Non-intrusive Model Combination in Learning Dynamics

Citation


@article{wu2024non,
  title={Non-intrusive model combination for learning dynamical systems},
  author={Wu, Shiqi and Chamoin, Ludovic and Li, Qianxiao},
  journal={Physica D: Nonlinear Phenomena},
  volume={463},
  pages={134152},
  year={2024},
  publisher={Elsevier}
}

Keywords

Learning Dynamics, Model Combination, Machine Learning, Koopman Operator, Iterative Projection

Summary

Motivation

Combining different modeling approaches improves performance in data-driven modeling of dynamical systems.

Traditional model combination techniques suffer from intrusiveness, requiring significant modifications to existing learning algorithms.

A non-intrusive method is desirable to facilitate seamless model combination without modifying individual learning algorithms.

Proposed Approach

Introduces an iterative, non-intrusive methodology to combine two model spaces for learning dynamical systems.

Uses iterative projections onto each model space instead of joint optimization.

Proven to be optimal in the linear setting, achieving solutions in the direct sum of two hypothesis spaces.

Converges linearly, with explicit a priori and a posteriori estimates.

Key Contributions

Framework for Non-Intrusive Model Combination:

Avoids modifying existing learning algorithms.

Iteratively projects residuals onto each model space.

Achieves a better solution than traditional residual learning.

Convergence Analysis:

Demonstrates that the method has linear convergence.

Defines convergence conditions and stopping criteria.

Application to Hybrid Models:

Demonstrates effectiveness in various dynamical systems.

Shows improvement over residual learning and joint training.

Methodology

Problem Formulation

Iterative Projection Algorithm

Initialize .

Iteratively update:

Continue until convergence criterion is met.

Acceleration Scheme

Enhances convergence by adaptively adjusting residual projections.

Uses a weighted combination of previous iterations for faster convergence.

Numerical Experiments

Experiment 1: Reaction–Diffusion Equation

A PDE system with a diffusion term (linear) and a reaction term (nonlinear).

Model Combination Approach:

Linear regression for diffusion term.
Koopman operator-based model for reaction term.

Results:

Iterative model combination significantly outperforms residual learning and single-model approaches.
Shows long-term stability and accuracy in predictions.

Experiment 2: Cardiac Electrophysiology Model

PDE-ODE coupled system modeling heart electrical activity.

Model Combination Approach:

Finite difference method for PDE.
Neural network for ODE modelling ion currents.

Results:

Improved accuracy and efficiency in solving coupled system dynamics.

Experiment 3: Control Problem in Parameterized Dynamical System

Problem.

In this section, we consider a parameterized discrete-time nonlinear controlled dynamical system

where denotes the state of the system,

is the control input,

represents the external input,

and is the transition mapping.

The control input remains subject to adjustments and monitoring by the controller,

whereas is influenced by external factors and cannot be controlled.

Example: Tailed-Accurate System

MPC Problem

We focus on the tracking problem. The problem is formulated as follows: given the initial value and external inputs , we would like to obtain a sequence of controlled inputs minimizing

where and are semi-positive matrices. We solve this tracking problem by Model Predictive Control (MPC), where the current control is obtained by solving a finite horizon open-loop optimal control problem at each sampling instant. The MPC formulation solves the following optimization problem:

subject to:

(43)

where

The MPC solver is outlined in Algorithm 3.

The fundamental challenge in an MPC solver lies in selecting the appropriate transition function of observables, denoted as in Algorithm 3. Unlike the previous two experiments, in this section, we compare the models with different structures for solving the control problem, highlighting the fact that our method of model combination can provide more accurate solutions when dealing with the control problem and guarantee the convexity of the optimization problem.

Methods

We compare the performance of the following four structures, where only the hybrid ones are trained using the proposed method:

Linear structure:

Hybrid structure (our methods, represented as hybrid structure 1 and hybrid structure 2 respectively):

Nonlinear structure:

Results

Conclusion

Proposed a non-intrusive model combination framework that allows efficient learning of dynamical systems.

Achieves better performance than residual learning while maintaining computational efficiency.

Validated across different dynamical system problems, showing broad applicability.

Future Work:

Extend to more complex hybrid models.
Combination on multi-step/Combination of controller