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Input Convex Lipschitz RNN: A Fast and Robust Approach for Engineering Tasks

Zihao Wang, Zhe Wu

TL;DR

This work tackles the dual goals of computational efficiency and robustness in neural network–based process modeling and control by introducing the Input Convex Lipschitz Recurrent Neural Network (ICLRNN). The ICLRNN enforces both input convexity and Lipschitz continuity through non-negative weights with spectral norm bounds and convex, non-decreasing, Lipschitz activations, ensuring that the network's outputs are convex in the inputs and globally 1-Lipschitz. The authors validate ICLRNN on a chemical process (CSTR) for modeling and MPC, showing faster convergence and lower FLOPs than competing recurrent units, and on real-world solar irradiance forecasting, where ICLRNN achieves superior accuracy and efficiency. The approach offers a practical pathway to deploy NN-based optimization in real-time engineering applications, with open-source code provided for replication and adoption.

Abstract

Computational efficiency and robustness are essential in process modeling, optimization, and control for real-world engineering applications. While neural network-based approaches have gained significant attention in recent years, conventional neural networks often fail to address these two critical aspects simultaneously or even independently. Inspired by natural physical systems and established literature, input convex architectures are known to enhance computational efficiency in optimization tasks, whereas Lipschitz-constrained architectures improve robustness. However, combining these properties within a single model requires careful review, as inappropriate methods for enforcing one property can undermine the other. To overcome this, we introduce a novel network architecture, termed Input Convex Lipschitz Recurrent Neural Networks (ICLRNNs). This architecture seamlessly integrates the benefits of convexity and Lipschitz continuity, enabling fast and robust neural network-based modeling and optimization. The ICLRNN outperforms existing recurrent units in both computational efficiency and robustness. Additionally, it has been successfully applied to practical engineering scenarios, such as modeling and control of chemical process and the modeling and real-world solar irradiance prediction for solar PV system planning at LHT Holdings in Singapore. Source code is available at https://github.com/killingbear999/ICLRNN.

Input Convex Lipschitz RNN: A Fast and Robust Approach for Engineering Tasks

TL;DR

This work tackles the dual goals of computational efficiency and robustness in neural network–based process modeling and control by introducing the Input Convex Lipschitz Recurrent Neural Network (ICLRNN). The ICLRNN enforces both input convexity and Lipschitz continuity through non-negative weights with spectral norm bounds and convex, non-decreasing, Lipschitz activations, ensuring that the network's outputs are convex in the inputs and globally 1-Lipschitz. The authors validate ICLRNN on a chemical process (CSTR) for modeling and MPC, showing faster convergence and lower FLOPs than competing recurrent units, and on real-world solar irradiance forecasting, where ICLRNN achieves superior accuracy and efficiency. The approach offers a practical pathway to deploy NN-based optimization in real-time engineering applications, with open-source code provided for replication and adoption.

Abstract

Computational efficiency and robustness are essential in process modeling, optimization, and control for real-world engineering applications. While neural network-based approaches have gained significant attention in recent years, conventional neural networks often fail to address these two critical aspects simultaneously or even independently. Inspired by natural physical systems and established literature, input convex architectures are known to enhance computational efficiency in optimization tasks, whereas Lipschitz-constrained architectures improve robustness. However, combining these properties within a single model requires careful review, as inappropriate methods for enforcing one property can undermine the other. To overcome this, we introduce a novel network architecture, termed Input Convex Lipschitz Recurrent Neural Networks (ICLRNNs). This architecture seamlessly integrates the benefits of convexity and Lipschitz continuity, enabling fast and robust neural network-based modeling and optimization. The ICLRNN outperforms existing recurrent units in both computational efficiency and robustness. Additionally, it has been successfully applied to practical engineering scenarios, such as modeling and control of chemical process and the modeling and real-world solar irradiance prediction for solar PV system planning at LHT Holdings in Singapore. Source code is available at https://github.com/killingbear999/ICLRNN.
Paper Structure (24 sections, 6 theorems, 22 equations, 7 figures, 4 tables, 1 algorithm)

This paper contains 24 sections, 6 theorems, 22 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Nonlinear Systems and Optimization
  3. Notations
  4. Class of Systems
  5. MPC Formulation
  6. Family of Recurrent Units
  7. Simple RNN
  8. High-performing RNNs
  9. Input Convex RNN
  10. Lipschitz RNN
  11. Input Convex Lipschitz Recurrent Neural Networks
  12. Introduction to Input Convex Lipschitz RNN
  13. Implementation of ICLRNN
  14. Lipschitz Continuity of ICLRNN
  15. Input Convexity of ICLRNN
  16. ...and 9 more sections

Key Result

Lemma 1

Consider a recurrence relation of the form, omitting the bias term: where $g$ represents a general transformation applied to the weighted inputs, $\boldsymbol{\mathbf{W}}^{(x)}$ and $\boldsymbol{\mathbf{U}}^{(h)}$ are weight matrices, and $\boldsymbol{\mathbf{x}}_t$ and $\boldsymbol{\mathbf{h}}_{t-1}$ are the inputs. The terms $\boldsymbol{\mathbf{W}}^{(x)}\boldsymb where $L(\boldsymbol{\mathbf{W

Figures (7)

  • Figure 1: Mean testing MSE vs. degree of additive noise over 3 random trials.
  • Figure 2: Mean Lipschitz constant of the trained model vs. degree of additive noise over 3 random trials.
  • Figure 3: MPC converging path in a fixed timeframe with an initial condition at $[-1.5~kmol/m^3, 70~K]$.
  • Figure 4: Concentration profile (top left), temperature profile (middle left), converging path (bottom left) in a fixed timeframe with an initial condition at $[-1.5~kmol/m^3, 70~K]$ under MPC, and concentration profile (top right), temperature profile (middle right), converging path (bottom right) in a fixed timeframe with an initial condition at $[1.5~kmol/m^3, -70~K]$.
  • Figure 5: Concentration profile (top left), temperature profile (middle left), converging path (bottom left) in a fixed timeframe with an initial condition at $[-1.25~kmol/m^3, 50~K]$, and concentration profile (top right), temperature profile (middle right), converging path (bottom right) in a fixed timeframe with an initial condition at $[1.25~kmol/m^3, -50~K]$.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Remark 1
  • Definition 1: eriksson2013applied
  • Definition 2
  • Lemma 1
  • Lemma 2: eriksson2013applied
  • Lemma 3: virmaux2018lipschitz
  • Proposition 1
  • proof
  • Remark 2
  • Definition 3: boyd2004convex
  • ...and 4 more