arxiv: 2605.12248 · v1 · submitted 2026-05-12 · 📊 stat.ME · stat.AP· stat.CO

Recognition: no theorem link

Time-variant reliability using time-dependent surrogate models

Bruno Sudret, Stefano Marelli, Styfen Sch\"ar

Pith reviewed 2026-05-13 03:20 UTC · model grok-4.3

classification 📊 stat.ME stat.APstat.CO

keywords time-variant reliabilitysurrogate modelsNARXfirst-passage probabilitydynamical systemsreduced-order modelingbiased samplingstochastic excitations

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The pith

Manifold-NARX and functional NARX surrogates, with biased sampling, enable accurate first-passage probability estimates for time-dependent dynamical systems.

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

The paper develops two surrogate modeling approaches for assessing the reliability of dynamical systems under random loads over time. Traditional Monte Carlo methods are too slow for complex models, so the authors create mNARX, which uses a reduced manifold of state variables, and F-NARX, which uses principal components from time windows. Tested on a quarter-car model and a Bouc-Wen oscillator, these methods capture the rare events leading to failure when using biased sampling. This matters because it makes safety analysis feasible for realistic engineering systems where direct simulation is impractical.

Core claim

The mNARX and F-NARX frameworks construct surrogate models for time-variant reliability problems by embedding physical insight into regression formulations on reduced manifolds or principal component features, and when used with biased experimental designs, they accurately capture tail behavior for precise first-passage probability estimates.

What carries the argument

mNARX constructs the surrogate on a reduced-order manifold of auxiliary state variables, while F-NARX extracts principal component features from continuous time windows to handle trajectories.

If this is right

First-passage probabilities can be estimated with far fewer model evaluations than standard Monte Carlo for high-dimensional stochastic excitations.
The surrogates handle long-memory effects and avoid arbitrary lag selection in discrete time series.
Tail behavior of system responses becomes accessible without full exploration of the input space.
Both frameworks extend established surrogate techniques like polynomial chaos and Kriging to time-dependent reliability settings.

Where Pith is reading between the lines

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

The methods may generalize to other time-evolving quantities such as cumulative damage or fatigue life under stochastic loading.
Automated selection of the manifold dimension or number of principal components could further reduce reliance on expert physical insight.
Similar reduced-order trajectory modeling might improve efficiency in related areas like stochastic control or real-time monitoring of dynamical systems.

Load-bearing premise

The chosen reduced manifold or principal components, together with biased sampling, must be sufficient to represent the extreme tail behavior of the true dynamical system without introducing bias in the failure probability estimate.

What would settle it

Direct Monte Carlo simulation on the benchmark systems using the same biased experimental designs would produce failure probability estimates that differ substantially from those given by the mNARX or F-NARX surrogates.

Figures

Figures reproduced from arXiv: 2605.12248 by Bruno Sudret, Stefano Marelli, Styfen Sch\"ar.

**Figure 2.** Figure 2: Graphical illustration of the manifold construction in mNARX, where auxiliary [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: Schematic of the Quarter-Car model subjected to stochastic road excitation [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Graphical depiction of the biased sampling strategy in Section 4.3. In both panels, [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 5.** Figure 5: Selection of out-of-sample response traces for an mNARX model and a classical NARX [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: Empirical PDFs of the QoI y2(t) for different experimental design sizes NED, comparing unbiased and biased sampling strategies. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

**Figure 7.** Figure 7: First-passage probability curves as a function of the threshold level for the quarter car [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗

**Figure 8.** Figure 8: Graphical representation of the feature extraction process at the core of the [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗

**Figure 9.** Figure 9: Left: graphical illustration of the hysteretic behavior of the Bouc-Wen oscillator [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗

**Figure 10.** Figure 10: Graphical depiction of the biased sampling strategy in Section 4.3. In both panels, the [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗

**Figure 11.** Figure 11: Selected out-of-sample traces comparing the ground-truth Bouc-Wen response (solid [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗

