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Physics-informed structured learning of a class of recurrent neural networks with guaranteed properties

Daniele Ravasio, Claudia Sbardi, Marcello Farina, Andrea Ballarino

Abstract

This paper proposes a physics-informed learning framework for a class of recurrent neural networks tailored to large-scale and networked systems. The approach aims to learn control-oriented models that preserve the structural and stability properties of the plant. The learning algorithm is formulated as a convex optimisation problem, allowing the inclusion of linear matrix inequality constraints to enforce desired system features. Furthermore, when the plant exhibits structural modularity, the resulting optimisation problem can be parallelised, requiring communication only among neighbouring subsystems. Simulation results show the effectiveness of the proposed approach.

Physics-informed structured learning of a class of recurrent neural networks with guaranteed properties

Abstract

This paper proposes a physics-informed learning framework for a class of recurrent neural networks tailored to large-scale and networked systems. The approach aims to learn control-oriented models that preserve the structural and stability properties of the plant. The learning algorithm is formulated as a convex optimisation problem, allowing the inclusion of linear matrix inequality constraints to enforce desired system features. Furthermore, when the plant exhibits structural modularity, the resulting optimisation problem can be parallelised, requiring communication only among neighbouring subsystems. Simulation results show the effectiveness of the proposed approach.

Paper Structure

This paper contains 30 sections, 9 theorems, 67 equations, 3 figures, 1 table, 8 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Statement of the problem
  4. Imposing the modular plant structure
  5. Imposing stability guarantees
  6. State of the art
  7. Paper objectives and structure
  8. Notation and preliminaries
  9. The selected recurrent neural network class
  10. The recurrent neural network model
  11. Hyperparameters definition
  12. The trained model
  13. Learning unstructured models
  14. Least-squares learning procedure
  15. The set membership approach
  16. ...and 15 more sections

Key Result

Theorem 1

If system eq:general_sys admits a dissipation-form $\delta$ISS Lyapunov function over the sets $\mathcal{X}$ and $\mathcal{U}$, then it is $\delta$ISS with respect to such sets, in the sense of Definition def:delta_ISS. $\square$

Figures (3)

  • Figure 1: Modular system structure for a simple case where $n_\mathrm s=2$.
  • Figure 2: Reactor 2 modelling: prediction of the RNN model obtained using least-square (blue) and set-membership (yellow) compared to the ground truth (red).
  • Figure 3: pH-neutralisation modelling: prediction of the RNN model compared to the ground truth.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • ...and 2 more