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arxiv: 2605.18905 · v1 · pith:AQJ7GUD2new · submitted 2026-05-17 · 💻 cs.LG · cs.AI· cs.NA· cs.NE· math.NA

Stability and Discretization Error of State Space Model Neural Operators

Abderrahim Bendahi , Adrien Fradin , Johan Peralez , Julie Digne , Madiha Nadri This is my paper

Pith reviewed 2026-05-20 14:54 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.NAcs.NEmath.NA
keywords neural operatorsdiscretization errorstate space modelsFourier neural operatorsPDE solvinginput-to-state stabilitynumerical stabilityoperator learning
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The pith

The paper proves analytical bounds linking PDE solution regularity to discretization error for neural operators, deriving a specific theorem for SS-NOs and FNOs along with stability analysis.

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

This paper establishes theoretical guarantees for discretization error and stability when neural operators approximate solutions to partial differential equations. It derives bounds that relate the regularity of the underlying PDE solutions to the accuracy achieved with discretized inputs. The analysis specializes these bounds into a new discretization error theorem for State Space Model-based Neural Operators and Fourier Neural Operators. An input-to-state stability study then quantifies how discretization affects the stability properties that hold in the continuous formulation. Experiments on 1D and 2D benchmarks confirm that the theoretical predictions match observed behavior across resolution changes.

Core claim

We prove analytical bounds that link solution regularity to input discretization, providing a formal quantification of neural operator accuracy under real-world numerical constraints. We derive these bounds to the specific cases of State Space Model-based Neural Operators (SS-NOs) and FNOs, thus providing a new discretization error theorem for these models. Additionally, through an input-to-state stability (ISS) analysis, we formally assess the impact of discretization on the stability of SS-NOs results obtained in the continuous domain.

What carries the argument

The discretization error theorem that bounds approximation error in terms of solution regularity and discretization level, together with the input-to-state stability analysis applied to SS-NOs.

If this is right

  • The discretization error theorem supplies explicit error controls that practitioners can use to select input resolutions for a given PDE regularity class.
  • SS-NOs retain their continuous-domain stability properties under discretization according to the ISS analysis.
  • The same regularity-to-error linkage applies to FNOs, adjusting their known algebraic convergence for finite-resolution inputs.
  • Empirical robustness across resolutions on 1D and 2D benchmarks follows directly from the theoretical bounds.

Where Pith is reading between the lines

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

  • Similar bounds could be derived for additional neural operator families once their continuous formulations are shown to satisfy comparable regularity assumptions.
  • The framework offers a way to trade off computational cost against guaranteed accuracy by choosing the coarsest discretization still compatible with a target solution class.
  • Relaxing the regularity hypothesis to weaker function spaces would broaden applicability to rougher real-world PDEs.

Load-bearing premise

The analysis assumes that the underlying PDE solutions possess sufficient regularity such as Sobolev or Hölder continuity.

What would settle it

Measure whether the derived error bounds are violated when SS-NOs or FNOs are applied to PDE solutions that lack the assumed regularity, for instance by using discontinuous or merely continuous but non-differentiable target functions and checking if observed errors exceed the predicted rates.

Figures

Figures reproduced from arXiv: 2605.18905 by Abderrahim Bendahi, Adrien Fradin, Johan Peralez, Julie Digne, Madiha Nadri.

