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Deep Operator Neural Network Model Predictive Control

Thomas Oliver de Jong, Khemraj Shukla, Mircea Lazar

TL;DR

This paper addresses learning-based predictive control for continuous-time nonlinear MIMO systems by modeling the input-to-output operator $G$ with DeepONet. It introduces Multi-Step DeepONet (MS-DeepONet) to produce multi-step, multi-output predictions over horizon $N$ with sampling time $T_s$ in a single evaluation, and proves a universal approximation property for multi-step operators. It also derives an adaptive-basis interpretation that facilitates data-enabled predictive control and provides hyperparameter-tuning strategies and PyTorch implementations. Numerical experiments on the van der Pol oscillator, a quadruple-tank process, and a pendulum-on-a-cart system show MS-DeepONet consistently outperforms the standard DeepONet in learning accuracy and MPC performance, with real-time feasible computation times.

Abstract

In this paper, we consider the design of model predictive control (MPC) algorithms based on deep operator neural networks (DeepONets). These neural networks are capable of accurately approximating real and complex valued solutions of continuous time nonlinear systems without relying on recurrent architectures. The DeepONet architecture is made up of two feedforward neural networks: the branch network, which encodes the input function space, and the trunk network, which represents dependencies on temporal variables or initial conditions. Utilizing the original DeepONet architecture as a predictor within MPC for Multi Input Multi Output (MIMO) systems requires multiple branch networks, to generate multi output predictions, one for each input. Moreover, to predict multiple time steps into the future, the network has to be evaluated multiple times. Motivated by this, we introduce a multi step DeepONet (MS-DeepONet) architecture that computes in one shot multi step predictions of system outputs from multi step input sequences, which is better suited for MPC. We prove that the MS DeepONet is a universal approximator in terms of multi step sequence prediction. Additionally, we develop automated hyper parameter selection strategies and implement MPC frameworks using both the standard DeepONet and the proposed MS DeepONet architectures in PyTorch. The implementation is publicly available on GitHub. Simulation results demonstrate that MS-DeepONet consistently outperforms the standard DeepONet in learning and predictive control tasks across several nonlinear benchmark systems: the van der Pol oscillator, the quadruple tank process, and a cart pendulum unstable system, where it successfully learns and executes multiple swing up and stabilization policies.

Deep Operator Neural Network Model Predictive Control

TL;DR

This paper addresses learning-based predictive control for continuous-time nonlinear MIMO systems by modeling the input-to-output operator with DeepONet. It introduces Multi-Step DeepONet (MS-DeepONet) to produce multi-step, multi-output predictions over horizon with sampling time in a single evaluation, and proves a universal approximation property for multi-step operators. It also derives an adaptive-basis interpretation that facilitates data-enabled predictive control and provides hyperparameter-tuning strategies and PyTorch implementations. Numerical experiments on the van der Pol oscillator, a quadruple-tank process, and a pendulum-on-a-cart system show MS-DeepONet consistently outperforms the standard DeepONet in learning accuracy and MPC performance, with real-time feasible computation times.

Abstract

In this paper, we consider the design of model predictive control (MPC) algorithms based on deep operator neural networks (DeepONets). These neural networks are capable of accurately approximating real and complex valued solutions of continuous time nonlinear systems without relying on recurrent architectures. The DeepONet architecture is made up of two feedforward neural networks: the branch network, which encodes the input function space, and the trunk network, which represents dependencies on temporal variables or initial conditions. Utilizing the original DeepONet architecture as a predictor within MPC for Multi Input Multi Output (MIMO) systems requires multiple branch networks, to generate multi output predictions, one for each input. Moreover, to predict multiple time steps into the future, the network has to be evaluated multiple times. Motivated by this, we introduce a multi step DeepONet (MS-DeepONet) architecture that computes in one shot multi step predictions of system outputs from multi step input sequences, which is better suited for MPC. We prove that the MS DeepONet is a universal approximator in terms of multi step sequence prediction. Additionally, we develop automated hyper parameter selection strategies and implement MPC frameworks using both the standard DeepONet and the proposed MS DeepONet architectures in PyTorch. The implementation is publicly available on GitHub. Simulation results demonstrate that MS-DeepONet consistently outperforms the standard DeepONet in learning and predictive control tasks across several nonlinear benchmark systems: the van der Pol oscillator, the quadruple tank process, and a cart pendulum unstable system, where it successfully learns and executes multiple swing up and stabilization policies.

Paper Structure

This paper contains 16 sections, 5 theorems, 56 equations, 11 figures, 3 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. THE MULTI-STEP DEEPONET ARCHITECTURE
  4. MS-DeepONet architecture
  5. The underlying structure of the MS-DeepONet architecture: an addaptive basis viewpoint
  6. MULTI-STEP DEEPONET LEARNING AND HYPER PARAMETER TUNING ALGORITHMS
  7. Data-generation
  8. Model learning and hyper parameter tuning
  9. NEURAL OPERATOR PREDICTIVE CONTROL
  10. Multi-step DeepONet MPC problem formulation
  11. Standard DeepONet model learning approach and MPC problem formulation
  12. NUMERICAL EXAMPLES
  13. Van der Pol oscillator
  14. Quadruple Tank Process
  15. Pendulum on a cart
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose that $\sigma\in(TW)$, $X$ is a Banach Space, $K_1 \subset X$, $K_2 \subset \mathbb{R}^d$ are two compact sets in $X$ and $\mathbb{R}^d$, and $V$ is a compact set in $C(K_1)$, $G$ is a nonlinear continuous operator, which maps $V$ into $C(K_2)$. Then for any $\epsilon > 0$, there are positive holds for all $u\in V$ and $z\in K_2$.

Figures (11)

  • Figure 1: Illustration of a fully connected feedforward neural network (FFN) and it's underlying basis.
  • Figure 2: Standard DeepONet formulations: (a) stacked and (b) unstacked architectures.
  • Figure 3: Illustration of the stacked MimoONet MimoONet for 2 inputs and 2 outputs.
  • Figure 4: Multi-step DeepONet architecture: the branch input at time instant $t = 0$ is the multi-step input sequence $\bar{\mathbf{u}}_0$; the trunk input is the measured state $x(0)$; the output is the predicted multi-step output sequence $\bar{\mathbf{u}}_0$.
  • Figure 5: Flow chart for model learning algorithms and predictive control. Red colors indicate online algorithms and blue colors offline.
  • ...and 6 more figures

Theorems & Definitions (14)

  • Theorem 1: Theorem 5, Chen_ONN
  • Theorem 2
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • ...and 4 more