

This work tackles bias in Multi-Physics nuclear simulations by fusing mathematical MP models with data through two data-driven ROM approaches: TR-GEIM and PBDW. It details offline-online workflows, stability considerations, and error analyses, and demonstrates the methods on the IAEA 2D PWR and TWIGL2D benchmarks under weakly coupled and linearly coupled coupling schemes. Results show that TR-GEIM and PBDW can accurately correct online state estimates using a limited number of sensors, bringing aFOM predictions in line with ground-truth FOM trajectories and field distributions. The findings support the viability of data-driven bias correction for real-time, multi-query nuclear analysis and indicate avenues for scaling to more realistic scenarios with surrogate coupling data.

Abstract

Nowadays, interest in combining mathematical knowledge about phenomena and data from the physical system is growing. Past research was devoted to developing so-called high-fidelity models, intending to make them able to catch most of the physical phenomena occurring in the system. Nevertheless, models will always be affected by uncertainties related, for example, to the parameters and inevitably limited by the underlying simplifying hypotheses on, for example, geometry and mathematical equations; thus, in a way, there exists an upper threshold of model performance. Now, research in many engineering sectors also focuses on the so-called data-driven modelling, which aims at extracting information from available data to combine it with the mathematical model. Focusing on the nuclear field, interest in this approach is also related to the Multi-Physics modelling of nuclear reactors. Due to the multiple physics involved and their mutual and complex interactions, developing accurate and stable models both from the physical and numerical point of view remains a challenging task despite the advancements in computational hardware and software, and combining the available mathematical model with data can further improve the performance and the accuracy of the former. This work investigates this aspect by applying two Data-Driven Reduced Order Modelling (DDROM) techniques, the Generalised Empirical Interpolation Method and the Parametrised-Background Data-Weak formulation, to literature benchmark nuclear case studies. The main goal of this work is to assess the possibility of using data to perform model bias correction, that is, verifying the reliability of DDROM approaches in improving the model performance and accuracy through the information provided by the data. The obtained numerical results are promising, foreseeing further investigation of the DDROM approach to nuclear industrial cases.
Paper Structure (26 sections, 2 theorems, 34 equations, 18 figures, 4 tables, 4 algorithms)

This paper contains 26 sections, 2 theorems, 34 equations, 18 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Reduced Order Modelling Techniques for Data Assimilation
  3. Generalised Empirical Interpolation Method
  4. Offline Greedy Algorithm
  5. Online Estimation
  6. Parameterised-Background Data-Weak formulation
  7. Offline Phase: Construction of the Spaces
  8. Online Estimation
  9. Data-Driven Model Bias Correction
  10. Multi-Physics Case Studies
  11. Governing Equations
  12. High-Fidelity Full Order Model
  13. Approximated Full Order Model: External Picard Coupling
  14. Linearised Full Order Model: Linear Coupling Dependence
  15. IAEA 2D PWR benchmark
  16. ...and 11 more sections

Key Result

Theorem 1

Let $\{y_m = v_m\left(u^{\text{true}}\right)\}_{m=1}^M$, let $\beta_{N,M}>0$ and let $U_M$ satisfy the unsolvency condition The following estimates hold given $\mathcal{U}$ an Hilbert space, $\left|\left|{\mathcal{I}_M}\right|\right|^2_{\mathcal{L}(\mathcal{U})}$ as the Lebesgue Constant in the $\mathcal{U}-$norm (it can be related to the inf-sup constant as in maday_generalized_2015maday_adapti

Figures (18)

  • Figure 1: Conceptual scheme of Data-Driven Reduced Order Modelling.
  • Figure 2: Reduced Order Modelling for Multi-Physics systems: state-of-the-art and proposed approach.
  • Figure 3: Approximated Full Order Model solution scheme.
  • Figure 4: Regions, boundary conditions and mesh of the IAEA 2D PWR reactor.
  • Figure 5: Regions of the TWIGL2D reactor.
  • ...and 13 more figures

Theorems & Definitions (2)

  • Theorem 1: PBDW a priori error estimates for perfect measurements
  • Theorem 2: PBDW a priori error estimates for imperfect measurements