Figure 1
Figure 1. Figure 1: Comparison of a SS￾NO and a FNO kernel (both randomly initialized) evalu￾ated at spatial resolution L = 2 15 on [0 , 1]. SS-NO exhibits continuous, full-band spectral structure without mode trun￾cation, while FNO enforces a hard cutoff frequency (at K = 16), resulting in compact spec￾tral support. As an intermediate step toward establishing discretization error bounds, we derive explicit estimates on the d… view at source ↗
Figure 2
Figure 2. Figure 2: Relative ℓ 2 error vs. resolution L for GRF inputs of varying smoothness s (mean ± std over N = 50 samples for (a),(b) and N = 5 samples for (c),(d)). Ground truth and layerwise comparison. As is standard in numerical analysis when the true continuous quantity is unavailable, we define a high-resolution discrete SS-NO evaluation as our reference (“ground truth”), and measure discrepancies when the same ope… view at source ↗
Figure 3
Figure 3. Figure 3: Stability of a single dis￾cretized SS-NO layer (mean ± std over N = 400 samples ((a)) and N = 8 samples ((b)). s Input regularity via Gaussian random fields. To assess the dependency on Sobolev regularity, we generate Gaussian ran￾dom fields (GRFs) inputs with fixed smoothness (Section I.2). We evaluate on several smoothness levels (as shown in the fig￾ure caption), spanning low-regularity fields to highly… view at source ↗
Figure 4
Figure 4. Figure 4: Representative solutions of the one-dimensional viscous Burgers’ equation with viscosity [PITH_FULL_IMAGE:figures/full_fig_p045_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: One-dimensional Gaussian random field realizations with varying smoothness parameters. [PITH_FULL_IMAGE:figures/full_fig_p045_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Two-dimensional Gaussian random field realizations with varying smoothness parameters. [PITH_FULL_IMAGE:figures/full_fig_p046_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Relative discretization error for trained 1D ReLU SS-NOs on the Burgers benchmark, for depths T ∈ {1, 2, 4, 8}. In each subplot, the same trained operator is evaluated on coarsened versions of the same test input and compared to its full-resolution prediction. The plotted quantity is the relative ℓ 2 error, shown as mean ± standard deviation over 30 test samples, together with a log–log fit. Two points are… view at source ↗
Figure 8
Figure 8. Figure 8: Output perturbation error for 1D SS-NOs on GRF inputs, for depths [PITH_FULL_IMAGE:figures/full_fig_p049_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Empirical stability versus depth T on a log–log scale. We plot the estimated Lipschitz factor of the full T-layer network (maximum output-to-input perturbation ratio over 200 random GRF pairs) together with the mean output perturbation error at ε = 0.8. The same model family and hyperparameters as in [PITH_FULL_IMAGE:figures/full_fig_p049_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Stress test with increasingly oscillatory inputs. For each input frequency [PITH_FULL_IMAGE:figures/full_fig_p050_10.png] view at source ↗
read the original abstract

Neural operators have emerged as a powerful, discretization-invariant framework for solving partial differential equations (PDEs). Although established approaches like the Deep Operator Network (DeepONet) have successfully achieved universal approximation for operators, and architectures such as Fourier Neural Operators (FNOs) have shown algebraic convergence rates, a precise theoretical connection between the continuous theory and its discrete numerical implementation remains a challenge. Specifically, the relationship between the continuous formulation and the discrete numerical stability has yet to be fully explored. In this paper, we address this gap by establishing theoretical guarantees for the discretization error and stability of neural operator approximation schemes. We prove analytical bounds that link solution regularity to input discretization, providing a formal quantification of neural operator accuracy under real-world numerical constraints. We derive these bounds to the specific cases of State Space Model-based Neural Operators (SS-NOs) and FNOs, thus providing a new discretization error theorem for these models. Additionally, through an input-to-state stability (ISS) analysis, we formally assess the impact of discretization on the stability of SS-NOs results obtained in the continuous domain. Our empirical experiments on 1D and 2D benchmarks validate our theoretical bounds and show the robustness of SS-NOs under varying resolutions.

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

3 major / 2 minor

Summary. The paper claims to derive a new discretization error theorem for State Space Model-based Neural Operators (SS-NOs) and FNOs that analytically bounds the gap between continuous neural operator outputs and their discrete implementations by linking solution regularity (Sobolev or Hölder) to input discretization. It further provides an input-to-state stability (ISS) analysis for SS-NOs and reports empirical validation on 1D and 2D PDE benchmarks showing robustness under varying resolutions.

Significance. If the central bounds are shown to apply without circularity or unverified assumptions, the work would supply a useful theoretical bridge between continuous neural operator theory and practical discrete implementations, quantifying accuracy under real-world grid constraints for SS-NOs and FNOs.

major comments (3)
  1. [Discretization error theorem (likely §3)] Discretization error theorem: the proof invokes Sobolev/Hölder regularity of PDE solutions to control the approximation error when passing from the continuous SSM or Fourier integral operator to its discretized version, yet the manuscript provides no verification that the specific 1D/2D benchmark solutions satisfy the precise regularity index required for the bounds to hold.
  2. [ISS analysis section] ISS stability analysis: the formal assessment of discretization impact on SS-NO stability inherits the same regularity dependence; without confirming that the benchmark problems meet the assumed regularity, the stability guarantees cannot be directly read as confirmation for the reported numerical results.
  3. [Experiments section] Empirical validation: the experiments on 1D and 2D benchmarks are presented as validating the theoretical bounds, but absent explicit regularity checks or error-bar reporting on the solution smoothness, the numerical results do not constitute a direct test of the theorem's hypotheses.
minor comments (2)
  1. [Theorem statement] Notation for the discretization parameter and regularity index should be introduced consistently in the theorem statement and reused in the ISS section to improve readability.
  2. [Abstract and introduction] The abstract states that bounds are 'derived to the specific cases' of SS-NOs and FNOs; a brief comparison table or remark clarifying which parts of the proof are model-specific versus general would help readers.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below, clarifying the role of regularity assumptions in our theorems and their relation to the benchmarks while committing to targeted revisions for improved transparency.

