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Physics-informed Discovery of State Variables in Second-Order and Hamiltonian Systems

Félix Chavelli, Zi-Yu Khoo, Dawen Wu, Jonathan Sze Choong Low, Stéphane Bressan

TL;DR

The paper tackles the challenge of discovering minimal, interpretable state variables for dynamical systems from data without overparameterization or reliance on external intrinsic-dimension estimates. It introduces physics-informed biases across three models—PI-AE, PI-VAE, and HPI-VAE—at observational, learning, and inductive levels to enforce second-order and Hamiltonian structure within a baseline autoencoder framework. Empirically, the approach improves DOF identification and yields non-redundant, more interpretable latent representations across five dynamical systems, outperforming Chen et al.'s baseline. This work provides a principled pathway to physics-grounded, compact state representations with potential impact on modeling, prediction, and control in physics and engineering contexts.

Abstract

The modeling of dynamical systems is a pervasive concern for not only describing but also predicting and controlling natural phenomena and engineered systems. Current data-driven approaches often assume prior knowledge of the relevant state variables or result in overparameterized state spaces. Boyuan Chen and his co-authors proposed a neural network model that estimates the degrees of freedom and attempts to discover the state variables of a dynamical system. Despite its innovative approach, this baseline model lacks a connection to the physical principles governing the systems it analyzes, leading to unreliable state variables. This research proposes a method that leverages the physical characteristics of second-order Hamiltonian systems to constrain the baseline model. The proposed model outperforms the baseline model in identifying a minimal set of non-redundant and interpretable state variables.

Physics-informed Discovery of State Variables in Second-Order and Hamiltonian Systems

TL;DR

The paper tackles the challenge of discovering minimal, interpretable state variables for dynamical systems from data without overparameterization or reliance on external intrinsic-dimension estimates. It introduces physics-informed biases across three models—PI-AE, PI-VAE, and HPI-VAE—at observational, learning, and inductive levels to enforce second-order and Hamiltonian structure within a baseline autoencoder framework. Empirically, the approach improves DOF identification and yields non-redundant, more interpretable latent representations across five dynamical systems, outperforming Chen et al.'s baseline. This work provides a principled pathway to physics-grounded, compact state representations with potential impact on modeling, prediction, and control in physics and engineering contexts.

Abstract

The modeling of dynamical systems is a pervasive concern for not only describing but also predicting and controlling natural phenomena and engineered systems. Current data-driven approaches often assume prior knowledge of the relevant state variables or result in overparameterized state spaces. Boyuan Chen and his co-authors proposed a neural network model that estimates the degrees of freedom and attempts to discover the state variables of a dynamical system. Despite its innovative approach, this baseline model lacks a connection to the physical principles governing the systems it analyzes, leading to unreliable state variables. This research proposes a method that leverages the physical characteristics of second-order Hamiltonian systems to constrain the baseline model. The proposed model outperforms the baseline model in identifying a minimal set of non-redundant and interpretable state variables.
Paper Structure (8 sections, 3 equations, 5 figures, 3 tables)

This paper contains 8 sections, 3 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Methodology
  4. Observational bias
  5. Learning bias
  6. Inductive bias
  7. Experiments
  8. Conclusion

Figures (5)

  • Figure 1: Chen et al.'s baseline (A and B) and our proposed models (C, D, E).
  • Figure 2: Sequences of frames of the five dynamical systems: from the top in each row, the reaction-diffusion system, single pendulum system, double pendulum system, elastic pendulum system, and swing stick system from Chen et al.
  • Figure 3: Values of the latent variables obtained by the models (y-axis) against time (x-axis) for one trajectory of the simple pendulum. The subplots show the respective latent variables for the (a) baseline, (b) PI-AE, (c) PI-VAE, and (d) HPI-VAE model.
  • Figure 4: Values of the latent variables obtained by the models (y-axis) against time (x-axis) for one trajectory of the double pendulum. The subplots show the respective latent variables for the (a) baseline, (b) PI-AE, (c) PI-VAE, and (d) HPI-VAE model.
  • Figure 5: Values of the latent variables obtained by the models (y-axis) against time (x-axis) for one trajectory of the elastic pendulum. The subplots show the respective latent variables for the (a) baseline, (b) PI-AE, (c) PI-VAE, and (d) HPI-VAE model.