Stability Analysis of a B-Spline Deep Neural Operator for Nonlinear Systems

This paper investigates stability properties of neural operators through a structured representation offered by Hybrid B-spline Deep Neural Operator (HBDNO). While existing stability-aware architectures typically enforce restrictive constraints that limit universality, HBDNO preserves full expressive power by representing outputs via B-spline control points. The authors show that these control points form a natural observable for post-training stability analysis. By applying Dynamic Mode Decomposition (DMD) and connecting the resulting discrete dynamics to the Koopman operator framework, they provide a principled approach to spectral characterization of learned operators. Numerical results demonstrate the ability to assess stability and reveal future directions for safety-critical applications.

The goal of this work is to leverage the structured Bspline representation of HBDNO to assess stability of the learned operator. Since a B-spline output is expressed as a weighted combination of basis functions and control points, the sequence of control points produced by the operator contains rich information about the latent dynamics learned from data. The authors exploit this structure by applying Dynamic Mode Decomposition (DMD) to the control-point trajectory generated by HBDNO. To ensure that this analysis captures meaningful dynamical properties of the underlying system, they establish a connection between the evolution of control points and [the] Koopman operator framework.

The learned operator G maps an initial condition x0 to an approximate trajectory on [0, T] and admits the B-spline representation:

(Gx)(t) = B̂d (t) c(x0),

where c(x0) ∈ R is the stacked control-point vector generated by the neural network Ψ, and B̂d (t) ≜ diag(Bd (t)), Bd (t) ≜ [B1,d (t) · · · Bl,d (t)]. This compactly encodes the entire trajectory using a common B-spline basis with component-specific control points.

The continuity at internal knots depends on multiplicity: if a knot appears m times, the spline is C d−m-continuous at that point. Furthermore, the B-spline basis satisfies Bj,d (t) ∈ C([0, T]), 0 ≤ Bj,d (t) ≤ 1, and forms a partition of unity. As a consequence, each scalar B-spline si (t) =l X ci,j Bj d (t) is a convex combination of the control points for every t ∈ [0, T], i.e., si(t) ∈ conv[ci,1,..., ci,l], ∀ t ∈ [0, T]. Hence, the entire trajectory (Gx)(t) lies in the componentwise convex hull of the corresponding control points.

The control points are generated by a neural network Ψ: Rn → Rnl, Ψ(x0) = c(x0), which outputs the stacked vector of B-spline coefficients described above. Substituting c(x0) into (5), the model produces a continuous trajectory on [0, T]. As established in Romagnoli et al. (2024), the HBDNO satisfies the universal approximation property: sup∥Px − Gx∥X < ε, for any compact K ⊂ S.

The authors aim to infer the asymptotic stability of the equilibrium of the continuous-time system (1) from the behavior of the control points generated by the HBDNO. By leveraging [the] convex hull property of the B-spline representation, asymptotic stability of the control-point sequence implies asymptotic stability of [the] continuous-time approximation (Gx)(t). Then, under the universal approximation property (4), [the] asymptotic stability of (Gx)(t) can be related to the asymptotic stability of the true solution x(t). The key step is to show that the control points evolve according to an underlying discrete-time dynamical system, so that their sequence provides a suitable representation (in the Koopman-theoretic sense) of a latent nonlinear map.

The authors establish Proposition 1: "Let Gx be the deep neural operator defined in (14). For a fixed B-spline degree d and number of control points l, assume that G satisfies the uniform approximation property (4) on a compact set of trajectories containing those generated by initial conditions x0 ∈ D. Then the sequence of control points (15) can be written in the quasi-Markovian form ĉ j+1 = F (ĉ j) + δ j,∥δ j∥ ≤ ε(l), where ε(l) → 0 as l → ∞."

The authors further establish Proposition 2: "Let Gx be the deep neural operator defined in (14) and let Ψ the mapping defined in (15) from the initial condition x0 of (1) to the corresponding sequence of control points. Assume that Ψ is injective and that the quasi-Markovian property (18) holds. Then the pair (ĉ, F) defines a faithful finite-dimensional (quasi)Markovian representation of the latent dynamics in the sense of Section 4.3."

Finally, they state Theorem 1: "Let the system (1) have an equilibrium point x = 0. Let G be a universal approximator of the Volterra operator P as in (4), and assume that the control points generated by G form a faithful, approximately Markovian finite-dimensional representation (ĉ, F) with ĉ = 0 an asymptotically stable equilibrium of the discrete-time map F. Then x = 0 is an asymptotically stable equilibrium of the original system (1)."

The numerical simulations provide a proof of concept for this framework, considering an asymptotically stable linear time invariant system of order n = 2 where the eigenvalues are given by [specific values]. The experiments suggest that quasi-Markovian effects vanish with enough control points, leading to effectively Markovian dynamics. For the example studied, the spectral radii of both Exact DMD and Hankel DMD operators were found to be well within the unit disk, indicating asymptotic decay of the control-point trajectories and suggesting that [the] observable induced by the B-spline representation is sufficiently robust for stability assessment. Moreover, the close agreement between the two spectral radii implies that quasi-Markovian effects do not significantly affect the reconstruction, supporting the idea that [the] chosen number of control points is adequate to treat the sequence as effectively Markovian. The Hankel DMD approximation error w.r.t the true solution was M SE = 7.5832e−6."

