arxiv: 2605.10685 · v3 · submitted 2026-05-11 · 💻 cs.AI

Recognition: unknown

GESR: A Genetic Programming-Based Symbolic Regression Method with Gene Editing

Jingyi Liu, Lina Yu, Liping Zhang, Mingzhu Wan, Min Wu, Weijun Li, Xin Ning, Yanjie Li, Yusong Deng

Authors on Pith no claims yet

Pith reviewed 2026-05-14 21:15 UTC · model grok-4.3

classification 💻 cs.AI

keywords symbolic regressiongenetic programmingBERT modelgene editingmutation guidancecrossover predictionevolutionary algorithmsformula discovery

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The pith

Two BERT models replace random mutations and crossovers in genetic programming to raise efficiency on symbolic regression tasks.

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

The paper presents GESR, which augments standard genetic programming for symbolic regression by training BERT models to direct gene-level changes rather than leaving mutation and crossover to chance. One BERT uses masked language modeling to suggest replacements for expression symbols during mutation; the second predicts suitable crossover points between parent expressions. The central idea is that these predictions reduce the fraction of detrimental edits that waste fitness evaluations in traditional GP. Experiments indicate the resulting search requires less computation while still recovering accurate symbolic expressions across several benchmark problems. A reader cares because symbolic regression is a core route from raw data to interpretable mathematical laws in science and engineering.

Core claim

GESR trains two BERT models to serve as gene editors inside a genetic programming loop. The first model guides mutation of individual expression symbols through its masked language modeling capability. The second model guides crossover by predicting the most useful cut point between two parent individuals. By substituting these directed operations for the random ones used in conventional GP, the method produces fewer low-fitness offspring and reaches target expressions in fewer generations.

What carries the argument

Two BERT models that predict beneficial mutations of symbols and crossover points between expression trees.

If this is right

Guided rather than random genetic operators can measurably lower the computational cost of evolving symbolic expressions.
The two-BERT editing scheme delivers competitive accuracy while cutting the number of generations needed on standard regression benchmarks.
Replacing stochastic variation steps with learned predictors inside evolutionary loops reduces the production of detrimental individuals.
The overall workflow remains a population-based search but executes each generation more productively.

Where Pith is reading between the lines

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

The same predictor-guided editing pattern could be tested on other tree-structured evolutionary problems such as program synthesis or circuit design.
Larger or domain-specific pretraining corpora of mathematical expressions might further improve the accuracy of the mutation and crossover predictors.
Hybrid systems that interleave deep-learning guidance with evolutionary search may scale to higher-dimensional or noisier data sets where pure random GP currently struggles.

Load-bearing premise

The BERT predictions actually steer the population toward higher-fitness expressions rather than merely reproducing patterns seen during their training.

What would settle it

A head-to-head run on the same symbolic regression benchmarks in which GESR requires at least as many fitness evaluations as ordinary GP to reach the same accuracy level.

read the original abstract

Mathematical formulas serve as a language through which humans communicate with nature. Discovering mathematical laws from scientific data to describe natural phenomena has been a long-standing pursuit of humanity for centuries. In the field of artificial intelligence, this challenge is known as the symbolic regression problem. Among existing symbolic regression approaches, Genetic Programming (GP) based on evolutionary algorithms remains one of the most classical and widely adopted methods. GP simulates the evolutionary process across generations through genetic mutation and crossover. However, mutations and crossovers in GP are entirely random. While this randomness effectively mimics natural evolution, it inevitably produces both beneficial and detrimental variations. If there existed a metaphorical `God` capable of foreseeing which genetic mutations or crossovers would yield superior outcomes and performing targeted gene editing accordingly, the efficiency of evolution could be substantially improved. Motivated by this idea, we propose in this paper a symbolic regression approach based on gene editing, termed GESR. In GESR, we trained two "hands of God" (two BERT models). Among them, the first leverages the BERT's masked language modeling capability to guide the mutation of genes (expression symbols). The other BERT model guides the crossover of individual genes by predicting the crossover point. Experimental results demonstrate that GESR significantly improves computational efficiency compared with traditional GP algorithms and achieves strong overall performance across multiple symbolic regression tasks.