read point-by-point responses

  1. Referee: [Discretization error theorem (likely §3)] Discretization error theorem: the proof invokes Sobolev/Hölder regularity of PDE solutions to control the approximation error when passing from the continuous SSM or Fourier integral operator to its discretized version, yet the manuscript provides no verification that the specific 1D/2D benchmark solutions satisfy the precise regularity index required for the bounds to hold.

    Authors: The discretization error theorem is explicitly conditional on the solution belonging to a Sobolev space of sufficient index or satisfying a Hölder condition; this is stated in the theorem statement and used to bound the operator approximation error via standard embedding and interpolation arguments. The 1D and 2D benchmarks (viscous Burgers equation, advection-diffusion, and 2D incompressible Navier-Stokes) are classical test problems whose solutions are known to possess the required regularity for the smooth initial conditions and forcing terms used in the literature. We will add a concise remark in Section 3 and the experiments section referencing these established regularity properties and citing the relevant PDE analysis to make the link explicit. revision: partial

  2. Referee: [ISS analysis section] ISS stability analysis: the formal assessment of discretization impact on SS-NO stability inherits the same regularity dependence; without confirming that the benchmark problems meet the assumed regularity, the stability guarantees cannot be directly read as confirmation for the reported numerical results.

    Authors: The ISS analysis for SS-NOs is derived under the identical regularity hypothesis as the discretization theorem, ensuring that the continuous-to-discrete stability margin remains controlled. Because the benchmark problems satisfy this hypothesis by standard PDE theory, the reported numerical robustness under grid refinement is consistent with the guarantees. We will revise the ISS section to restate the assumption clearly and note its satisfaction for the chosen benchmarks, thereby allowing the numerical results to be read as supporting evidence under the theorem's hypotheses. revision: partial

  3. Referee: [Experiments section] Empirical validation: the experiments on 1D and 2D benchmarks are presented as validating the theoretical bounds, but absent explicit regularity checks or error-bar reporting on the solution smoothness, the numerical results do not constitute a direct test of the theorem's hypotheses.

    Authors: The experiments primarily illustrate practical robustness of SS-NOs across resolutions rather than serving as a direct numerical test of the regularity index. We agree that adding explicit references to solution smoothness and, where feasible, error-bar information on discretization-induced variations would tighten the connection to the theorem. In the revised manuscript we will include such discussion and supplementary plots that quantify how observed errors behave with respect to the regularity parameters appearing in the bounds. revision: yes

Circularity Check

0 steps flagged

No circularity: discretization error theorem relies on standard external regularity assumptions

full rationale

The paper derives analytical bounds linking PDE solution regularity (Sobolev/Hölder) to discretization error for SS-NOs and FNOs, then performs ISS stability analysis. These steps invoke classical functional analysis results on approximation under grid refinement rather than defining regularity via the error bounds themselves or fitting parameters to data and relabeling them as predictions. No load-bearing self-citations, ansatz smuggling, or renaming of known results appear in the derivation chain; the central theorem remains independent of the paper's own inputs or empirical fits. Benchmarks illustrate applicability but do not close the proof.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard mathematical assumptions about PDE solution regularity and neural operator approximation properties drawn from prior literature; no free parameters, invented entities, or ad-hoc axioms are described in the abstract.

axioms (2)
  • domain assumption PDE solutions possess sufficient regularity (e.g., Sobolev or Hölder) to apply the error bounds
    Invoked when linking solution regularity to input discretization in the discretization error theorem
  • standard math Standard neural operator approximation properties hold in the continuous setting
    Background assumption for extending continuous theory to discrete implementations

pith-pipeline@v0.9.0 · 5772 in / 1291 out tokens · 35079 ms · 2026-05-20T14:54:13.267758+00:00 · methodology

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Reference graph

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