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arxiv: 2604.06081 · v1 · submitted 2026-04-07 · 💻 cs.LG · cs.CE· math.DS

A machine learning framework for uncovering stochastic nonlinear dynamics from noisy data

Matteo Bosso , Giovanni Franzese , Kushal Swamy , Maarten Theulings , Alejandro M. Arag\'on , Farbod Alijani This is my paper

Pith reviewed 2026-05-10 19:52 UTC · model grok-4.3

classification 💻 cs.LG cs.CEmath.DS
keywords symbolic regressionstochastic differential equationsGaussian processesnonlinear dynamicsmachine learningnoise modelingoscillatorsbiological synchronization
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The pith

A hybrid method recovers the exact symbolic form of stochastic differential equations from noisy data while quantifying parameter uncertainty.

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

The paper sets out to show that noisy observations of nonlinear systems can be decomposed into a deterministic governing equation and a separate noise model without assuming the form of either in advance. It does so by combining deep symbolic regression, which searches for compact symbolic expressions, with Gaussian process maximum likelihood estimation that captures uncertainty and the stochastic component. A sympathetic reader would care because many real processes, from synchronized cells to vibrating structures, are governed by both fixed rules and intrinsic fluctuations, yet existing tools handle one or the other but not both together. The approach is shown to succeed on standard oscillator benchmarks and on real experimental data from coupled biological oscillators using only a few hundred points.

Core claim

The central claim is that a hybrid symbolic regression and probabilistic machine learning framework recovers the symbolic deterministic dynamics of stochastic differential equations while simultaneously inferring the structure and uncertainty of the noise from the same noisy time series, without requiring prior assumptions on the functional forms of either part.

What carries the argument

Hybrid framework pairing deep symbolic regression for deterministic equation discovery with Gaussian-process maximum likelihood estimation for separate noise and parameter uncertainty modeling.

Load-bearing premise

The deterministic dynamics and the noise can be modeled separately from the same observations without any assumed functional forms for either, and the symbolic regression step will still recover the correct equations despite the noise.

What would settle it

Apply the method to synthetic trajectories generated from a known stochastic van der Pol oscillator with controlled noise amplitude and check whether the recovered symbolic equation matches the true deterministic part while the noise model correctly isolates the added stochastic term.

Figures

Figures reproduced from arXiv: 2604.06081 by Alejandro M. Arag\'on, Farbod Alijani, Giovanni Franzese, Kushal Swamy, Maarten Theulings, Matteo Bosso.

Figure 1
Figure 1. Figure 1: Overview of our methodology for discovering SDEs with structured diffusion. Here the methodology [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Linear oscillator with additive noise. Predicted coefficients for the drift (left column) and diffusion (right column) as a function of noise level. Red lines indicate the model predictions, while black dashed lines denote the ground truth. Error bars (where shown) correspond to the 2.5%–97.5% quantiles across realizations, reflecting the framework’s uncertainty when applied to five independent datasets ob… view at source ↗
Figure 3
Figure 3. Figure 3: Multiplicative frequency noise. Estimation of the natural frequency distribution for the linear system with 5% frequency noise. Left: Ground truth frequency variations (blue) compared with estimated variations (black) using the Euler–Maruyama approximation with ∆t = 0.01. Right: Ground truth probability density function of the frequency parameter (blue), approximated distribution from the stochastic proces… view at source ↗
Figure 4
Figure 4. Figure 4: Linear and Duffing oscillators with noise in all parameters. Violin plots show distributions of recovered coefficients at 1% and 5% noise. Blue bars denote ground truth, while red bars indicate predictions. Each distribution is modeled as Gaussian, with the mean corresponding to the drift and the standard deviation to the diffusion of the coefficients. In both panels, C denotes the additive noise constant … view at source ↗
Figure 5
Figure 5. Figure 5: The experiment setup. Microscope image of two E. coli bacteria trapped in separate cavities connected by a narrow channel [40]. As derived in Supplementary Material F, the phase dynamics of coupled bacterial oscillators can be reduced to the noisy Adler equation: dφ dt = ∆ω − K sin(φ) + ξ(t) (10) where φ = ϕ2 − ϕ1 is the phase difference, ∆ω the frequency mismatch, K the coupling strength, and ξ(t) the add… view at source ↗
Figure 6
Figure 6. Figure 6: Comparison between experimental data and discovered models for two synchronization datasets. Left: Violin plots show the distributions of coupling strength (k) and angular velocity (ω) for the two oscillators. Blue bars represent empirical estimates obtained with the method of Japaridze et al. [40], while red bars indicate predictions from our algorithm. Each distribution is modeled as Gaussian, with the m… view at source ↗
Figure 7
Figure 7. Figure 7: The extracted noisy process (blue dots) is plotted against the fit of our tailored GP (shaded grey [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Schematic overview of Deep Symbolic Regression (DSR). A recurrent neural network (RNN) [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Results from applying the GP-DSR framework to four benchmark SDEs under broad DSR [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Bar plots showing the occurrence frequency of each diffusion token in the model predictions, for [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Frequency noise estimation comparison across various time steps [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: σL (Lorenz coupling parameter) noise estimation comparison across various time steps ∆t (illustrated in red) against the ground truth distribution (in blue). 12, the estimated distribution from the Euler–Maruyama approximation matches the original distribution (shown in blue) even for time steps up to ∆t = 10−2 s. This indicates that the underestimation bias observed in our experiments is due to the super… view at source ↗
read the original abstract

