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Automated transport separation using the neural shifted proper orthogonal decomposition

Beata Zorawski, Shubhaditya Burela, Philipp Krah, Arthur Marmin, Kai Schneider

TL;DR

This work addresses the separation of multiple transports in transport-dominated fields by jointly learning shifts $\Delta^k(t)$ and co-moving fields $q^k$ through NsPOD, a two-network neural architecture that preserves translation symmetry. It casts the problem as a continuous optimization and its discretization with a nuclear-norm regularization to promote low-rank co-moving fields, then solves it with two neural networks: ShapeNet for $q^k$ and ShiftNet for $\Delta^k$. The approach demonstrates effective transport separation on synthetic Crossing waves data and a 1D wildland-fire model, achieving reconstruction errors on the order of a few percent and enabling low-rank representations in each transport frame. By warming ShapeNet with shifts from ShiftNet and leveraging existing sPOD methods, NsPOD provides a practical and scalable path to transport-aware ROMs without requiring a priori transport operators, with code available for reproducibility.

Abstract

This paper presents a neural network-based methodology for the decomposition of transport-dominated fields using the shifted proper orthogonal decomposition (sPOD). Classical sPOD methods typically require an a priori knowledge of the transport operators to determine the co-moving fields. However, in many real-life problems, such knowledge is difficult or even impossible to obtain, limiting the applicability and benefits of the sPOD. To address this issue, our approach estimates both the transport and co-moving fields simultaneously using neural networks. This is achieved by training two sub-networks dedicated to learning the transports and the co-moving fields, respectively. Applications to synthetic data and a wildland fire model illustrate the capabilities and efficiency of this neural sPOD approach, demonstrating its ability to separate the different fields effectively.

Automated transport separation using the neural shifted proper orthogonal decomposition

TL;DR

This work addresses the separation of multiple transports in transport-dominated fields by jointly learning shifts and co-moving fields through NsPOD, a two-network neural architecture that preserves translation symmetry. It casts the problem as a continuous optimization and its discretization with a nuclear-norm regularization to promote low-rank co-moving fields, then solves it with two neural networks: ShapeNet for and ShiftNet for . The approach demonstrates effective transport separation on synthetic Crossing waves data and a 1D wildland-fire model, achieving reconstruction errors on the order of a few percent and enabling low-rank representations in each transport frame. By warming ShapeNet with shifts from ShiftNet and leveraging existing sPOD methods, NsPOD provides a practical and scalable path to transport-aware ROMs without requiring a priori transport operators, with code available for reproducibility.

Abstract

This paper presents a neural network-based methodology for the decomposition of transport-dominated fields using the shifted proper orthogonal decomposition (sPOD). Classical sPOD methods typically require an a priori knowledge of the transport operators to determine the co-moving fields. However, in many real-life problems, such knowledge is difficult or even impossible to obtain, limiting the applicability and benefits of the sPOD. To address this issue, our approach estimates both the transport and co-moving fields simultaneously using neural networks. This is achieved by training two sub-networks dedicated to learning the transports and the co-moving fields, respectively. Applications to synthetic data and a wildland fire model illustrate the capabilities and efficiency of this neural sPOD approach, demonstrating its ability to separate the different fields effectively.
Paper Structure (18 sections, 13 equations, 3 figures, 2 tables)

This paper contains 18 sections, 13 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation:
  3. Shifted proper orthogonal decomposition
  4. Variational formulation and discretization
  5. Continuous problem description
  6. Discretization
  7. Neural network approach for joint learning
  8. ShapeNet architecture.
  9. ShiftNet architecture.
  10. Training.
  11. Refinement.
  12. Numerical results
  13. Stopping criterion.
  14. Performance evaluation.
  15. Implementation.
  16. ...and 3 more sections

Figures (3)

  • Figure 1: Architecture of NsPOD.
  • Figure 2: Results of NsPOD on crossing waves test case for two different initialization seeds (a) and (b). Furthermore, (c) shows the results of the sPOD-ALM using the shifts and co-moving fields determined by NsPOD.
  • Figure 3: Temperature field of the wildland-fire model test case decomposed with NsPOD.