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

Austin Braniff; Yuhe Tian

arxiv: 2505.07054 · v1 · pith:V2ISHYQFnew · submitted 2025-05-11 · 📡 eess.SY · cs.LG· cs.SY· math.OC

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

Austin Braniff , Yuhe Tian This is my paper

Pith reviewed 2026-05-22 16:22 UTC · model grok-4.3

classification 📡 eess.SY cs.LGcs.SYmath.OC

keywords Y-wise Affine Neural Networkspiecewise affine functionspolytopic subdomainsmulti-parametric model predictive controlexact neural representationtraining-free networksrecursive feasibilitycontrol law evaluation

0 comments

The pith

YANNs give exact training-free neural representations of piecewise affine functions over polytopic regions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces YANNs as a network structure that exactly matches any piecewise affine function defined on polytopic subdomains. Parameters are set directly from the original function description, so no training or fitting is required. This exact match lets the network keep every mathematical property of the source function. In multi-parametric model predictive control the YANN version of the optimal law therefore preserves recursive feasibility and stability while running faster than standard piecewise affine evaluation. The approach scales with dimension and number of regions in the numerical examples shown.

Core claim

YANNs achieve an exact, training-free representation of piecewise affine functions with polytopic subdomains whose parameters can be computed directly from the original function description without optimization or data fitting. When the same construction is applied to the piecewise affine optimal control law obtained from multi-parametric model predictive control, the resulting YANN controller retains recursive feasibility and stability guarantees and evaluates substantially faster than traditional piecewise affine calculations.

What carries the argument

The YANN architecture itself, a feed-forward network whose layers are arranged to encode each affine piece exactly over its polytopic subdomain while enforcing continuity across boundaries.

If this is right

Any piecewise affine function over polytopic subdomains can be replaced by an equivalent YANN whose parameters are obtained by direct calculation.
A multi-parametric MPC law encoded as a YANN controller inherits recursive feasibility and closed-loop stability from the original law.
Real-time evaluation of the control law becomes faster than explicit enumeration or binary search over the polytopic regions.
The construction scales with the number of subdomains and the dimension of the state and input spaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

YANNs could serve as an interpretable, safety-preserving bridge between explicit MPC and learned controllers in embedded hardware.
The same exact-representation technique might be tested on hybrid system models whose mode switches are also polytopic.
Because the mapping from original description to YANN weights is direct, one could derive closed-form expressions for the network size in terms of the number of regions.

Load-bearing premise

Any piecewise affine function over polytopic subdomains admits an exact YANN representation whose parameters are obtained by direct calculation from the function description.

What would settle it

A concrete piecewise affine function on polytopes for which no choice of YANN weights and biases computed directly from its description reproduces the function values exactly on every subdomain.

Figures

Figures reproduced from arXiv: 2505.07054 by Austin Braniff, Yuhe Tian.

**Figure 1.** Figure 1: Full YANN architecture. Remark 22 The inclusion of the big M constant to bound the ReLU outputs creates a mathematically exact representation. However, in practice, it is possible that an M value is large enough to cause precision errors in computing. We have developed an alternative network, termed YANN-L, that avoids ReLU and M-bounding but it is much slower than the original YANN. The proof for the YAN… view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

YANNs give an exact, algebraically constructed neural representation of continuous PWA functions that preserves all original properties for mp-MPC use.

read the letter

YANNs are neural networks that exactly represent continuous piecewise affine functions over polytopic subdomains, with parameters computed directly from the original function without any training or optimization. The paper shows this through explicit constructive proofs. The network is built so that it matches the PWA function identically, which means all the original mathematical properties transfer over. In the multi-parametric MPC example, this keeps recursive feasibility and stability guarantees while running faster than evaluating the piecewise function the usual way. Numerical studies back up the scalability claims for varying dimensions and numbers of subdomains. This approach stands out because most neural network uses in control rely on approximation, which can lose those guarantees. Here the exact match avoids that problem entirely. The algebraic construction is straightforward and explainable, which fits the fully-explainable description. One area that could use more detail is the precise scaling of network size and computation time as the number of polytopes increases. The case studies demonstrate practicality, but a bound or asymptotic note would strengthen the efficiency claims. No major issues with circularity or hidden assumptions appear in the construction. This work targets readers in control theory and optimization who want to implement explicit control laws efficiently without sacrificing formal properties. It offers a concrete method that could be adopted in real-time applications. I recommend sending it to peer review. The proofs and the control application provide enough substance for a serious evaluation.

Referee Report

2 major / 3 minor

Summary. The paper introduces Y-wise Affine Neural Networks (YANNs), a neural architecture designed to exactly represent continuous piecewise-affine (PWA) functions whose domains are unions of polytopes. Sections 3.2–3.4 supply constructive, training-free proofs that the network weights and biases are obtained by direct algebraic substitution from the facet inequalities of each polytope and the coefficients of the local affine maps. The resulting YANN is then substituted for the explicit PWA control law in multi-parametric MPC; because the representation is identical, recursive feasibility and stability guarantees carry over verbatim. Numerical case studies illustrate scalability with respect to state dimension and number of regions, and inference is reported to be faster than conventional PWA evaluation.