Modeling real-world systems requires accounting for noise - whether it arises from unpredictable fluctuations in financial markets, irregular rhythms in biological systems, or environmental variability in ecosystems. While the behavior of such systems can often be described by stochastic differential equations, a central challenge is understanding how noise influences the inference of system parameters and dynamics from data. Traditional symbolic regression methods can uncover governing equations but typically ignore uncertainty. Conversely, Gaussian processes provide principled uncertainty quantification but offer little insight into the underlying dynamics. In this work, we bridge this gap with a hybrid symbolic regression-probabilistic machine learning framework that recovers the symbolic form of the governing equations while simultaneously inferring uncertainty in the system parameters. The framework combines deep symbolic regression with Gaussian process-based maximum likelihood estimation to separately model the deterministic dynamics and the noise structure, without requiring prior assumptions about their functional forms. We verify the approach on numerical benchmarks, including harmonic, Duffing, and van der Pol oscillators, and validate it on an experimental system of coupled biological oscillators exhibiting synchronization, where the algorithm successfully identifies both the symbolic and stochastic components. The framework is data-efficient, requiring as few as 100-1000 data points, and robust to noise - demonstrating its broad potential in domains where uncertainty is intrinsic and both the structure and variability of dynamical systems must be understood.

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 / 2 minor

Summary. The paper proposes a hybrid framework that combines deep symbolic regression (DSR) with Gaussian process maximum likelihood estimation (GP-MLE) to recover the symbolic form of governing stochastic differential equations from noisy time-series data while simultaneously inferring uncertainty in the system parameters. The approach is presented as requiring no prior assumptions on the functional forms of either the deterministic drift or the noise structure. Validation is reported on numerical benchmarks (harmonic, Duffing, and van der Pol oscillators) and on experimental data from coupled biological oscillators, with claims of data efficiency (100–1000 points) and robustness to noise.

Significance. If the separation of deterministic dynamics from noise structure can be achieved reliably, the work would usefully bridge symbolic regression methods (which typically ignore uncertainty) and probabilistic approaches (which typically lack interpretable structure). The experimental validation on biological synchronization data and the reported data efficiency are concrete strengths that would support applicability in domains where both equation discovery and uncertainty quantification matter. The hybrid construction itself is a reasonable engineering response to the limitations of each component method taken alone.

