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YANNs: Y-wise Affine Neural Networks for Exact and Efficient Representations of Piecewise Linear Functions

Austin Braniff, Yuhe Tian

TL;DR

YANNs introduce a five-layer neural architecture that exactly represents continuous piecewise affine functions over polytopic subdomains without training, enabling exact representation of explicit mp-MPC laws and preserving control guarantees such as recursive feasibility and stability. The method uses indicator-based constraint checking (layers 1–3) and affine evaluation (layers 4–5), with a Big-$M$ gate ensuring exactness. Compared to traditional mp-MPC computations, YANNs offer substantially faster inference while maintaining accuracy and scalability to thousands of subdomains; a slower, more precise variant (YANN-L) avoids Big-$M$ at the cost of speed. Numerical case studies on a double-integrator mp-MPC and a CSTR demonstrate speedups and the tradeoffs between speed and precision, supporting YANNs as an interpretable, efficient starting point for data-driven control.

Abstract

This work formally introduces Y-wise Affine Neural Networks (YANNs), a fully-explainable network architecture that continuously and efficiently represent piecewise affine functions with polytopic subdomains. Following from the proofs, it is shown that the development of YANNs requires no training to achieve the functionally equivalent representation. YANNs thus maintain all mathematical properties of the original formulations. Multi-parametric model predictive control is utilized as an application showcase of YANNs, which theoretically computes optimal control laws as a piecewise affine function of states, outputs, setpoints, and disturbances. With the exact representation of multi-parametric control laws, YANNs retain essential control-theoretic guarantees such as recursive feasibility and stability. This sets YANNs apart from the existing works which apply neural networks for approximating optimal control laws instead of exactly representing them. By optimizing the inference speed of the networks, YANNs can evaluate substantially faster in real-time compared to traditional piecewise affine function calculations. Numerical case studies are presented to demonstrate the algorithmic scalability with respect to the input/output dimensions and the number of subdomains. YANNs represent a significant advancement in control as the first neural network-based controller that inherently ensures both feasibility and stability. Future applications can leverage them as an efficient and interpretable starting point for data-driven modeling/control.

YANNs: Y-wise Affine Neural Networks for Exact and Efficient Representations of Piecewise Linear Functions

TL;DR

YANNs introduce a five-layer neural architecture that exactly represents continuous piecewise affine functions over polytopic subdomains without training, enabling exact representation of explicit mp-MPC laws and preserving control guarantees such as recursive feasibility and stability. The method uses indicator-based constraint checking (layers 1–3) and affine evaluation (layers 4–5), with a Big- gate ensuring exactness. Compared to traditional mp-MPC computations, YANNs offer substantially faster inference while maintaining accuracy and scalability to thousands of subdomains; a slower, more precise variant (YANN-L) avoids Big- at the cost of speed. Numerical case studies on a double-integrator mp-MPC and a CSTR demonstrate speedups and the tradeoffs between speed and precision, supporting YANNs as an interpretable, efficient starting point for data-driven control.

Abstract

This work formally introduces Y-wise Affine Neural Networks (YANNs), a fully-explainable network architecture that continuously and efficiently represent piecewise affine functions with polytopic subdomains. Following from the proofs, it is shown that the development of YANNs requires no training to achieve the functionally equivalent representation. YANNs thus maintain all mathematical properties of the original formulations. Multi-parametric model predictive control is utilized as an application showcase of YANNs, which theoretically computes optimal control laws as a piecewise affine function of states, outputs, setpoints, and disturbances. With the exact representation of multi-parametric control laws, YANNs retain essential control-theoretic guarantees such as recursive feasibility and stability. This sets YANNs apart from the existing works which apply neural networks for approximating optimal control laws instead of exactly representing them. By optimizing the inference speed of the networks, YANNs can evaluate substantially faster in real-time compared to traditional piecewise affine function calculations. Numerical case studies are presented to demonstrate the algorithmic scalability with respect to the input/output dimensions and the number of subdomains. YANNs represent a significant advancement in control as the first neural network-based controller that inherently ensures both feasibility and stability. Future applications can leverage them as an efficient and interpretable starting point for data-driven modeling/control.
Paper Structure (16 sections, 77 equations, 1 figure, 4 tables)

This paper contains 16 sections, 77 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. Multi-parametric model predictive control
  4. Neural networks
  5. Further notation
  6. Problem formulation
  7. Exact representation
  8. Subdomain identification
  9. Function Evaluation
  10. Full Theorem
  11. Numerical case studies
  12. Conclusions
  13. Corollary to Lemma \ref{['Lemma8']}
  14. Bounded function theorem
  15. Proof for YANN-L
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Full YANN architecture.