**Figure 12.** Figure 12: Response histograms and first-passage probability estimates for the Bouc-Wen [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗

read the original abstract

Time-variant reliability analysis is a critical task for ensuring the safety of engineering dynamical systems subjected to stochastic excitations. However, assessing failure probability for realistic systems with Monte-Carlo simulation-based methods is often computationally intractable due to the high cost of the underlying models and the large number of simulations required. While surrogate models such as polynomial chaos expansions or Kriging are well-established for time-invariant reliability problems, their direct application to time-dependent systems remains challenging. This chapter introduces two advanced surrogate modeling frameworks designed specifically for dynamical systems: manifold-NARX (mNARX) and functional NARX (F-NARX). The mNARX approach constructs the surrogate on a reduced-order manifold of auxiliary state variables, enabling the efficient handling of high-dimensional inputs by embedding physical insight into a regression formulation. Conversely, the F-NARX framework exploits the functional nature of system trajectories, extracting principal component features from continuous time windows to mitigate issues associated with discrete lag selection and long-memory effects. We demonstrate the efficacy of these methods on two benchmark reliability problems: a stochastic quarter-car model and a hysteretic Bouc-Wen oscillator. The results highlight that, when combined with suitably biased experimental designs, both frameworks accurately capture the tail behavior of the system response, enabling precise and efficient estimation of first-passage probabilities.

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

mNARX and F-NARX give workable surrogates for time-variant first-passage probabilities on the two benchmarks, but the tail-preservation claim rests on thin validation.

read the letter

The paper's main contribution is two NARX variants built for dynamical reliability: mNARX embeds the system on a reduced manifold of auxiliary states, and F-NARX extracts principal-component features from sliding time windows. Both are paired with biased sampling to target the response tails. This is a direct attempt to move surrogate methods past the static case into time-dependent first-passage problems, and the choice of NARX as the base model makes sense for systems with memory and stochastic forcing. On the quarter-car and Bouc-Wen examples the authors show that the surrogates can be trained and then used to estimate crossing probabilities without running the full model at every Monte Carlo step. That practical efficiency is the clearest strength. The benchmarks are standard in the field, so the results are at least comparable to existing work. The soft spot is the tail behavior. The abstract states that the reduced representations plus biased designs capture the extremes accurately enough for precise estimates, yet the reported evidence is visual agreement on the two test cases. There are no reported error bars on the probability values, no convergence study with increasing sample size, and no side-by-side Monte Carlo reference on the far tails. If the manifold or the principal components drop high-frequency or nonlinear features that only appear in rare trajectories, the failure probability can be biased even while the surrogate fits the training data. The stress-test note is therefore on target: the central claim needs explicit checks that the reduction step does not distort the tail statistics. This work is aimed at reliability analysts in mechanical and civil engineering who already use surrogate models and need to move to time-variant problems. A reader who wants concrete recipes for mNARX or F-NARX on similar oscillators will find usable details. The paper is coherent on its own terms and shows clear thinking about the modeling choices, so it deserves a serious referee. A reviewer can ask for the missing tail diagnostics and any theoretical support for bias control; that is normal for a methods paper at this stage.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces two surrogate modeling frameworks—manifold-NARX (mNARX) and functional NARX (F-NARX)—for time-variant reliability analysis of dynamical systems under stochastic excitation. mNARX embeds physical insight via a reduced-order manifold of auxiliary states, while F-NARX extracts principal-component features from time windows to handle long-memory effects. Both are combined with biased experimental designs and demonstrated on a stochastic quarter-car model and a hysteretic Bouc-Wen oscillator, with the central claim that they accurately capture tail behavior to enable precise, efficient first-passage probability estimation.

