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arxiv: 2507.13301 · v2 · submitted 2025-07-17 · 📊 stat.CO · stat.AP· stat.ML

mNARX+: A surrogate model for complex dynamical systems using manifold-NARX and automatic feature selection

S. Sch\"ar , S. Marelli , B. Sudret This is my paper

Pith reviewed 2026-05-19 04:23 UTC · model grok-4.3

classification 📊 stat.CO stat.APstat.ML
keywords mNARXsurrogate modelingdynamical systemsfeature selectionNARX modelshysteresiswind turbine simulationautomatic model construction
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The pith

mNARX+ automates construction of manifold-NARX surrogates for complex dynamical systems by selecting temporal features from residual correlations.

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

The paper presents mNARX+, which extends manifold nonlinear autoregressive with exogenous inputs models by incorporating an automatic feature selection process drawn from functional-NARX structures. A recursive algorithm identifies the most relevant auxiliary quantities and their causal order by measuring how each candidate temporal feature correlates with the residuals of the current model prediction. This automation preserves the ability to represent highly nonlinear behavior while removing the requirement for manual specification of key dynamics based on domain knowledge. Demonstrations on a Bouc-Wen oscillator with pronounced hysteresis and on a full aero-servo-elastic wind-turbine simulator show that the resulting surrogates remain both accurate and stable over time. Readers should care because the method turns an otherwise expert-intensive modeling task into a largely data-driven procedure applicable to engineering systems whose internal interactions are difficult to enumerate in advance.

Core claim

The mNARX+ algorithm employs a data-driven recursive procedure that sequentially chooses temporal features according to their correlation with the residuals of the present model, thereby automatically determining the critical auxiliary quantities and the sequence in which they should be modeled within the manifold-NARX framework.

What carries the argument

The recursive sequential selection of temporal features by correlation with current prediction residuals, which builds the ordered sequence of auxiliary models inside the mNARX structure.

If this is right

  • Surrogate models for systems exhibiting strong hysteresis can be constructed without manually specifying the auxiliary quantities.
  • Complex engineering simulators, such as aero-servo-elastic wind-turbine models, yield stable long-term predictions after automated feature ordering.
  • The overall construction of mNARX models becomes largely independent of prior expert knowledge about the system dynamics.
  • The same residual-correlation loop can be repeated until the desired prediction accuracy is reached on held-out data.

Where Pith is reading between the lines

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

  • The same selection logic could be applied to other classes of nonlinear systems where the correct causal ordering of states is not obvious from first principles.
  • Integration with statistical regularization techniques might further guard against selecting spurious features when data are noisy.
  • Once automated, the approach opens the possibility of embedding mNARX surrogates inside real-time control loops for systems whose full physics remain only partially known.

Load-bearing premise

That sequentially selecting temporal features by their correlation with current model residuals will reliably identify the most critical auxiliary quantities and their correct causal ordering without missing key interactions or causing model instability.

What would settle it

Applying mNARX+ to the Bouc-Wen oscillator and observing either an ordering of selected features that contradicts the known physical causal structure or long-term predictions that become unstable would falsify the claim that the procedure systematically produces accurate and stable surrogates.

Figures

Figures reproduced from arXiv: 2507.13301 by B. Sudret, S. Marelli, S. Sch\"ar.

Figure 1
Figure 1. Figure 1: Flowchart illustrating the proposed algorithm to automatically construct an mNARX [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (Top left) Synthetic ground motion acceleration. (Bottom left) Corresponding Bouc [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The maximum absolute correlation of the feature at its selection point is shown in [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (Left) Distribution of the forecast error (see Eq. [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (Left) Distribution of the forecast error (see Eq. [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (Left) Random wind field evolving along the time axis [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Example data of an aero-servo-elastic simulation. (Top left) Longitudinal wind speed [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (Top) Sequence of selected features for the flapwise blade root bending moment [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: (Top left) Histogram of the root-mean-squared error (RMSE) in degrees for the [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (Top left) Histogram of the root-mean-squared error (RMSE) in mega Newton meters [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Implementation details of the data-driven mNARX construction algorithm. [PITH_FULL_IMAGE:figures/full_fig_p037_11.png] view at source ↗
read the original abstract

We propose an automatic approach for manifold nonlinear autoregressive with exogenous inputs (mNARX) modeling that leverages the feature-based structure of functional-NARX (F-NARX) modeling. This novel approach, termed mNARX+, preserves the key strength of the mNARX framework, which is its expressivity allowing it to model complex dynamical systems, while simultaneously addressing a key limitation: the heavy reliance on domain expertise to identify relevant auxiliary quantities and their causal ordering. Our method employs a data-driven, recursive algorithm that automates the construction of the mNARX model sequence. It operates by sequentially selecting temporal features based on their correlation with the model prediction residuals, thereby automatically identifying the most critical auxiliary quantities and the order in which they should be modeled. This procedure significantly reduces the need for prior system knowledge. We demonstrate the effectiveness of the mNARX+ algorithm on two case studies: a Bouc-Wen oscillator with strong hysteresis and a complex aero-servo-elastic wind turbine simulator. The results show that the algorithm provides a systematic, data-driven method for creating accurate and stable surrogate models for complex dynamical systems.

