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arxiv: 2511.13595 · v3 · submitted 2025-11-17 · 📡 eess.SY · cs.AI· cs.SY

Physics-Informed Neural Networks for Nonlinear Output Regulation

Sebastiano Mengozzi , Giovanni B. Esposito , Michelangelo Bin , Andrea Acquaviva , Andrea Bartolini , Lorenzo Marconi This is my paper

Pith reviewed 2026-05-17 20:25 UTC · model grok-4.3

classification 📡 eess.SY cs.AIcs.SY
keywords physics-informed neural networksnonlinear output regulationregulator equationszero-error manifoldfeedforward inputhelicopter synchronizationgeneralization across exosystems
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The pith

A physics-informed neural network learns the zero-error manifold and feedforward input to solve nonlinear output regulation problems without data or analytic PDE solutions.

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

This paper introduces a PINN method to approximate the functions that define the invariant manifold and steady-state input for nonlinear plants driven by an exosystem. The network minimizes the residuals of the regulator equations directly, using boundary conditions and feasibility constraints but no precomputed trajectories. The resulting mapping from exosystem state to plant state and input supports real-time evaluation and generalizes to different initial conditions and parameters within families of exosystems. Validation occurs on synchronizing a helicopter's vertical motion to an oscillating platform, where the learned manifold achieves high-fidelity reconstruction and sustained regulation under variations. A reader would care because the approach offers a practical solver for a class of control problems that are otherwise intractable by hand.

Core claim

The paper claims that a physics-informed neural network can accurately approximate the pair (π(w), c(w)) satisfying the regulator equations for a nonlinear plant, thereby constructing the zero-regulation-error manifold and the required feedforward input. This learned operator maps exosystem states to the corresponding steady-state plant states and inputs, enables real-time inference, and generalizes across families of the exosystem that differ in initial conditions and parameters, as shown by sustained performance on the helicopter synchronization example.

What carries the argument

A physics-informed neural network trained by minimizing the residuals of the regulator equations (the PDE system with algebraic constraint) subject to boundary and feasibility conditions, thereby approximating the invariant manifold π(w) and feedforward input c(w).

If this is right

  • The learned operator supports real-time computation of the control input at each instant.
  • Regulation performance is maintained when the exosystem varies in initial conditions and parameters within a family.
  • The zero-error manifold is reconstructed with high fidelity on the helicopter vertical dynamics task.
  • The same solver framework applies to any nonlinear system that admits a solution to the regulator equations.

Where Pith is reading between the lines

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

  • Combining the PINN with a state observer could extend the method to partial-information settings.
  • The residual-minimization approach may transfer to other control problems whose solutions are defined by invariant manifolds or PDE constraints.
  • Scaling tests on higher-dimensional plants would reveal whether the network size and training cost grow favorably with state dimension.

Load-bearing premise

The plant and exosystem states are fully known and the nonlinear system admits a solution to the output regulation problem.

What would settle it

Run the trained PINN on the helicopter example with exosystem frequencies or amplitudes outside the range used during training and check whether regulation error remains near zero or grows substantially.

Figures

Figures reproduced from arXiv: 2511.13595 by Andrea Acquaviva, Andrea Bartolini, Giovanni B. Esposito, Lorenzo Marconi, Michelangelo Bin, Sebastiano Mengozzi.

Figure 1
Figure 1. Figure 1: Loss landscape of the trained PINN evaluated over [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Control structure for the vertical landing problem. The [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Grid of experiments over different exosystem configurations. The [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Histogram of the vertical tracking error over all grid [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Trajectories of the exosystem reference signal and the [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Trajectories of the exosystem reference signal and the [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Behavior of the learned mapping cb(w) as a function of time for different values of the initial condition w1(0). 0 2 4 6 8 10 12 14 16 Time (s) 0 1 2 3 4 5 Radius (m) 0.044 0.045 0.046 0.047 0.048 0.049 πφ [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Behavior of the learned mapping πϕ(w) as a function of time for different values of the initial condition w1(0). 0 2 4 6 8 10 12 14 16 Time (s) 0 1 2 3 4 5 Radius (m) 0.012 0.014 0.016 0.018 0.020 0.022 πθ [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Behavior of the learned mapping πθ(w) as a function of time for different values of the initial condition w1(0). reconstructs the zero-error manifold with high accuracy. The controller maintains small tracking error over a wide range of amplitudes and frequencies, including conditions not seen during training. The method is computationally efficient: the training stage requires only a few minutes on a cons… view at source ↗
read the original abstract

This work addresses the full-information output regulation problem for nonlinear systems, assuming the states of both the plant and the exosystem are known. In this setting, perfect tracking or rejection is achieved by constructing a zero-regulation-error manifold $\pi(w)$ and a feedforward input $c(w)$ that render such manifold invariant. The pair $(\pi(w), c(w))$ is characterized by the regulator equations, i.e., a system of PDEs with an algebraic constraint. We focus on accurately solving the regulator equations introducing a physics-informed neural network (PINN) approach that directly approximates $\pi(w)$ and $c(w)$ by minimizing the residuals under boundary and feasibility conditions, without requiring precomputed trajectories or labeled data. The learned operator maps exosystem states to steady state plant states and inputs, enables real-time inference and, critically, generalizes across families of the exosystem with varying initial conditions and parameters. The framework is validated on a regulation task that synchronizes a helicopter's vertical dynamics with a harmonically oscillating platform. The resulting PINN-based solver reconstructs the zero-error manifold with high fidelity and sustains regulation performance under exosystem variations, highlighting the potential of learning-enabled solvers for nonlinear output regulation. The proposed approach is broadly applicable to nonlinear systems that admit a solution to the output regulation problem.

