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Bond Graphs for multi-physics informed Neural Networks for multi-variate time series

Alexis-Raja Brachet, Pierre-Yves Richard, Céline Hudelot

TL;DR

The paper tackles forecasting in multi-physics time series by embedding physical knowledge into neural models. It introduces the Neural Bond Graph Encoder (NBgE), which converts Bond Graphs into dual graphs and applies Bond Graph Convolution on frequency-domain representations via MPGNNs to produce physics-informed latent features. Across a simulated DC motor and a partially known respiratory system dataset, NBgE improves baseline models and demonstrates robustness when physics knowledge is incomplete. The work positions NBgE as a task-agnostic encoder that can feed diverse downstream predictors and points to future directions such as unsupervised pretraining and integration with PINNs to fully leverage multi-physics inductive biases.

Abstract

In the trend of hybrid Artificial Intelligence techniques, Physical-Informed Machine Learning has seen a growing interest. It operates mainly by imposing data, learning, or architecture bias with simulation data, Partial Differential Equations, or equivariance and invariance properties. While it has shown great success on tasks involving one physical domain, such as fluid dynamics, existing methods are not adapted to tasks with complex multi-physical and multi-domain phenomena. In addition, it is mainly formulated as an end-to-end learning scheme. To address these challenges, we propose to leverage Bond Graphs, a multi-physics modeling approach, together with Message Passing Graph Neural Networks. We propose a Neural Bond graph Encoder (NBgE) producing multi-physics-informed representations that can be fed into any task-specific model. It provides a unified way to integrate both data and architecture biases in deep learning. Our experiments on two challenging multi-domain physical systems - a Direct Current Motor and the Respiratory System - demonstrate the effectiveness of our approach on a multivariate time-series forecasting task.

Bond Graphs for multi-physics informed Neural Networks for multi-variate time series

TL;DR

The paper tackles forecasting in multi-physics time series by embedding physical knowledge into neural models. It introduces the Neural Bond Graph Encoder (NBgE), which converts Bond Graphs into dual graphs and applies Bond Graph Convolution on frequency-domain representations via MPGNNs to produce physics-informed latent features. Across a simulated DC motor and a partially known respiratory system dataset, NBgE improves baseline models and demonstrates robustness when physics knowledge is incomplete. The work positions NBgE as a task-agnostic encoder that can feed diverse downstream predictors and points to future directions such as unsupervised pretraining and integration with PINNs to fully leverage multi-physics inductive biases.

Abstract

In the trend of hybrid Artificial Intelligence techniques, Physical-Informed Machine Learning has seen a growing interest. It operates mainly by imposing data, learning, or architecture bias with simulation data, Partial Differential Equations, or equivariance and invariance properties. While it has shown great success on tasks involving one physical domain, such as fluid dynamics, existing methods are not adapted to tasks with complex multi-physical and multi-domain phenomena. In addition, it is mainly formulated as an end-to-end learning scheme. To address these challenges, we propose to leverage Bond Graphs, a multi-physics modeling approach, together with Message Passing Graph Neural Networks. We propose a Neural Bond graph Encoder (NBgE) producing multi-physics-informed representations that can be fed into any task-specific model. It provides a unified way to integrate both data and architecture biases in deep learning. Our experiments on two challenging multi-domain physical systems - a Direct Current Motor and the Respiratory System - demonstrate the effectiveness of our approach on a multivariate time-series forecasting task.
Paper Structure (24 sections, 2 theorems, 5 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 24 sections, 2 theorems, 5 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Theoretical preliminaries
  3. Preliminaries on Bond graphs
  4. Preliminaries on MPGNNs
  5. NBgE : a multi-physics-informed encoder
  6. Working hypotheses
  7. A matrix representation of a bond graph
  8. The Dual graph generation process from the Bond Matrix
  9. a. Discrete frequency matrix operators
  10. b. Dual graph generation process
  11. The Bond Graph Convolution
  12. Building the Neural Bond graph Encoder
  13. Bond graph knowledge distillation in NBgE
  14. Experimental study
  15. Datasets
  16. ...and 9 more sections

Key Result

Lemma 1

Any bond is connected to a junction.

Figures (3)

  • Figure 1: The NBgE pipeline. A dual graph is generated from the input data and the Bond Graph of the system, gathering the data and the physical knowledge. Then, NBgE encodes the input data with the Bond Graph Convolution defined on top of the dual graph. The multi-physics-informed representation can be taken as the input of any task-specific prediction model, such as here, in this paper, a time-series forecasting model.
  • Figure 2: From the physical system to the dual graph. The case of the DC Motor with 7 bonds is illustrated here. From the bond graph of the system, the Bond Matrix is developed. Then, through the 7-step process, the dual graph is generated. The nodes represent the effort and flow variables, the edges represent the way the power is distributed. Once the dual graph is initialized, the Bond Graph Convolution updates the node features, being the input of any task-specific model.
  • Figure 3: Predictions on three windows covering the studied physical systems: the DC Motor's angular speed and the RS's two air channels. The input is in gray, and the target is in dashed gray. NBgE-informed model predictions, on the white background, are shown against non-informed ones, on the gray background.

Theorems & Definitions (2)

  • Lemma 1
  • Theorem 1: Expressiveness of the Bond Matrix