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A data-driven model-free physical-informed deep operator network for solving nonlinear dynamic system

Jieming Sun, Lichun Li

TL;DR

A data-driven model-free physical-informed Deep Operator Network (DeepOnet) framework to learn the nonlinear dynamic systems from few available data and uses a surrogate machine learning model to extract the short-term dependence.

Abstract

The existing physical-informed Deep Operator Networks are mostly based on either the well-known mathematical formula of the system or huge amounts of data for different scenarios. However, in some cases, it is difficult to get the exact mathematical formula and vast amounts of data in some dynamic systems, we can only get a few experimental data or limited mathematical information. To address the cases, we propose a data-driven model-free physical-informed Deep Operator Network (DeepOnet) framework to learn the nonlinear dynamic systems from few available data. We first explore the short-term dependence of the available data and use a surrogate machine learning model to extract the short-term dependence. Then, the surrogate machine learning model is incorporated into the DeepOnet as the physical information part. Then, the constructed DeepOnet is trained to simulate the system's dynamic response for given control inputs and initial conditions. Numerical experiments on different systems confirm that our DeepOnet framework learns to approximate the dynamic response of some nonlinear dynamic systems effectively.

A data-driven model-free physical-informed deep operator network for solving nonlinear dynamic system

TL;DR

A data-driven model-free physical-informed Deep Operator Network (DeepOnet) framework to learn the nonlinear dynamic systems from few available data and uses a surrogate machine learning model to extract the short-term dependence.

Abstract

The existing physical-informed Deep Operator Networks are mostly based on either the well-known mathematical formula of the system or huge amounts of data for different scenarios. However, in some cases, it is difficult to get the exact mathematical formula and vast amounts of data in some dynamic systems, we can only get a few experimental data or limited mathematical information. To address the cases, we propose a data-driven model-free physical-informed Deep Operator Network (DeepOnet) framework to learn the nonlinear dynamic systems from few available data. We first explore the short-term dependence of the available data and use a surrogate machine learning model to extract the short-term dependence. Then, the surrogate machine learning model is incorporated into the DeepOnet as the physical information part. Then, the constructed DeepOnet is trained to simulate the system's dynamic response for given control inputs and initial conditions. Numerical experiments on different systems confirm that our DeepOnet framework learns to approximate the dynamic response of some nonlinear dynamic systems effectively.
Paper Structure (13 sections, 1 theorem, 11 equations, 18 figures, 1 table)

This paper contains 13 sections, 1 theorem, 11 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Physical-informed Neural Networks
  4. Deep Operator Network
  5. Short-term Dependency
  6. Methodology
  7. Using Machine Learning Models to Learn Short-Term Dependency
  8. Model-free physical-informed deep operator network
  9. Result
  10. Pendulum System
  11. Driven Oscillator System
  12. Chaotic Jerk System
  13. Conclusion

Key Result

Theorem 2.1

The approximated sampled discrete time system (sys:euler) is $n$-step dependent, i.e. there exists a function $\gamma$ mapping from $x(t-n),\ldots,x(t-1)$ and $u(t-n),\ldots,u(t-1)$ to $x(t)$.

Figures (18)

  • Figure 1: The DeepOnet architecture
  • Figure 2: Using machine learning model to learn the short-term dependency.
  • Figure 3: The proposed model-free physical-informed DeepOnet architecture
  • Figure 4: Test Results of DeepOnet on the Pendulum System without Noise
  • Figure 5: Test Results of Our Approach on the Pendulum System without Noise
  • ...and 13 more figures

Theorems & Definitions (1)

  • Theorem 2.1