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Unsupervised Physics-Informed Neural Network-based Nonlinear Observer design for autonomous systems using contraction analysis

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Pith's one-line read An unsupervised physics-informed neural network can learn contraction-based observer gains that guarantee global exponential convergence of the estimation error for autonomous nonlinear systems.

desk verdict The PINN idea is new, but the training target is infeasible for the paper's own examples, and the loss function's sign error makes the central claim unsupported. read the letter →

arxiv 2411.09237 v1 pith:QFDJJVQM submitted 2024-11-14 eess.SY cs.SY

Yasmine Marani , Israel Filho , Tareq Al-Naffouri , Taous-Meriem Laleg-Kirati This is my paper · ORCID
classification eess.SYcs.SY MSC 93B5393C1068T0793D30
keywords physics-informedneuralnetworksnonlinearobserverscontractionanalysismatrixpartialdifferentialinequalityunsupervisedlearningexponentialstabilityinput-to-stateautonomoussystems
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Contraction-based observer design has long been held back by the need to solve a matrix partial differential inequality (MPDI) for the observer's correction gain. This paper replaces that analytic step with an unsupervised physics-informed neural network (PINN) that learns the gain by penalizing violations of the inequality in its loss function. The authors claim the learned gain enforces the contraction condition enough to guarantee global exponential convergence of the estimation error, and they prove an exponential input-to-state-stability bound under bounded measurement noise and approximation error. If true, this gives a practical, data-free route to nonlinear observers with strong guarantees.

What carries the argument

The central object is the symmetric matrix $D(\hat{x},y)$ from the contraction condition (7), whose negative semi-definiteness is enforced via Sylvester's criterion using its leading principal minors. The loss function $L = \mu_1 L_{\text{MPDI}} + \mu_2 L_{\text{BC}}$ penalizes the signed principal minors (positive for odd indices, negative for even indices) and the boundary condition $\hat{k}_\theta(\hat{x}, h(\hat{x}))=0$, with collocation points sampled uniformly over the domain of interest. Automatic differentiation provides the Jacobian $\partial \hat{k}_\theta / \partial \hat{x}$ needed to form $D$.

What would settle it

Take the trained gain and evaluate the largest eigenvalue of $D(\hat{x},y) = \operatorname{He}\{\frac{\partial f}{\partial \hat{x}} + \frac{\partial \hat{k}_\theta}{\partial \hat{x}}\} + 2\lambda I$ on a dense grid over the domain $\mathcal{X}\times\mathcal{Y}$; if any eigenvalue is positive at a grid point, the MPDI (7) is violated and the claimed exponential convergence guarantee is not applicable to that gain.

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Extended reading notes

Core claim

The paper's central claim is that the loss function (6)-(8), built from the leading principal minors of the matrix $D(\hat{x},y)=\operatorname{He}\{\frac{\partial f}{\partial \hat{x}} + \frac{\partial \hat{k}_\theta}{\partial \hat{x}}\} + 2\lambda I$, enforces the contraction MPDI (7) on the training domain, so the learned gain $\hat{k}_\theta$ yields a globally exponentially convergent observer for the autonomous nonlinear system (1). Theorem 3 further states that under a Lipschitz gain and bounded noise and approximation error, the estimation error satisfies the exponential input-to-state bound (14) whenever the contraction rate $\lambda > 2$. The claim is supported by simulation on the Van der Pol and reverse Duffing oscillators, where increasing $\lambda$ reduces the mean squared estimation error under 15% measurement noise.

Load-bearing premise

The whole guarantee rests on the unverified Assumption 3(A2) that the trained network's gain is uniformly close to some ideal gain that satisfies the strict contraction inequality with $P=I$ and $\lambda>2$; if no such ideal gain exists or the training is not $\varepsilon$-close to it, the exponential bounds in Theorem 3 do not follow.

Editorial extensions

If this is right

  • The approach removes the need to solve the MPDI analytically; any smooth autonomous system with a known Jacobian can be plugged into the training loop directly.
  • The learned observer is robust to bounded measurement noise and network approximation error, with the steady-state error bound shrinking as the contraction rate $\lambda$ increases.
  • Because the loss is unsupervised (no state trajectories required), the method needs only collocation points, not data, to design an observer.
  • The exponential input-to-state bound (14) gives a quantitative guarantee that the estimation error remains within a neighborhood determined by noise level and approximation error.
  • Higher $\lambda$ improves noise rejection but makes it harder to satisfy the MPDI, revealing a design trade-off in choosing the contraction rate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension is to verify post-training whether the learned gain actually satisfies the MPDI on a dense grid; if not, the theoretical guarantee is void even if simulations look good. This check is not performed in the paper.
  • The method could be extended to non-autonomous systems by making the gain depend explicitly on time, though the paper leaves this as future work.
  • The choice $P=I$ simplifies the MPDI, but allowing a non-constant metric $P(x)$ could widen the class of acceptable systems; the same PINN framework might be adapted to learn both $P$ and $k$ together.
  • The ISS bound depends on the Lipschitz constant $L$ of the gain and the bounds of noise and error; a post-training computation of these constants could turn the qualitative guarantee into a computable error certificate.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes an unsupervised physics-informed neural network (PINN) approach for designing the correction term of a nonlinear observer for autonomous systems. The observer gain is learned by minimizing a loss that penalizes violation of the contraction matrix partial differential inequality (MPDI) from Theorem 1, with the metric fixed to the identity, plus a boundary-condition loss. The authors also state a robustness result (Theorem 3) bounding the estimation error under measurement noise and neural-network approximation error, and they report simulations on a Van der Pol oscillator and a reverse Duffing oscillator. The central claim is that the learned gain enforces the contraction conditions and therefore yields a globally exponentially convergent observer whose robustness is quantified by Theorem 3.