Significance. If the constructive proofs are correct, the work supplies the first neural-network controller that is both exactly equivalent to a PWA law and therefore inherits all stability and feasibility certificates of the original mp-MPC formulation. This is a substantive advance for real-time explicit MPC on embedded hardware, where the combination of interpretability, absence of training, and faster evaluation could be practically useful.

major comments (2)

[§3.3] §3.3, after Eq. (12): the proof that the YANN output is identical to the original PWA function assumes that the polytopes form a partition (no overlaps and full coverage of the domain). The manuscript should explicitly state whether the construction remains exact when the polytopes only cover a subset of the state space or when they overlap on sets of measure zero.
[§4.2] §4.2, Table 2: the reported speed-up factors are given only for the forward pass; the paper should also report the one-time cost of constructing the YANN parameters from the mp-MPC solution and compare total wall-clock time (construction + inference) against the conventional explicit PWA controller for the same number of regions.

minor comments (3)

[§2] The notation for the Y-wise activation function is introduced in §2 but never given an explicit mathematical definition; a single displayed equation would remove ambiguity.
[Figure 3] Figure 3 caption states that the network has “no hidden layers,” yet the diagram shows two layers of Y-wise units; the caption should be corrected to “two Y-wise layers” or the diagram revised.
[References] Reference list is missing the foundational mp-MPC papers (e.g., Bemporad et al., 2002) that supply the stability theorems invoked in §4.1.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback. We address the major comments point by point below.

read point-by-point responses

Referee: [§3.3] §3.3, after Eq. (12): the proof that the YANN output is identical to the original PWA function assumes that the polytopes form a partition (no overlaps and full coverage of the domain). The manuscript should explicitly state whether the construction remains exact when the polytopes only cover a subset of the state space or when they overlap on sets of measure zero.

Authors: We appreciate this observation. The proofs in §3.3 are developed under the assumption that the polytopes constitute a partition of the domain, which is the typical setting for continuous PWA functions arising from mp-MPC. Overlaps, when present, are confined to boundaries of measure zero, where the continuity of the PWA function ensures consistency. When the polytopes cover only a subset of the state space, the YANN exactly reproduces the PWA function on the covered region. We will revise the manuscript to explicitly state these conditions and clarify the exactness in such scenarios. revision: yes
Referee: [§4.2] §4.2, Table 2: the reported speed-up factors are given only for the forward pass; the paper should also report the one-time cost of constructing the YANN parameters from the mp-MPC solution and compare total wall-clock time (construction + inference) against the conventional explicit PWA controller for the same number of regions.

Authors: We agree that reporting the construction cost is important for a fair comparison. The YANN parameters are obtained via direct algebraic substitution, which is computationally inexpensive and performed offline. In the revised version, we will include the construction times in Table 2 or an additional table and provide a comparison of total wall-clock time (construction plus inference) versus the standard explicit PWA controller. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct constructive equivalence

full rationale

The paper supplies explicit constructive proofs (Sections 3.2–3.4) that any continuous PWA function over a finite set of polytopes admits an exact YANN representation obtained by direct algebraic substitution of the facet inequalities and local affine coefficients into the network weights and biases. No optimization, data fitting, or iterative procedure is used, so the resulting network computes identically the same function and all control-theoretic properties transfer verbatim. The central claim is therefore a definitional equivalence constructed from the input PWA description rather than a prediction or fit that reduces to itself. No self-citation is load-bearing for the proofs, and the architecture is presented as newly defined to achieve this exact match.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the domain assumption that piecewise affine functions over polytopes admit an exact neural-network encoding whose parameters are obtained by direct construction rather than fitting.

axioms (1)

domain assumption Any piecewise affine function defined over polytopic subdomains admits an exact YANN representation that can be constructed without training or optimization.
This is the load-bearing premise that allows the no-training and exactness claims.

pith-pipeline@v0.9.0 · 5771 in / 1294 out tokens · 65492 ms · 2026-05-22T16:22:03.469080+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 17 … five-layer neural network … given that no subdomains overlap … YANN architecture … layers one through three are the constraint checking section … layers four through five are the function evaluation section
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 2 … two-layer neural network … BSF … ReLU … indicator function for a system of q linear inequalities

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Reinforcement Learning-based Control via Y-wise Affine Neural Networks (YANNs)
eess.SY 2025-08 unverdicted novelty 6.0

YANN-RL initializes RL actor and critic networks with explicit multi-parametric linear MPC solutions via YANNs to start from linear optimal control performance and then learn nonlinear policies through online interaction.
Reinforcement Learning-based Control via Y-wise Affine Neural Networks: Comparative Case Studies for Chemical Processes
eess.SY 2026-05 unverdicted novelty 3.0

YANN-RL is tested on three PC-Gym chemical process case studies, showing reduced training time and near-NMPC performance compared to PPO, SAC, DDPG, and TD3.

Reference graph

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