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.

Referee Report

2 major / 2 minor

Summary. The paper proposes a physics-informed neural network (PINN) approach to solve the regulator equations for the full-information output regulation problem in nonlinear systems. It directly approximates the zero-error manifold π(w) and feedforward input c(w) by minimizing PDE residuals under boundary and feasibility conditions, without labeled data or precomputed trajectories. The method is claimed to enable real-time inference and to generalize across exosystem families with varying initial conditions and parameters, with validation on synchronizing helicopter vertical dynamics to a harmonic exosystem.

Significance. If the generalization and accuracy claims hold with rigorous support, the work would provide a useful data-free solver for nonlinear regulator equations that could support real-time control applications where classical analytic or numerical methods are intractable. The direct residual-minimization strategy without requiring trajectories is a clear methodological strength that aligns with standard PINN practice and avoids circularity in the approximation.

major comments (2)
  1. [Abstract] Abstract and numerical validation: the central generalization claim—that the learned operator 'generalizes across families of the exosystem with varying initial conditions and parameters'—is load-bearing for the contribution, yet the reported validation is confined to a single helicopter example with harmonic exosystem variations inside the training distribution; no quantitative out-of-distribution tests, parameter sweeps, or a-posteriori error bounds on manifold invariance are supplied.
  2. [Numerical Results] Numerical results section: while 'high fidelity' reconstruction is stated, no quantitative metrics (e.g., L² or pointwise residual norms for the regulator PDEs, or tracking error under exosystem perturbations) are reported, preventing assessment of whether the approximation error remains small enough to preserve invariance for the claimed real-time and generalization use cases.
minor comments (2)
  1. [Problem Formulation] The assumption that the nonlinear system admits a solution to the regulator equations is stated but could be cross-referenced to a specific theorem or condition in the problem formulation section for clarity.
  2. [Abstract] Notation for the exosystem parameters and their variation ranges should be introduced explicitly when discussing generalization, to avoid ambiguity in what 'families' means.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments highlight important aspects of our generalization claims and the presentation of numerical results. We address each major comment below and have revised the manuscript to strengthen the supporting evidence.

read point-by-point responses

  1. Referee: [Abstract] Abstract and numerical validation: the central generalization claim—that the learned operator 'generalizes across families of the exosystem with varying initial conditions and parameters'—is load-bearing for the contribution, yet the reported validation is confined to a single helicopter example with harmonic exosystem variations inside the training distribution; no quantitative out-of-distribution tests, parameter sweeps, or a-posteriori error bounds on manifold invariance are supplied.

    Authors: We agree that the generalization claim benefits from more extensive validation. The helicopter example demonstrates the PINN's performance across a range of initial conditions and exosystem parameters that were included in the training distribution, showing that the learned mapping sustains regulation under these variations. To strengthen the evidence, the revised manuscript now includes explicit out-of-distribution tests with exosystem parameters outside the training range, additional parameter sweeps, and a-posteriori error bounds obtained by evaluating the regulator PDE residuals on held-out samples. These additions provide quantitative support for the invariance properties under the tested conditions. revision: yes

  2. Referee: [Numerical Results] Numerical results section: while 'high fidelity' reconstruction is stated, no quantitative metrics (e.g., L² or pointwise residual norms for the regulator PDEs, or tracking error under exosystem perturbations) are reported, preventing assessment of whether the approximation error remains small enough to preserve invariance for the claimed real-time and generalization use cases.

    Authors: We accept that the original numerical results section lacked sufficient quantitative detail. The phrase 'high fidelity' was intended to reflect close visual agreement between the approximated manifold and the expected steady-state behavior, along with successful regulation in simulation. In the revised version, we have added explicit quantitative metrics, including the L² norm and maximum pointwise norms of the PDE residuals for both the manifold and input equations, as well as the closed-loop tracking error under exosystem perturbations. These metrics are now reported for multiple test cases and confirm that the residual errors remain small enough to maintain practical invariance of the zero-error manifold. revision: yes

Circularity Check

0 steps flagged

No significant circularity; PINN directly minimizes regulator equation residuals

full rationale

The paper formulates the output regulation problem via the standard regulator equations (PDEs with algebraic constraint) and trains a PINN to approximate the manifold π(w) and input c(w) by penalizing those residuals plus boundary conditions. This is a direct residual-minimization procedure with no precomputed trajectories or labeled data, so the learned mapping is not equivalent to its inputs by construction. No self-citations appear as load-bearing premises, no uniqueness theorems are imported from the authors' prior work, and no fitted parameters are relabeled as predictions. The generalization claim across exosystem families is an empirical assertion resting on the trained network's behavior rather than a definitional identity. The derivation chain is therefore self-contained against the classical regulator equations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of solutions to the regulator equations and the assumption that full state information is available; no free parameters or invented entities are introduced in the abstract description.

axioms (2)
  • domain assumption The nonlinear system admits a solution to the output regulation problem
    Explicitly stated as the condition under which the framework is broadly applicable.
  • domain assumption Full state information of plant and exosystem is available
    The setting is defined as the full-information case.

pith-pipeline@v0.9.0 · 5546 in / 1217 out tokens · 26997 ms · 2026-05-17T20:25:22.110904+00:00 · methodology

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Reference graph

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