Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

ControlSynth Neural ODEs: Modeling Dynamical Systems with Guaranteed Convergence

Wenjie Mei, Dongzhe Zheng, Shihua Li

TL;DR

It is shown that despite their highly nonlinear nature, convergence can be guaranteed via tractable linear inequalities in CSODEs, and it is illustrated that CSODEs have better learning and predictive abilities in these settings.

Abstract

Neural ODEs (NODEs) are continuous-time neural networks (NNs) that can process data without the limitation of time intervals. They have advantages in learning and understanding the evolution of complex real dynamics. Many previous works have focused on NODEs in concise forms, while numerous physical systems taking straightforward forms, in fact, belong to their more complex quasi-classes, thus appealing to a class of general NODEs with high scalability and flexibility to model those systems. This, however, may result in intricate nonlinear properties. In this paper, we introduce ControlSynth Neural ODEs (CSODEs). We show that despite their highly nonlinear nature, convergence can be guaranteed via tractable linear inequalities. In the composition of CSODEs, we introduce an extra control term for learning the potential simultaneous capture of dynamics at different scales, which could be particularly useful for partial differential equation-formulated systems. Finally, we compare several representative NNs with CSODEs on important physical dynamics under the inductive biases of CSODEs, and illustrate that CSODEs have better learning and predictive abilities in these settings.

ControlSynth Neural ODEs: Modeling Dynamical Systems with Guaranteed Convergence

TL;DR

It is shown that despite their highly nonlinear nature, convergence can be guaranteed via tractable linear inequalities in CSODEs, and it is illustrated that CSODEs have better learning and predictive abilities in these settings.

Abstract

Neural ODEs (NODEs) are continuous-time neural networks (NNs) that can process data without the limitation of time intervals. They have advantages in learning and understanding the evolution of complex real dynamics. Many previous works have focused on NODEs in concise forms, while numerous physical systems taking straightforward forms, in fact, belong to their more complex quasi-classes, thus appealing to a class of general NODEs with high scalability and flexibility to model those systems. This, however, may result in intricate nonlinear properties. In this paper, we introduce ControlSynth Neural ODEs (CSODEs). We show that despite their highly nonlinear nature, convergence can be guaranteed via tractable linear inequalities. In the composition of CSODEs, we introduce an extra control term for learning the potential simultaneous capture of dynamics at different scales, which could be particularly useful for partial differential equation-formulated systems. Finally, we compare several representative NNs with CSODEs on important physical dynamics under the inductive biases of CSODEs, and illustrate that CSODEs have better learning and predictive abilities in these settings.

Paper Structure

This paper contains 57 sections, 1 theorem, 21 equations, 8 figures, 13 tables.

Table of Contents

  1. Introduction
  2. ControlSynth Neural Ordinary Differential Equations
  3. Related Work
  4. SONODEs, ANODEs, and NCDEs vs CSODEs
  5. Convergence and Stability
  6. Physical Information Based Dynamics
  7. Theoretical Results: Convergence Analysis
  8. Convergence Conditions
  9. Preliminary Experiments
  10. Convergence and Stability
  11. Generalization and Extrapolation
  12. Computational Performance
  13. Complex Systems Time Series Prediction Experiments
  14. Model Architectures for Experiments
  15. Experimental Tasks and Datasets Description
  16. ...and 42 more sections

Key Result

Theorem 1

Let Assumptions asssum_nonlinear-assum:error_term be satisfied. If there exist positive semidefinite symmetric matrices $P,\tilde{P}$; positive semidefinite diagonal matrices $\{\Lambda^{j}=\mathrm{diag}(\Lambda^{j}_1,\dots,\Lambda^{j}_n)\}{}_{j=1}^{M},\;\{\tilde{\Lambda}^{j}=\mathrm{diag}(\tilde{\L

Figures (8)

  • Figure 1: Schematic of the CSODEs solver, showing integration via NNs at one time step. Using the forward Euler method as an example, it shows how $u_t$ and $x_t$ evolve through the update neural function $h(\cdot)$ and NNs $g(\cdot)$, $A_1 f_1(W_1 \cdot)$, ..., $A_M f_M(W_M \cdot)$ to yield $u_{t+\Delta t}$ and $x_{t+\Delta t}$ as the next variables.
  • Figure 2: Comparison of Mean Absolute Error (MAE) loss reduction curves across 1500 training epochs between CSODE and NODE models until convergence.
  • Figure 3: Qualitative comparison of the CSODE (top) and the NODE (bottom) predictions against ground truth trajectories at 400, 600, and 800 training epochs.
  • Figure 4: Qualitative results of Hindmarsh-Rose model prediction
  • Figure 5: Qualitative results of Reaction-Diffusion system prediction
  • ...and 3 more figures

Theorems & Definitions (6)

  • Remark 1
  • Definition 1
  • Theorem 1
  • Definition 2
  • Definition 3
  • Proof 1: Proof of Theorem \ref{['thm:convergence']}