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A Fast Learning-Based Surrogate of Electrical Machines using a Reduced Basis

Alejandro Ribés, Nawfal Benchekroun, Théo Delagnes

TL;DR

This paper addresses the need for fast surrogates of parameterized PDEs on non-regular meshes suitable for real-time digital twins. It introduces a two-stage POD-SVR framework where a POD-based reduced basis is learned from a snapshot ensemble and multiple SVRs map time and parameter inputs to reduced coefficients to produce full-field predictions in a direct-time fashion. A reconstruction-error bound of the form $\| \mathbf{X}_p-\hat{\mathbf{X}}_p \|_2 \le K_p \mathbf{e}$ is derived, supporting reliability, with $K_p$ defined using POD components. The method is validated on two industrial electrical-machine use cases, achieving low relative RMSE/AME (a few percent) and inference times around a few milliseconds on CPU, indicating strong potential for real-time interactive exploration in digital twins.

Abstract

A surrogate model approximates the outputs of a solver of Partial Differential Equations (PDEs) with a low computational cost. In this article, we propose a method to build learning-based surrogates in the context of parameterized PDEs, which are PDEs that depend on a set of parameters but are also temporal and spatial processes. Our contribution is a method hybridizing the Proper Orthogonal Decomposition and several Support Vector Regression machines. This method is conceived to work in real-time, thus aimed for being used in the context of digital twins, where a user can perform an interactive analysis of results based on the proposed surrogate. We present promising results on two use cases concerning electrical machines. These use cases are not toy examples but are produced an industrial computational code, they use meshes representing non-trivial geometries and contain non-linearities.

A Fast Learning-Based Surrogate of Electrical Machines using a Reduced Basis

TL;DR

This paper addresses the need for fast surrogates of parameterized PDEs on non-regular meshes suitable for real-time digital twins. It introduces a two-stage POD-SVR framework where a POD-based reduced basis is learned from a snapshot ensemble and multiple SVRs map time and parameter inputs to reduced coefficients to produce full-field predictions in a direct-time fashion. A reconstruction-error bound of the form is derived, supporting reliability, with defined using POD components. The method is validated on two industrial electrical-machine use cases, achieving low relative RMSE/AME (a few percent) and inference times around a few milliseconds on CPU, indicating strong potential for real-time interactive exploration in digital twins.

Abstract

A surrogate model approximates the outputs of a solver of Partial Differential Equations (PDEs) with a low computational cost. In this article, we propose a method to build learning-based surrogates in the context of parameterized PDEs, which are PDEs that depend on a set of parameters but are also temporal and spatial processes. Our contribution is a method hybridizing the Proper Orthogonal Decomposition and several Support Vector Regression machines. This method is conceived to work in real-time, thus aimed for being used in the context of digital twins, where a user can perform an interactive analysis of results based on the proposed surrogate. We present promising results on two use cases concerning electrical machines. These use cases are not toy examples but are produced an industrial computational code, they use meshes representing non-trivial geometries and contain non-linearities.

Paper Structure

This paper contains 16 sections, 1 theorem, 26 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Proposed Method
  4. Finding a Reduced Basis
  5. Training the SVRs
  6. Tuning the hyperparameters of the SVRs
  7. Description of the use cases
  8. An induction plate
  9. A three-phase transformer
  10. Results
  11. Real-Time Inference
  12. Conclusion
  13. Appendix: inference error upper bound
  14. Notation
  15. Introduction
  16. ...and 1 more sections

Key Result

Theorem 1.1

$\forall p \leq n$$\exists K_p >0$ that verifies

Figures (5)

  • Figure 1: A SVR takes as input a time step $t$ and a vector of parameters $\lambda$. It outputs a prediction $\hat{c}_i$, corresponding to the i-th coefficient on the reduced space, $i=1...r$.
  • Figure 2: Mesh of the induction plate use case.
  • Figure 3: Mesh of the three-phase transformer use case.
  • Figure 4: Non-linear behavior law of the transformer core.
  • Figure 5: Visual comparison between the modulus of a reconstructed magnetic field (top image) and the reference magnetic field (bottom image), on the test set of the three-phase transformer use case introduced in Section \ref{['subsec:use-case-Transfo3D']}.

Theorems & Definitions (1)

  • Theorem 1.1