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

GESR slots two BERT models into GP for symbolic regression to guide mutations and crossovers, but the models learn only symbol statistics with no fitness signal, so claimed efficiency gains rest on an untested transfer.

read the letter

The main thing here is the specific setup: one BERT trained with masked language modeling to suggest mutations on expression symbols, and a second to predict crossover points, all inside a standard genetic programming loop for symbolic regression. That pairing of two specialized BERTs for the genetic operators is not a routine extension of prior GP or language-model work. It does a clean job spelling out the motivation—random edits waste effort, so learned guidance from expression patterns might cut down on bad variations and speed things up. If the predictions actually steer toward lower error, the approach could be a practical tweak for automated formula discovery. The soft spot is exactly what the stress test flags: both BERTs optimize pure predictive likelihood on symbols, with zero regression loss or fitness feedback during training. Any speedup therefore hinges on whether higher-likelihood edits also produce better fits on the target data, which is not guaranteed and could collapse to distribution matching if the training expressions overlap with the benchmarks. The abstract asserts significant efficiency gains and strong performance, yet the description gives no numbers, baselines, training corpus details, or statistical tests, leaving the central claim unsupported on the evidence shown. This is aimed at researchers blending evolutionary search with modern predictors for scientific discovery tasks. A reader hunting for hybrid architecture ideas would pick up the concrete design choices. I would bring it to reading group to walk through whether the language-model objective actually improves search dynamics. I would not cite it yet. It deserves peer review so the experiments and data choices can be checked directly.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces GESR, a genetic programming (GP) approach to symbolic regression in which two BERT models replace random mutation and crossover operators. One BERT uses masked language modeling to guide mutations of expression symbols; the second predicts crossover points. The central claim is that this learned 'gene editing' yields substantially higher computational efficiency than standard GP while maintaining strong performance across multiple symbolic regression benchmarks.

Significance. If the empirical claims are substantiated, the work would demonstrate a practical way to inject statistical priors from expression corpora into evolutionary search, addressing a long-standing inefficiency in GP-based symbolic regression. The hybrid use of language-model guidance is novel in this domain and, if shown to transfer beyond training distributions, could influence subsequent research on learned operators in evolutionary algorithms.

major comments (3)

[Abstract] Abstract: The claims that GESR 'significantly improves computational efficiency' and 'achieves strong overall performance' are presented without any quantitative metrics (runtime, RMSE, R², success rate), baseline algorithms, number of benchmarks, or statistical tests. The experimental section must supply these details with tables or figures to allow assessment of the central claim.
[Method] Method (BERT training): Both BERT models are trained exclusively with masked-language-modeling objectives on expression symbols and contain no regression loss or fitness signal from the target dataset. This design choice means any efficiency gain rests on an untested assumption that higher-likelihood edits also produce lower regression error; the manuscript should either add a fitness-aware fine-tuning stage or provide an ablation showing that MLM guidance outperforms random valid edits on held-out regression tasks.
[Experiments] Experimental setup: No information is given on the corpus used to train the BERT models (source, size, diversity, overlap with evaluation benchmarks), GP hyperparameters, baseline implementations (standard GP, other SR methods), or evaluation protocol (number of runs, statistical significance). These omissions prevent verification that reported gains are not artifacts of distribution matching.

minor comments (2)

[Abstract/Introduction] The informal metaphor of 'hands of God' and 'God' in the abstract and introduction should be replaced with precise technical language for a formal journal.
[Method] Notation for the two BERT models and their inputs/outputs should be defined consistently in a single table or figure to improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed review. The comments highlight important areas for clarification and strengthening. We address each major comment below and indicate the revisions planned for the next version of the manuscript.

read point-by-point responses

Referee: [Abstract] The claims that GESR 'significantly improves computational efficiency' and 'achieves strong overall performance' are presented without any quantitative metrics (runtime, RMSE, R², success rate), baseline algorithms, number of benchmarks, or statistical tests. The experimental section must supply these details with tables or figures to allow assessment of the central claim.