major comments (2)
  1. [Abstract / framework description] Abstract and framework description: The repeated claim that the method proceeds 'without requiring prior assumptions about their functional forms' is load-bearing for the stated novelty. Deep symbolic regression searches over a discrete, fixed library of operators and terminals, while GP-MLE conditions on a chosen kernel family; both choices constitute functional priors. The manuscript must clarify the scope of this claim, discuss the generality of the chosen library and kernels, and address failure modes when the true drift or noise process lies outside those families. Without such discussion the hybrid method reduces to a library-based procedure augmented with uncertainty quantification rather than a prior-free discovery procedure.
  2. [Methods (framework description)] Methods section on component separation: The central modeling assumption—that deterministic dynamics (via DSR) and noise structure (via GP-MLE) can be separately recovered from the same noisy observations without one contaminating the other—requires explicit justification and algorithmic detail. It is unclear how the two sub-problems are coupled or decoupled during optimization and whether noise can be misattributed to the drift term (or vice versa). A precise description of the joint or alternating inference procedure, together with any convergence or identifiability arguments, is needed to support the separation claim.
minor comments (2)
  1. [Abstract] The abstract is lengthy and contains several overlapping sentences; condensing it would improve readability while preserving the core claims.
  2. [Numerical experiments] Quantitative recovery metrics (e.g., equation recovery rate, parameter error with uncertainty intervals, comparison against pure DSR or pure GP baselines) should be reported more prominently for the numerical benchmarks to allow direct assessment of performance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. These have highlighted important points regarding the framing of our claims and the clarity of the methodological description. We address each major comment below and commit to revisions that will strengthen the manuscript without altering its core contributions.

read point-by-point responses

  1. Referee: [Abstract / framework description] Abstract and framework description: The repeated claim that the method proceeds 'without requiring prior assumptions about their functional forms' is load-bearing for the stated novelty. Deep symbolic regression searches over a discrete, fixed library of operators and terminals, while GP-MLE conditions on a chosen kernel family; both choices constitute functional priors. The manuscript must clarify the scope of this claim, discuss the generality of the chosen library and kernels, and address failure modes when the true drift or noise process lies outside those families. Without such discussion the hybrid method reduces to a library-based procedure augmented with uncertainty quantification rather than a prior-free discovery procedure.

    Authors: We agree that the phrasing 'without requiring prior assumptions about their functional forms' is imprecise and risks overstating the method's generality. The approach does rely on a fixed operator library for DSR and a kernel family for GP-MLE. In the revised manuscript we will update the abstract, introduction, and framework description to state that the method avoids strong parametric assumptions on the specific forms of the drift and diffusion terms, while still depending on the choice of a general library and kernel. We will explicitly specify the libraries and kernels used in the experiments, discuss their breadth for common classes of stochastic nonlinear systems, and add a dedicated limitations subsection that examines failure modes (including misidentification) when the true functions lie outside these families. These changes will better contextualize the hybrid method's novelty. revision: yes

  2. Referee: [Methods (framework description)] Methods section on component separation: The central modeling assumption—that deterministic dynamics (via DSR) and noise structure (via GP-MLE) can be separately recovered from the same noisy observations without one contaminating the other—requires explicit justification and algorithmic detail. It is unclear how the two sub-problems are coupled or decoupled during optimization and whether noise can be misattributed to the drift term (or vice versa). A precise description of the joint or alternating inference procedure, together with any convergence or identifiability arguments, is needed to support the separation claim.

    Authors: We acknowledge that the current Methods section provides insufficient detail on the separation procedure and its robustness. The revised manuscript will expand this section with a precise algorithmic description of the inference workflow, clarifying the sequence or alternation between DSR and GP-MLE steps and how they are decoupled in practice. We will add discussion of potential noise-drift misattribution, supported by additional analysis of the benchmark results, and include a brief treatment of identifiability under the modeling assumptions. While full theoretical convergence guarantees may remain limited, we will strengthen the empirical justification and note this as an area for future analysis. revision: yes

Circularity Check

0 steps flagged

No circularity; hybrid framework derivation remains independent of its inputs

full rationale

The paper describes a hybrid approach that combines deep symbolic regression for deterministic dynamics with Gaussian process MLE for noise structure. No step in the abstract or described framework reduces by construction to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The central separation of deterministic and stochastic components is presented as an empirical procedure whose correctness is checked against external numerical benchmarks (harmonic, Duffing, van der Pol) and independent experimental oscillator data rather than being tautological with the method's own library or kernel choices. The 'no prior assumptions' phrasing is an overstatement given the fixed operator library and kernel family, but this is a claim-correctness issue, not a circular reduction of the derivation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on established capabilities of deep symbolic regression for equation discovery and Gaussian processes for uncertainty quantification. No new physical entities are postulated. The separability of deterministic and stochastic components is a key domain assumption.

axioms (1)
  • domain assumption Data is generated from a stochastic differential equation whose deterministic and noise components can be modeled separately
    The framework design explicitly separates modeling of deterministic dynamics and noise structure.

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