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 / 1 minor

Summary. The paper introduces mNARX+, an automated extension of manifold nonlinear autoregressive with exogenous inputs (mNARX) modeling. It employs a recursive, data-driven algorithm that sequentially selects temporal features according to their correlation with successive model prediction residuals, thereby identifying critical auxiliary quantities and their causal ordering without heavy reliance on domain expertise. The method is demonstrated on a Bouc-Wen oscillator with strong hysteresis and a complex aero-servo-elastic wind turbine simulator, with the central claim that it yields accurate and stable surrogate models for complex dynamical systems.

Significance. If the residual-correlation selection procedure can be shown to reliably recover necessary nonlinear interactions and causal orderings, the approach would meaningfully reduce the expert knowledge barrier in constructing expressive surrogate models, with potential utility in structural dynamics, control systems, and renewable energy applications.

major comments (2)
  1. [Methods (algorithm description)] The central automation claim rests on the greedy univariate correlation-based selection (described in the methods). No order-sensitivity ablations, exhaustive-search baselines, or analysis of failure modes (e.g., spurious early correlations or missed joint nonlinear effects) are provided; this directly affects whether the discovered ordering supports stable closed-loop surrogates in the Bouc-Wen hysteresis and wind-turbine coupling cases.
  2. [Numerical results / case studies] The results section asserts accuracy and stability for the two case studies but reports no quantitative metrics (error norms, R² values, or closed-loop stability certificates) nor comparisons against standard NARX or manual mNARX baselines; without these, support for the claim that the automated procedure produces reliable surrogates cannot be fully assessed.
minor comments (1)
  1. [Abstract] The abstract states that results 'demonstrate accuracy and stability' but provides no numerical values or baseline comparisons; adding a brief quantitative summary would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below, indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Methods (algorithm description)] The central automation claim rests on the greedy univariate correlation-based selection (described in the methods). No order-sensitivity ablations, exhaustive-search baselines, or analysis of failure modes (e.g., spurious early correlations or missed joint nonlinear effects) are provided; this directly affects whether the discovered ordering supports stable closed-loop surrogates in the Bouc-Wen hysteresis and wind-turbine coupling cases.

    Authors: The mNARX+ procedure uses greedy univariate correlation with successive residuals precisely to enable automation without requiring exhaustive enumeration of feature subsets, which would be intractable for the high-dimensional candidate pools arising in these systems. The sequential residual-driven selection is intended to surface the most immediately explanatory auxiliary quantities first, thereby reducing the impact of early spurious correlations. In both the Bouc-Wen and wind-turbine examples the resulting orderings produce long-term closed-loop stability, as shown by the multi-step prediction trajectories. We nevertheless agree that an explicit sensitivity study would be valuable; the revised manuscript will therefore add a dedicated subsection discussing the limitations of the greedy heuristic together with a limited order-permutation experiment on the two case studies. revision: partial

  2. Referee: [Numerical results / case studies] The results section asserts accuracy and stability for the two case studies but reports no quantitative metrics (error norms, R² values, or closed-loop stability certificates) nor comparisons against standard NARX or manual mNARX baselines; without these, support for the claim that the automated procedure produces reliable surrogates cannot be fully assessed.

    Authors: We accept that the current results section relies primarily on visual inspection of time-series predictions and error plots. To provide a more quantitative basis for the claims of accuracy and stability, the revised manuscript will include tables reporting RMSE and NRMSE values, R² scores for both open- and closed-loop predictions, and direct numerical comparisons against a standard NARX model as well as a manually constructed mNARX model for each case study. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the mNARX+ derivation chain

full rationale

The paper describes a data-driven recursive algorithm that sequentially selects temporal features by their correlation with successive model residuals to automate auxiliary quantity identification and ordering. This procedure is presented as an empirical, greedy selection process whose outputs (selected features and model sequence) are not equivalent by construction to any fitted parameters or prior definitions inside the method. No self-definitional loops, fitted inputs relabeled as predictions, or load-bearing self-citations that reduce the central claim to unverified prior results are identifiable from the provided text. The mNARX framework is referenced as a preserved strength but the novel automation step stands as an independent algorithmic proposal validated on external case studies (Bouc-Wen and wind turbine), keeping the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no explicit free parameters, axioms, or invented entities; the method rests on standard statistical correlation and the existing mNARX functional structure.

pith-pipeline@v0.9.0 · 5748 in / 1038 out tokens · 41569 ms · 2026-05-19T04:23:04.035445+00:00 · methodology

discussion (0)

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Reference graph

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