Authors: We agree that the abstract would be strengthened by including concrete quantitative support. The experiments section already presents tables and figures with runtime comparisons, RMSE and R² values, success rates, baseline comparisons (including standard GP), and statistical tests across the benchmark suite. In the revised manuscript we will update the abstract to summarize key metrics, such as average runtime reduction and the fraction of benchmarks on which GESR matches or exceeds baseline performance. revision: yes
Referee: [Method] Both BERT models are trained exclusively with masked-language-modeling objectives on expression symbols and contain no regression loss or fitness signal from the target dataset. This design choice means any efficiency gain rests on an untested assumption that higher-likelihood edits also produce lower regression error; the manuscript should either add a fitness-aware fine-tuning stage or provide an ablation showing that MLM guidance outperforms random valid edits on held-out regression tasks.

Authors: The MLM-only training is intentional: it learns general syntactic priors over mathematical expressions from a broad corpus, supporting transfer across different regression problems. A full fitness-aware fine-tuning stage would risk overfitting to particular datasets and reduce generality. To directly test the value of the learned guidance, we will add an ablation study in the revised manuscript that compares MLM-guided edits against random but syntactically valid edits on held-out benchmarks, reporting the resulting differences in efficiency and regression accuracy. revision: partial
Referee: [Experiments] No information is given on the corpus used to train the BERT models (source, size, diversity, overlap with evaluation benchmarks), GP hyperparameters, baseline implementations (standard GP, other SR methods), or evaluation protocol (number of runs, statistical significance). These omissions prevent verification that reported gains are not artifacts of distribution matching.

Authors: We apologize for these descriptive omissions in the initial submission. The revised manuscript will contain a dedicated experimental-setup subsection that specifies the source, size, and diversity of the expression corpus used for BERT training, the steps taken to avoid overlap with evaluation benchmarks, all GP hyperparameters, the concrete implementations of the baseline algorithms, and the full evaluation protocol (number of independent runs per benchmark together with the statistical significance tests employed). revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces GESR as a GP variant that augments mutation and crossover with two BERT models trained via masked language modeling on expression symbols. The central claims rest on comparative experimental results across symbolic regression benchmarks rather than any closed-form derivation or self-referential definition. No equations are presented that reduce a prediction to a fitted input by construction, and no uniqueness theorem or ansatz is imported via self-citation to force the architecture. The method is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on standard assumptions of evolutionary computation and transformer language models; no new physical entities or ad-hoc constants are introduced beyond typical hyper-parameters of BERT fine-tuning.

axioms (2)

domain assumption BERT masked-language-model pre-training transfers useful priors for predicting beneficial edits to symbolic expressions.
Invoked when the first BERT is used to guide mutation.
domain assumption Predicting a single crossover point is sufficient to produce fitter offspring on average.
Invoked when the second BERT selects crossover locations.

pith-pipeline@v0.9.0 · 5558 in / 1264 out tokens · 43983 ms · 2026-05-14T21:15:56.228754+00:00 · methodology

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discussion (0)

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Hyper-Lorenz System (Four-Dimensional Hyperchaos) [44, 45] The Hyper-Lorenz system is a four-dimensional extension of the classical Lorenz system, governed by:    ˙x=a(y−x) +w, ˙y=cx−y−xz, ˙z=xy−bz, ˙w=dw−xz, a= 10, b= 2.667, c= 28, d= 1.1.(8) By retaining the core nonlinear convection termsxzandxyfrom the Lorenz system while introducing an addit...
[63]

Hyper-Jha System (Four-Dimensional Hyperchaos) [46] The Hyper-Jha system is another Lorenz-type four-dimensional hyperchaotic system described by:    ˙x=a(y−x) +w, ˙y=cx−y−xz, ˙z=xy−bz, ˙w=−dy, (9) Unlike the Hyper-Lorenz system, Hyper-Jha introduces a linear decay term−dyin the auxiliary state equation, altering energy feedback pathways among st...
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This results in phase- space projections that differ markedly from those of Hyper-Lorenz and Hyper-Jha, while preserving the Lorenz-type nonlinear backbone

Hyper-Pang System (Four-Dimensional Hyperchaos) [47] The Hyper-Pang system is another Lorenz-type hyperchaotic model governed by:    ˙x=a(y−x), ˙y=cx−y−xz+w, ˙z=xy−bz, ˙w=−d(x+y), (10) A distinguishing feature of this system is that the auxiliary variablewcou- ples simultaneously to multiple original state variables through linear combinations, r...
[65]