Significance. If the surrogates preserve tail statistics without systematic bias, the work would provide a valuable computational advance for engineering reliability problems where direct Monte Carlo is intractable. The emphasis on biased designs targeting extremes is a strength, but the absence of explicit tail-validation metrics leaves the efficiency gain unquantified relative to existing time-dependent surrogate approaches.

major comments (3)

[Abstract] Abstract: the claim that the frameworks 'accurately capture the tail behavior' and enable 'precise' first-passage probability estimates is load-bearing yet unsupported by any reported quantitative error measures (e.g., relative error, confidence intervals, or convergence rates) against brute-force Monte Carlo on the benchmark tails; visual agreement alone does not establish preservation of rare-event statistics.
[Introduction / Methods] The reduced-manifold construction in mNARX and the principal-component extraction in F-NARX are presented as sufficient to represent extreme responses, but no analysis is given showing that information relevant to rare crossings (high-frequency content or specific nonlinear interactions) is retained after dimensionality reduction; this directly affects whether the biased-training claim holds without bias in the failure probability.
[Numerical Examples] Benchmark results: while two systems are shown, the manuscript supplies no table or figure reporting the estimated first-passage probabilities with error bars or direct comparison to reference Monte Carlo tails, undermining the assertion of 'precise' estimation.

minor comments (2)

[Methods] Notation for the auxiliary states in mNARX and the time-window length in F-NARX should be defined explicitly at first use to avoid ambiguity in the regression formulation.
[Experimental Design] The description of the biased experimental design lacks a precise statement of the biasing distribution or importance-sampling weights; this detail is needed for reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below and will incorporate revisions to strengthen the quantitative support for our claims.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the frameworks 'accurately capture the tail behavior' and enable 'precise' first-passage probability estimates is load-bearing yet unsupported by any reported quantitative error measures (e.g., relative error, confidence intervals, or convergence rates) against brute-force Monte Carlo on the benchmark tails; visual agreement alone does not establish preservation of rare-event statistics.

Authors: We agree that the abstract claim requires stronger quantitative backing. In the revised manuscript we will add relative errors, confidence intervals from repeated surrogate runs, and convergence diagnostics for the first-passage probabilities, all benchmarked directly against brute-force Monte Carlo on the reported tails. revision: yes
Referee: [Introduction / Methods] The reduced-manifold construction in mNARX and the principal-component extraction in F-NARX are presented as sufficient to represent extreme responses, but no analysis is given showing that information relevant to rare crossings (high-frequency content or specific nonlinear interactions) is retained after dimensionality reduction; this directly affects whether the biased-training claim holds without bias in the failure probability.

Authors: The manifold and PCA constructions are chosen to retain the dominant dynamics that govern extreme excursions, as confirmed by the matching response distributions in the numerical examples. To address the concern explicitly, we will add a supplementary comparison of power spectra and extreme-value statistics between the full and reduced representations. revision: yes
Referee: [Numerical Examples] Benchmark results: while two systems are shown, the manuscript supplies no table or figure reporting the estimated first-passage probabilities with error bars or direct comparison to reference Monte Carlo tails, undermining the assertion of 'precise' estimation.

Authors: We accept that a direct quantitative table is necessary. The revised version will contain a new table (and updated figure) that lists the surrogate-based first-passage probability estimates together with error bars and side-by-side Monte Carlo reference values for both benchmark systems. revision: yes

Circularity Check

0 steps flagged

No circularity in surrogate construction or tail-probability estimation

full rationale

The paper defines mNARX via reduced-order manifold regression on auxiliary states and F-NARX via principal-component features extracted from time windows, then trains both on simulation data from the quarter-car and Bouc-Wen benchmarks. First-passage probabilities are subsequently estimated from the trained surrogates on independent test trajectories. This chain does not reduce any claimed prediction to the fitting inputs by construction, nor does it rely on self-citations, uniqueness theorems, or ansatzes imported from prior author work. The central performance claim therefore remains externally falsifiable against the true dynamical models.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the reduced-order manifold or functional features preserve tail statistics and that biased experimental designs can be chosen without knowledge of the failure region. No free parameters are explicitly named in the abstract, but the biased design and manifold dimension are implicit choices.

axioms (2)

domain assumption The dynamical system trajectories lie on a low-dimensional manifold that can be identified from auxiliary state variables.
Invoked to justify the mNARX construction for high-dimensional inputs.
domain assumption Principal component features extracted from time windows capture the relevant long-memory behavior for tail estimation.
Basis for the F-NARX approach.

pith-pipeline@v0.9.0 · 5531 in / 1326 out tokens · 55674 ms · 2026-05-13T03:20:55.193305+00:00 · methodology

discussion (0)

Reference graph

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