arxiv: 2605.04337 · v1 · submitted 2026-05-05 · 🧮 math.DS · eess.SP· stat.ML

Recognition: unknown

Symbolic Regression via Neural Networks

Jeff Moehlis, Nibodh Boddupalli, Timothy Matchen

Pith reviewed 2026-05-08 16:47 UTC · model grok-4.3

classification 🧮 math.DS eess.SPstat.ML

keywords symbolic regressionneural networksdynamical systemsgoverning equationsdeep learningmodel discovery

0 comments

The pith

A deep neural network generates symbolic expressions for the governing equations of dynamical systems from data.

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

The paper develops a neural network that outputs symbolic mathematical expressions describing a dynamical system's evolution, rather than numerical predictions alone. Traditional deep learning provides accurate forecasts but limited insight into the rules, while fixed-dictionary symbolic methods typically demand prior system knowledge or produce overly complex results. By training the network end-to-end on trajectory data, the approach seeks to recover concise, interpretable equations without those limitations. A sympathetic reader would care because recovering explicit governing laws from observations alone would improve analysis, prediction, and control across fields that rely on dynamical models.

Core claim

The authors describe a deep neural network architecture that, when trained on data from a dynamical system, produces a symbolic expression for its governing equations. They demonstrate the model's performance by recovering accurate symbolic forms across several classical dynamical systems.

What carries the argument

A deep neural network architecture trained to map observed trajectories directly to symbolic differential equations.

Load-bearing premise

A neural network can be trained to output valid, parsimonious symbolic expressions for governing equations without prior knowledge of the system or overfitting to the data.

What would settle it

Train the network on data generated from the Lorenz system and verify whether it outputs the exact symbolic equations x' = sigma(y - x), y' = x(r - z) - y, z' = x y - b z.

Figures

Figures reproduced from arXiv: 2605.04337 by Jeff Moehlis, Nibodh Boddupalli, Timothy Matchen.

**Figure 1.** Figure 1: FIG. 1. Network architecture of view at source ↗

**Figure 2.** Figure 2: FIG. 2. Architecture of the view at source ↗

**Figure 3.** Figure 3: FIG. 3. (a) State-space comparison of the trajectory generated view at source ↗

**Figure 4.** Figure 4: FIG. 4. (a) State-space comparison of the trajectory generated view at source ↗

**Figure 5.** Figure 5: FIG. 5. (a) State-space visualization of the trajectory generated by view at source ↗

**Figure 6.** Figure 6: FIG. 6. (a) State-space visualization of the trajectory generated by view at source ↗

**Figure 8.** Figure 8: FIG. 8. (a) State-space comparison of trajectory generated (dashed view at source ↗

**Figure 9.** Figure 9: FIG. 9. State-space trajectory generated by Eqs. (31a-31b) (dashed view at source ↗

**Figure 10.** Figure 10: FIG. 10. (a) State-space comparison of the trajectory generated view at source ↗

read the original abstract

Identifying governing equations for a dynamical system is a topic of critical interest across an array of disciplines, from mathematics to engineering to biology. Machine learning -- specifically deep learning -- techniques have shown their capabilities in approximating dynamics from data, but a shortcoming of traditional deep learning is that there is little insight into the underlying mapping beyond its numerical output for a given input. This limits their utility in analysis beyond simple prediction. Simultaneously, a number of strategies exist which identify models based on a fixed dictionary of basis functions, but most either require some intuition or insight about the system, or are susceptible to overfitting or a lack of parsimony. Here we present a novel approach that combines the flexibility and accuracy of deep learning approaches with the utility of symbolic solutions: a deep neural network that generates a symbolic expression for the governing equations. We first describe the architecture for our model, then show the accuracy of our algorithm across a range of classical 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.

Desk Editor's Note private letter to a colleague

The paper uses a neural network to output symbolic governing equations for dynamical systems, which is a reasonable idea but the abstract gives almost no implementation or comparison details to assess if it works.

read the letter

The main point is a deep neural network that produces symbolic expressions for the governing equations of dynamical systems instead of just numerical predictions. This tries to fix the black-box problem in standard deep learning while avoiding the need for a hand-crafted dictionary of basis functions that other symbolic regression approaches often require. They outline an architecture and report accuracy on classical systems, which is a straightforward way to test the concept on known cases like oscillators or chaotic attractors. That combination of flexibility and interpretability could matter for applications in biology or engineering where the form of the equations helps with analysis or control. The abstract does a clear job laying out the motivation and the contrast with existing methods. The soft spots are the missing pieces on how the network actually represents and generates valid symbolic output, what loss or regularization keeps expressions parsimonious, how data is generated or split, and any direct benchmarks against methods like SINDy or genetic programming. Without those, the accuracy claims are hard to weigh. The work is aimed at people doing data-driven modeling in dynamical systems who want interpretable results rather than pure simulation. A reader focused on scientific machine learning would get value from the full methods and experiments if they hold up. I would send it for peer review because the core direction addresses a real need and the basic setup is coherent enough for referees to evaluate and improve the evidence.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a novel deep neural network architecture that generates symbolic expressions for the governing equations of dynamical systems from data. It describes the model architecture and reports empirical accuracy on a range of classical dynamical systems, aiming to combine the flexibility of deep learning with the interpretability of symbolic models while avoiding the limitations of dictionary-based approaches.

Significance. If the method reliably produces accurate, parsimonious symbolic expressions without requiring prior system insight or suffering from overfitting, it would represent a meaningful advance in data-driven modeling of dynamical systems by bridging numerical approximation and symbolic insight. However, the abstract provides no quantitative metrics, error measures, data generation details, or baseline comparisons, so the significance cannot be assessed from the available information.

major comments (1)

[Abstract] Abstract: the assertion that the algorithm demonstrates 'accuracy ... across a range of classical dynamical systems' supplies no metrics, error handling, data generation protocol, or comparisons to dictionary-based methods, rendering it impossible to evaluate whether the central claim of combining flexibility and utility is supported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and constructive feedback on our manuscript. We address the single major comment below and will revise the manuscript to improve clarity.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that the algorithm demonstrates 'accuracy ... across a range of classical dynamical systems' supplies no metrics, error handling, data generation protocol, or comparisons to dictionary-based methods, rendering it impossible to evaluate whether the central claim of combining flexibility and utility is supported.

Authors: We agree that the abstract is insufficiently specific and does not allow readers to evaluate the strength of the results. The body of the manuscript contains quantitative metrics, error measures, and data generation details for the classical systems tested, along with discussion of how the approach avoids the need for a predefined dictionary. We will revise the abstract to summarize these elements (including key error metrics and data protocols) and to briefly note the advantages relative to dictionary-based methods, thereby making the central claim more readily assessable. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract and context describe a neural network trained on data from classical dynamical systems to output symbolic governing equations, with subsequent empirical accuracy checks. No load-bearing derivations, self-citations, fitted parameters renamed as predictions, or ansatzes imported via prior work are present in the provided material. The central claim rests on external data and architecture design rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No specific free parameters, axioms, or invented entities are identifiable from the abstract alone; the method is described at a high level without detailing training losses, network hyperparameters, or symbolic grammar constraints.

pith-pipeline@v0.9.0 · 5460 in / 1131 out tokens · 46308 ms · 2026-05-08T16:47:11.752450+00:00 · methodology

discussion (0)

Reference graph

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