The conclusion is that B-splines provide a natural observable space that enables post-training spectral assessment, and Exact DMD reveals asymptotic and Lyapunov behavior, while Hankel DMD improves spectral robustness. Experiments suggest that quasi-Markovian effects vanish with enough control points, leading to effectively Markovian dynamics. The stability transfer from the latent discrete-time representation to the original continuous-time system is established through the analysis of control-point evolution via Koopman theory and DMD. [The] B-spline representation is shown to be sufficiently robust for stability assessment.

Keywords: Nonlinear Systems, Machine Learning, Dynamic Mode Decomposition, Koopman Operator Theory.

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Summary:

This paper investigates the stability properties of a learned operator using a Hybrid B-spline Deep Neural Operator (HBDNO). The HBDNO preserves full expressive power while offering a structured output space amenable to post-training analysis. The core idea is that the B-spline control points generated by the operator serve as rich observables for latent dynamics. These control points are analyzed by applying Dynamic Mode Decomposition (DMD) and connecting the resulting discrete dynamics to the Koopman operator framework, providing a principled approach to spectral characterization of learned operators.

The HBDNO maps an initial condition x0 to an approximate trajectory (Gx)(t) via:

(Gx)(t) = B̂d (t) c(x0),

where c(x0) is the stacked control-point vector generated by a neural network Ψ, and B̂d (t) is a diagonal matrix of B-spline basis functions. The trajectory lies in the componentwise convex hull of its control points.

The authors establish that the sequence of control points (15) can be written in a quasi-Markovian form:

ĉ j+1 = F (ĉ j) + δ j, where ε(l) → 0 as l → ∞. This implies that as the number of control points increases, the sequence becomes strictly Markovian. The authors then show that this structure allows for a faithful finite-dimensional (quasi)Markovian representation of the latent dynamics in the sense of Section 4.3, where (ĉ, F) defines a faithful representation.

Theorem 1 proves that if G is a universal approximator and the control points form a faithful, approximately Markovian finite-dimensional representation (ĉ, F) with ĉ = 0 an asymptotically stable equilibrium of the discrete-time map F, then x = 0 is an asymptotically stable equilibrium of the original system (1).

The stability analysis relies on showing that while the trajectory approximation x̂(t) → 0 as t → ∞ due to the asymptotic stability of [the] latent representation, this can be related to the true solution x(t) using the universal approximation property. The numerical results demonstrate that both Exact DMD and Hankel DMD operators yield spectral radii well within the unit disk (e.g., ρ(ADMD) = 0.8901 and ρ(AHDMD) = 0.8996), indicating asymptotic decay of the control-point trajectories and suggesting that quasi-Markovian effects do not significantly affect the reconstruction, supporting the idea that the chosen number of control points is adequate to treat the sequence as effectively Markovian. The Hankel DMD approximation error w.r.t the true solution was M SE = 7.5832e−6. The framework successfully transfers stability from the latent discrete-time representation to the original continuous-time system, demonstrating that B-splines provide a natural observable space for principled spectral assessment of learned operators in nonlinear systems.

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Here are the specific improvements for AI systems based on this research:

1. The HBDNO architecture allows neural operators to represent outputs via a structured B-spline basis, providing a natural observable space for post-training analysis that does not impose restrictive architectural constraints common in stability-aware models.

2. The control points generated by the HBDNO can be treated as observables of a latent discrete-time dynamical system. By applying Dynamic Mode Decomposition (DMD) or Hankel DMD to these control points, the underlying nonlinear dynamics can be approximated by a finite-dimensional linear Koopman operator representation (ADMD).

3. The stability of the original continuous-time system's equilibrium point is directly inferred from the asymptotic stability of the latent discrete-time control-point map. This inference is robust because, as the number of control points increases, quasi-Markovian effects vanish, leading to strictly Markovian dynamics in the limit.

4. The resulting ADMD operator (or its Hankel variant) provides a spectral characterization of the learned operator's dynamics in a finite-dimensional space. The eigenvalues of this approximation directly indicate the dominant temporal scales and stability properties (i.e., whether trajectories converge to an equilibrium or exhibit sustained oscillations).

5. The system can be used for safety-critical applications (like Model Predictive Control or PDE solutions) by providing theoretical guarantees on the learned operator's stability, moving AI from purely predictive tools toward certifiable components.

The improved AI system can:

1. Predict the long-term behavior (stability and convergence rate) of complex nonlinear systems solved by neural operators with higher confidence due to spectral analysis of control points.

2. Perform automated sensitivity analysis on learned operators by analyzing the dynamics of their B-spline control points, identifying unstable or stable regions without retraining.

3. Develop robust, certifiable Model Predictive Control (MPC) strategies for nonlinear systems by ensuring that the underlying learned operator is asymptotically stable in a quantifiable sense derived from the latent dynamics.

4. Provide a data-driven surrogate model of complex continuous dynamics that is characterized spectrally via DMD/Hankel DMD, allowing engineers to understand which dynamic modes dominate the system's behavior.