The nonlinear termsxzandx 2 introduce modulation effects in fast and slow chan- nels, enabling complex chaotic behavior even in low dimensions

Shimizu–Morioka System [48] The Shimizu–Morioka system is described by the following three-dimensional autonomous ODE:    ˙x=y, ˙y=x−ay−xz, ˙z=−bz+x 2, a= 0.85, b= 0.5.(11) Originating from low-dimensional laser dynamics modeling, this system can be interpreted as a state-dependent feedback oscillator coupled with a slow variable. The nonlinear terms...
[66]

Its simple yet explicit nonlinear source makes it an important benchmark for evaluating the trade-off between nonlinear identification accuracy and model compactness

Genesio–Tesi System [49] The Genesio–Tesi system is a polynomial chaotic model governed by:    ˙x=y, ˙y=z, ˙z=−cx−by−az+x 2, a= 1.2, b= 2.92, c= 6.(12) This system exhibits an integration-chain structure, with chaos driven solely by the quadratic termx2. Its simple yet explicit nonlinear source makes it an important benchmark for evaluating the trade...
[67]

The bilinear termsxzandxydescribe nonlinear interactions between light and matter

Laser System [50] The Laser system originates from single-mode laser rate equations and is described by:    ˙x=a(y−x), ˙y=bx−y−xz, ˙z=−cz+xy, (13) Here, state variables typically correspond to electric field intensity, polarization, and population inversion. The bilinear termsxzandxydescribe nonlinear interactions between light and matter. Under suit...
[68]

Although structurally simple, the system can exhibit complex phase-space behavior under appropriate damping parameters, making it a foundational prototype in nonlinear dynamics

Duffing Autonomous System [59] The Duffing system is a classical model of nonlinear vibration, expressed in autonomous form as:    ˙x=y, ˙y=x−x 3 −ay, ˙z=−bz, (14) Here,xdenotes displacement andyvelocity, whilex 3 captures nonlinear stiffness effects. Although structurally simple, the system can exhibit complex phase-space behavior under appropriate ...
[69]

This system is a canonical benchmark in nonequilibrium thermodynamics and chemical kinetics

Brusselator System [51] The Brusselator system models autocatalytic chemical reactions:    ˙x=A+x 2y−(B+ 1)x, ˙y=Bx−x 2y, ˙z=−z+x, (15) The nonlinear termx 2yintroduces positive feedback, enabling oscillatory and chaotic behavior far from equilibrium. This system is a canonical benchmark in nonequilibrium thermodynamics and chemical kinetics
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Kawczynski–Strizhak System [60] The Kawczynski–Strizhak system is a third-order polynomial chaotic model:    ˙x=y, ˙y=z, ˙z=−az−by−x+x 2, (16) 53 Its chaos is driven entirely by the quadratic termx 2, making it suitable for analyzing how nonlinear forcing influences stability and bifurcation behavior
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Rucklidge System [52] The Rucklidge system, originally developed for thermal convection instability, is defined as:    ˙x=−kx+ay−yz, ˙y=x, ˙z=y 2 −z, (17) It can be viewed as a Lorenz-type variant, with nonlinear coupling termsyzand y2 shaping its stretching-and-folding phase-space geometry
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FitzHugh–Nagumo System [53, 54] The FitzHugh–Nagumo system is a reduced model of Hodgkin–Huxley neuron dynamics:    ˙x=c x− x3 3 +y , ˙y=− 1 c (x−a+by), ˙z=−z+x, (18) The cubic nonlinearity governs neuronal excitation and recovery dynamics and can produce oscillatory and chaotic firing patterns
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Finance System [55] The Finance system models nonlinear macroeconomic interactions:    ˙x=z+ (y−a)x, ˙y= 1−by−x 2, ˙z=−x−cz, (19) The quadratic termx 2 captures saturation effects and economic feedback loops, producing chaotic market-like fluctuations
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Dequan–Li System [56] The Dequan–Li system is a Lorenz-type chaotic system:    ˙x=a(y−x) +yz, ˙y=cx−xz+dy, ˙z=xy−bz, (20) Additional bilinear couplings reshape attractor geometry, making it suitable for testing modeling under mixed linear–nonlinear structures. 54
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Hadley Circulation Model [57] The Hadley circulation model describes large-scale atmospheric circulation:    ˙x=−y 2 −z 2 −ax+ac, ˙y=xy−bxz−y+d, ˙z=bxy+xz−z, (21) Its nonlinear energy-exchange terms allow complex low-dimensional climate dynamics
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(22) Chaos arises from the nonsmooth nonlinearitysign(x), making this system a challenging benchmark for symbolic regression under nonsmooth dynamics

