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arxiv: 1907.00821 · v1 · pith:5SYEON4Pnew · submitted 2019-07-01 · 💻 cs.LG · cs.SY· eess.SY· math.DS· stat.ML

Equation Discovery for Nonlinear System Identification

Nikola Simidjievski , Ljup\v{c}o Todorovski , Ju\v{s} Kocijan , Sa\v{s}o D\v{z}eroski This is my paper

Pith reviewed 2026-05-25 12:12 UTC · model grok-4.3

classification 💻 cs.LG cs.SYeess.SYmath.DSstat.ML
keywords equation discoverynonlinear system identificationprocess-based modelinggray-box modelingmodel structure identificationparameter estimationcontinuous-time systems
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The pith

Process-based modeling reconstructs both the structure and parameters of nonlinear systems from measured data.

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

The paper presents process-based modeling as an equation discovery method for nonlinear system identification that combines user-specified domain knowledge with automated structure search. One part selects and composes model elements from the provided building blocks; the second estimates numerical parameters. The approach is demonstrated on synthetic data and on real measurements from a standard benchmark, showing recovery of the underlying model. A reader would care because the method produces gray-box models that remain interpretable while fitting data, rather than treating the system as a black box. If the supplied domain knowledge is sufficient, the interlaced structure and parameter steps yield usable dynamical models directly from observations.

Core claim

Process-based modeling performs computer-aided identification of a model structure composed of elements selected from user-specified domain-specific modeling knowledge, interlaced with parameter estimation. When applied to continuous-time nonlinear systems, this reconstructs both the model structure and the parameters from measured data, as shown in a synthetic case study and a standard system-identification benchmark with real measurements.

What carries the argument

Process-based modeling, which identifies model structure by composing user-specified domain-specific building blocks and then estimates parameters.

If this is right

  • The method produces gray-box models that incorporate partial domain knowledge rather than purely data-driven black-box fits.
  • It recovers both structure and parameters on continuous-time nonlinear systems when the supplied elements are appropriate.
  • It supports model construction for tasks where expert knowledge about component processes is available but the exact combination is unknown.

Where Pith is reading between the lines

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

  • The same structure-identification step could be tested on discrete-time or hybrid systems to check whether the interlacing with parameter estimation generalizes.
  • If the building-block library is expanded automatically from first principles, the method might reduce reliance on manual domain input.
  • Recovered models could be directly inserted into simulation or control software without additional translation steps.

Load-bearing premise

The user-specified domain-specific modeling knowledge contains the correct building-block elements from which the true underlying model structure can be composed.

What would settle it

A dataset generated from a dynamical system whose governing equations require building blocks absent from the supplied domain knowledge, on which the method returns an incorrect structure, would falsify the reconstruction claim.

Figures

Figures reproduced from arXiv: 1907.00821 by Ju\v{s} Kocijan, Ljup\v{c}o Todorovski, Nikola Simidjievski, Sa\v{s}o D\v{z}eroski.

Figure 1
Figure 1. Figure 1: (a) The diagram of the three components of the water-tanks system; (b) The process-based model of the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Workflow of the ProBMoT algorithm for inducing process-based models from knowledge and data. Given the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: depicts the errors of the models learned with Multi-stage ProBMoT. The top two graphs in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Single-stage ProBMoT : (a) Performance compar￾ison of the summed validation errors of both signals h1 and h2 (RRMSEh1 +RRMSEh2) from the 9 obtained models across 6 different synthetic cases. (b) Performance compar￾ison test errors of the output signal h2 (RRMSEh2) from the 9 obtained models across 6 different synthetic cases. Single-stage ProBMoT is able to accurately identify the correct structure based o… view at source ↗
Figure 5
Figure 5. Figure 5: comparison of the errors of the three candi [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Relative error between the ground-truth parameters and the fitted parameters estimated on various levels of noise [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Simulation responses on the test data of the two [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Simulation responses on the test data of the con [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Estimated values of the two parameters Ph1 and Ph2 across the six synthetic data sets. ProBMoT is able to accurately estimate the values of both parameters in the range of 0.5 which yield an identical model structure to the correct one i.e. S-S. test this hypothesis, we performed an additional parameter estimation experiment for estimating the values of Ph1 and Ph2 using the synthetic data sets. For this t… view at source ↗
read the original abstract

Equation discovery methods enable modelers to combine domain-specific knowledge and system identification to construct models most suitable for a selected modeling task. The method described and evaluated in this paper can be used as a nonlinear system identification method for gray-box modeling. It consists of two interlaced parts of modeling that are computer-aided. The first performs computer-aided identification of a model structure composed of elements selected from user-specified domain-specific modeling knowledge, while the second part performs parameter estimation. In this paper, recent developments of the equation discovery method called process-based modeling, suited for nonlinear system identification, are elaborated and illustrated on two continuous-time case studies. The first case study illustrates the use of the process-based modeling on synthetic data while the second case-study evaluates on measured data for a standard system-identification benchmark. The experimental results clearly demonstrate the ability of process-based modeling to reconstruct both model structure and parameters from measured data.

