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Neural DDEs with Learnable Delays for Partially Observed Dynamical Systems

Thibault Monsel, Emmanuel Menier, Onofrio Semeraro, Lionel Mathelin, Guillaume Charpiat

TL;DR

It is demonstrated that Constant Lag Neural Delay Differential Equations (NDDEs) naturally serve as suitable models for partially observed states and are shown to outperform existing methods on both synthetic and experimental data.

Abstract

Many successful methods to learn dynamical systems from data have recently been introduced. Such methods often rely on the availability of the system's full state. However, this underlying hypothesis is rather restrictive as it is typically not confirmed in practice, leaving us with partially observed systems. Utilizing the Mori-Zwanzig (MZ) formalism from statistical physics, we demonstrate that Constant Lag Neural Delay Differential Equations (NDDEs) naturally serve as suitable models for partially observed states. In empirical evaluation, we show that such models outperform existing methods on both synthetic and experimental data.

Neural DDEs with Learnable Delays for Partially Observed Dynamical Systems

TL;DR

It is demonstrated that Constant Lag Neural Delay Differential Equations (NDDEs) naturally serve as suitable models for partially observed states and are shown to outperform existing methods on both synthetic and experimental data.

Abstract

Many successful methods to learn dynamical systems from data have recently been introduced. Such methods often rely on the availability of the system's full state. However, this underlying hypothesis is rather restrictive as it is typically not confirmed in practice, leaving us with partially observed systems. Utilizing the Mori-Zwanzig (MZ) formalism from statistical physics, we demonstrate that Constant Lag Neural Delay Differential Equations (NDDEs) naturally serve as suitable models for partially observed states. In empirical evaluation, we show that such models outperform existing methods on both synthetic and experimental data.
Paper Structure (23 sections, 4 theorems, 52 equations, 17 figures, 13 tables)

This paper contains 23 sections, 4 theorems, 52 equations, 17 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Modelling Partially Observed Dynamical Systems
  3. The Mori-Zwanzig (MZ) formalism
  4. Approximations of Integro-Differential Equations (IDE)
  5. Exact representation with Neural DDE
  6. Neural Delay Differential Equations with Learnable Delays
  7. Experiments
  8. Dynamical systems
  9. Results
  10. Conclusion
  11. Derivation of Theorem \ref{['th:mztheorem']}
  12. Cancelling out the noise term $F(x,t)$
  13. t-model derivation
  14. Learning the delays
  15. Neural IDE and Neural DDE Benchmark
  16. ...and 8 more sections

Key Result

Theorem 2.1

Mori-Zwanzig equation formalism Let us consider a nonlinear system evolving on a smooth manifold $\mathcal{S} \subset \mathbb{R}^n$: where the full state $x \in \mathcal{S}$ can be accessed only through the lens of an arbitrary number of scalar-valued observables $g_i : \mathcal{S} \xrightarrow[]{}\mathbb{R}$. Then, under reasonable assumptions (see Appendix ap:noise_term), the dynamics of the ve

Figures (17)

  • Figure 1: The MZ equation DDE approximation used to model partially observed systems. Here $h(\cdot)$ is a measurement sensing operator.
  • Figure 2: Modelling possibilities
  • Figure 3: Sketch of open cavity flow. Sensor placed in P.
  • Figure 4: Toy dataset random test sample
  • Figure 5: Toy dataset delay evolution during training
  • ...and 12 more figures

Theorems & Definitions (4)

  • Theorem 2.1
  • Proposition 2.2: Exact representation with delays
  • Theorem 3.1
  • Theorem H.1