Significance. The problem is relevant: numerically solving the contraction-based observer MPDI is genuinely challenging, and a learning-based unsupervised formulation is a reasonable idea to explore. I credit the authors for avoiding dependence on labeled trajectory data, for using automatic differentiation to compute the required Jacobians, and for attempting an explicit robustness analysis with noise and approximation error. However, the formal core of the paper is not sound as written. The loss function does not correctly encode negative semidefiniteness, and, more seriously, for the two numerical examples there exists no C1 gain satisfying the MPDI with P=I, so the assumptions underlying Theorem 3 are vacuous in the very systems used for validation. These are load-bearing issues that invalidate the main claims. The paper would need substantial reworking—correcting the loss, choosing a feasible metric and/or examples, and repairing Theorem 3—before its contribution could be credited.

major comments (3)
  1. [Section IV.B, Eq. (8)] The loss for even principal minors has the wrong sign. Since (7) requires D to be negative semidefinite, every even principal minor must be nonnegative, so the penalty should grow when Δ_i is negative. Writing l_i = min(0, Δ_i) for even i makes the loss decrease as Δ_i becomes more negative, thereby actively encouraging violation of the contraction condition. Independently, Theorem 2 as stated is false: for symmetric matrices, alternation of the leading principal minors is necessary but not sufficient for negative semidefiniteness; for example, diag(0,1) has leading minors Δ1=0 and Δ2=0, satisfying (−1)^1Δ1≥0 and (−1)^2Δ2≥0, yet the matrix is not negative semidefinite. A correct criterion requires all principal minors, not only the leading ones. Thus minimizing (6) does not enforce (7).
  2. [Sections IV.B and VI, Eqs. (7), (19), (20)] With P=I and h(x)=x1, no C1 gain satisfying (7) exists for either test system. The boundary condition k(x,h(x))=0 in (4), together with C1 regularity, forces ∂k₂/∂x̂₂=0 at every point of the form (x̂₁,x̂₂,x̂₁). For the reverse Duffing oscillator (20), ∂f₂/∂x₂=0, so the (2,2) entry of D in (7) is 2λ>0. For the Van der Pol oscillator (19), at x₁=0 that entry is 1+2λ>0. A symmetric matrix with a strictly positive diagonal entry cannot be negative semidefinite, and the condition (7) is required to hold for all (x̂,y), including y=h(x̂). Therefore Assumption 3(A2) is vacuous in the examples, and the learned observer cannot inherit the guarantees of Theorem 1 or Theorem 3. The boundary loss (9) does not repair the obstruction, because it penalizes the value of k at the diagonal, not its derivative along x̂₂.
  3. [Section V, Theorem 3 and Eq. (14)] The bound in (14) does not follow from the proof. From (17) and the claimed inequality (18), the differential inequality is ˙V ≤ −(λ−2)V + (1/2)∥ε∥² + (L²/2)∥v∥². The solution of this inequality yields an estimation bound with decay rate (λ−2)/2 and asymptotic gain proportional to (∥ε̄∥² + L²∥v̄∥²)^{1/2}/(λ−2)^{1/2}, not the stated e^{−ηt} factor and the term (ε̄+Lv̄)/(2√η). The step from (17) to (18) is also insufficiently justified: the contraction condition gives, at best, e^T[f(x)−f(x̂)−k(x̂,y)] ≤ −λ∥e∥², and the additional ∥e∥² term from Young's inequality changes the effective rate; the citation to [31, Theorem 4.3, p. 231] is too terse. Since Theorem 3 is presented as the formal robustness guarantee, this gap is significant.
minor comments (6)
  1. [Throughout] The manuscript inconsistently uses both “MPDI” and “MDPI” for the matrix partial differential inequality; please standardize.
  2. [Section IV.B] “Principle minors” should be “principal minors” in the text describing Eq. (8).
  3. [Section IV.C] The training algorithm is said to be “provided upon request,” but for a methods paper the pseudocode should be fully self-contained, including batch size and the exact stopping criteria, or the code should be released.
  4. [Section VI] Figure 3 is captioned “Inverse Duffing” while the text and Eq. (20) call the system the “reverse Duffing” oscillator; the nomenclature should be consistent.
  5. [Section VI] The dataset region is written as “(X , Y) = ([ −1, 1]², [−1, 1])”; this notation is ambiguous and should be clarified, since X is a subset of R² and Y is a subset of R.
  6. [Section V, proof of Theorem 3] The “identity” from [31] that converts the path term into the contraction-rate term should be stated explicitly, including its assumptions, because the proof depends on it.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the contraction loss enforces an imported sufficient condition, and the stability/ISS claims follow from external contraction-observer theorems, not from fitted outputs.