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 presents recent developments in process-based modeling, an equation discovery method for nonlinear system identification in a gray-box setting. It consists of two interlaced computer-aided parts: structure identification from user-specified domain-specific modeling primitives, followed by parameter estimation. The method is illustrated on two continuous-time case studies—one using synthetic data and one using measured data from a standard system-identification benchmark—with the central claim that the experimental results demonstrate successful reconstruction of both model structure and parameters from measured data.

Significance. If the central claim holds under appropriate user-supplied primitives, the approach provides a structured way to incorporate partial domain knowledge into nonlinear system identification, bridging black-box data-driven methods and fully mechanistic modeling. This could be valuable in domains with known building-block processes but unknown compositions.

major comments (2)
  1. [Abstract / Case studies] Abstract and case-study sections: The assertion that 'the experimental results clearly demonstrate the ability of process-based modeling to reconstruct both model structure and parameters' is presented without any quantitative metrics (e.g., parameter error, prediction RMSE, structure recovery rates), error bars, or baseline comparisons, leaving the strength of the demonstration unevaluated.
  2. [Method] Method description: The central claim depends on the user supplying a set of primitives that contains the correct building blocks; while this gray-box assumption is stated, the experiments do not include a sensitivity test or ablation on incomplete or noisy domain knowledge to bound the method's robustness.
minor comments (1)
  1. [Method] Notation for the two interlaced modeling parts could be clarified with explicit pseudocode or a diagram showing the information flow between structure search and parameter estimation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We provide point-by-point responses to the major comments below.

read point-by-point responses

  1. Referee: [Abstract / Case studies] Abstract and case-study sections: The assertion that 'the experimental results clearly demonstrate the ability of process-based modeling to reconstruct both model structure and parameters' is presented without any quantitative metrics (e.g., parameter error, prediction RMSE, structure recovery rates), error bars, or baseline comparisons, leaving the strength of the demonstration unevaluated.

    Authors: We agree that the presentation of results in the abstract and case studies relies on qualitative assessment and visual comparisons rather than explicit quantitative metrics. The manuscript includes plots showing the fit of the discovered models to the data, but does not report numerical values such as RMSE or parameter errors. To address this, we will add quantitative metrics, including parameter estimation errors, prediction RMSE, and where possible structure recovery indicators, along with comparisons to baseline methods in the revised version. revision: yes

  2. Referee: [Method] Method description: The central claim depends on the user supplying a set of primitives that contains the correct building blocks; while this gray-box assumption is stated, the experiments do not include a sensitivity test or ablation on incomplete or noisy domain knowledge to bound the method's robustness.

    Authors: The method is explicitly designed as a gray-box approach where the user provides the domain-specific primitives, and the claim is made under the assumption that these include the necessary building blocks. The experiments illustrate successful reconstruction when this assumption holds. We acknowledge that no sensitivity analysis to incomplete or noisy primitives is provided. Adding such an analysis would require additional experiments defining various levels of incompleteness, which we can include in a revision to better bound the method's applicability. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a gray-box equation discovery method (process-based modeling) that takes as explicit external inputs both user-specified domain primitives and measured time-series data, then performs structure search followed by parameter estimation. No derivation step reduces a claimed result to a quantity defined by the method's own fitted outputs, nor does any load-bearing premise rest on a self-citation chain that itself lacks independent verification. The reported reconstructions are evaluated on held-out synthetic and benchmark data under the stated assumption that the supplied building blocks are adequate; this is an empirical demonstration rather than a self-referential identity. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the supplied domain knowledge is sufficient to compose the true model; no free parameters are introduced by the discovery method itself beyond those estimated in the second stage, and no new entities are postulated.

axioms (1)
  • domain assumption User-provided domain knowledge contains the elementary processes or components needed to form the correct model structure.
    Invoked in the description of the first modeling stage that selects elements from user-specified knowledge.

pith-pipeline@v0.9.0 · 5713 in / 1122 out tokens · 32304 ms · 2026-05-25T12:12:21.567400+00:00 · methodology

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Reference graph

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