Sprott–Jerk System [58] The Sprott–Jerk system is a piecewise nonsmooth chaotic model:    ˙x=y, ˙y=z, ˙z=−az−y+ sign(x). (22) Chaos arises from the nonsmooth nonlinearitysign(x), making this system a challenging benchmark for symbolic regression under nonsmooth dynamics. J.2 Comprehensive Evaluation Protocol Based on Per-DimensionR 2 and Short-Horizo...
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cos(x)−1U(−1,1,20) Nguyen-6sin(x 1) + sin(x1 +x 2 1)U(−1,1,20) Nguyen-7log(x 1 + 1) + log(x2 1 + 1)U(0,2,20) Nguyen-8 √xU(0,4,20) Nguyen-9sin(x) + sin(x 2 2)U(0,1,20) Nguyen-102 sin(x) cos(x 2)U(0,1,20) Nguyen-11x x2 1 U(0,1,20) Nguyen-12x 4 1 −x 3 1 + 1 2 x2 2 −x 2 U(0,1,20) Nguyen-2′ 4x4 1 + 3x3 1 + 2x2 1 +xU(−1,1,20) Nguyen-5′ sin(x2
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cos(x)−2U(−1,1,20) Nguyen-8′ 3√xU(0,4,20) Nguyen-8′′ 3 q x2 1 U(0,4,20) Nguyen-1c 3.39x3 1 + 2.12x2 1 + 1.78xU(−1,1,20) Nguyen-5c sin(x2
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65 Name Expression Dataset Neat-1x 4 1 +x 3 1 +x 2 1 +xU(−1,1,20) Neat-2x 5 1 +x 4 1 +x 3 1 +x 2 1 +xU(−1,1,20) Neat-3sin(x 2

cos(x)−0.75U(−1,1,20) Nguyen-7c log(x+ 1.4) + log(x 2 1 + 1.3)U(0,2,20) Nguyen-8c √ 1.23xU(0,4,20) Nguyen-10c sin(1.5x) cos(0.5x2)U(0,1,20) Korns-11.57 + 24.3∗x 4 1 U(−1,1,20) Korns-20.23 + 14.2 (x4+x1) (3x2) U(−1,1,20) Korns-34.9 (x2−x1+ x1 x3 (3x3)) −5.41U(−1,1,20) Korns-40.13sin(x 1)−2.3U(−1,1,20) Korns-53 + 2.13log(|x 5|)U(−1,1,20) Korns-61.3 + 0.13 p...
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66 Name Expression Dataset Livermore-174sin(x 1)cos(x2)U(−3,3,100) Livermore-18sin(x 2 1)∗cos(x 1)−5U(−3,3,100) Livermore-19x 5 1 +x 4 1 +x 2 1 +x 1 U(−3,3,100) Livermore-20e (−x2

cos(x)−1U(−1,1,20) Neat-4log(x+ 1) + log(x 2 1 + 1)U(0,2,20) Neat-52 sin(x) cos(x 2)U(−1,1,100) Neat-6 Px k=1 1 k E(1,50,50) Neat-72−2.1 cos(9.8x 1) sin(1.3x2)E(−50,50,10 5) Neat-8 e−(x1 )2 1.2+(x2−2.5)2 U(0.3,4,100) Neat-9 1 1+x−4 1 + 1 1+x−4 2 E(−5,5,21) Keijzer-10.3x 1sin(2πx1)U(−1,1,20) Keijzer-22.0x 1sin(0.5πx1)U(−1,1,20) Keijzer-30.92x 1sin(2.41πx1)...

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