full rationale

The derivation chain is not circular. The learned gain k̂θ is trained by minimizing a physics loss (6)-(9) that directly penalizes violation of the matrix partial differential inequality (7), i.e., the contraction condition from Theorem 1 with P = I. This is a standard PDE-constrained learning design: the learned object is constructed to satisfy a sufficient condition, and the exponential-convergence claim (Section IV and Theorem 3) is a consequence of that enforced inequality plus the external theorem [31], not an output inferred from fitted data. The parameter λ is a user-chosen design constant, and Table II is an empirical consistency check of Theorem 3, not a fit used to produce the bound (14). The paper does cite works by the same authors ([7], [26], [27]), but only as related work on algebraic estimators and KKL observer design; none of those citations carries the central existence or stability argument. The robustness proof invokes the 'identity in [31, see Theorem 4.3 p. 231]', an external survey result, so the paper is not relying on a self-citation chain. The known concerns—that the leading-principal-minor criterion in Theorem 2 is not by itself sufficient for negative semidefiniteness, and that Assumption 3(A2)'s existence of an ideal gain with P = I may fail for the example systems—are correctness/falsifiability issues, not circular reductions: they question whether the imported sufficient condition is satisfied, not whether the paper's derivation re-labels its own inputs.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard contraction theory, on the correctness of the loss-based enforcement (which has a sign error), and on the unverified existence and closeness of an ideal gain. The paper adds no new physical entities.

free parameters (3)
  • lambda (contraction rate) = 2.5, 4, 5
    Chosen by hand in Section VI; appears in the MPDI (7) and in the ISS bound (14); a larger λ makes the inequality harder to satisfy.
  • loss weights mu1, mu2, rho1, rho2 = VdP: mu1=1e-3, mu2=1, rho1=1, rho2=1e-1; Duffing: mu1=1, mu2=1, rho1=1, rho2=1
    Hand-selected in Table I; they balance the contraction loss and boundary loss and scale the principal-minor penalties.
  • Training region X, Y = VdP: X=[-2,2]x[-3,3], Y=[-2,2]; Duffing: X=[-1,1]^2, Y=[-1,1]
    Chosen by hand in Section VI; the learned gain is only enforced on this compact set, so the guarantees are semi-global.
assumptions (6)
  • standard math Theorem 1 (observer contraction condition)
    The existence of P, k, lambda satisfying (4) is taken from Bernard et al. [31] and used as the target for the neural loss.
  • standard math Theorem 2 (Sylvester's criterion) applied to negative semidefiniteness
    Used to justify checking leading principal minors only; however, for semidefinite matrices leading principal minors are insufficient, and the paper's sign convention for even minors is inverted.
  • domain assumption Assumption 1: forward invariance on compact sets X, Y
    Restricts validity to trajectories staying in the training region; supports the use of bounded training data.
  • domain assumption Assumption 2: learned gain is Lipschitz in its second argument
    Needed for the noise term in Theorem 3; justified by choosing Lipschitz activations, but not verified for the trained network.
  • domain assumption Assumption 3: bounded noise and bounded approximation error
    Defines v_bar and epsilon_bar; the latter bounds the difference between learned and ideal gain but is never quantified or verified.
  • ad hoc to paper Existence of an ideal gain k satisfying (4) with P=I and lambda>2 for the considered systems
    Not proven for Van der Pol or reverse Duffing; the numerical success of the observers depends on this existence, and Remark 1 fixes P=I without justification.

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Cite this review

Pith. "Pith review of Unsupervised Physics-Informed Neural Network-based Nonlinear Observer design for autonomous systems using contraction analysis." pith.science (2026). https://pith.science/paper/QFDJJVQM

@misc{pith2026241109237,
  author       = {Pith},
  title        = {Pith review of: Unsupervised Physics-Informed Neural Network-based Nonlinear Observer design for autonomous systems using contraction analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QFDJJVQM}},
  note         = {Machine review of arXiv:2411.09237}
}
read the original abstract

Contraction analysis offers, through elegant mathematical developments, a unified way of designing observers for a general class of nonlinear systems, where the observer correction term is obtained by solving an infinite dimensional inequality that guarantees global exponential convergence. However, solving the matrix partial differential inequality involved in contraction analysis design is both analytically and numerically challenging and represents a long-lasting challenge that prevented its wide use. Therefore, the present paper proposes a novel approach that relies on an unsupervised Physics Informed Neural Network (PINN) to design the observer's correction term by enforcing the partial differential inequality in the loss function. The performance of the proposed PINN-based nonlinear observer is assessed in numerical simulation as well as its robustness to measurement noise and neural network approximation error.

Figures

Figures reproduced from arXiv: 2411.09237 by the authors.

Figure 1
Figure 1. Block diagram of the proposed unsupervised learning-based [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. State estimation of the Van der Pol Oscillator under 15 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. State estimation of the Inverse Duffing Oscillator under 15 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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