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arxiv: 2605.08130 · v1 · submitted 2026-05-01 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

Additive Atomic Forests for Symbolic Function and Antiderivative Discovery

Reda Belaiche
Authors on Pith no claims yet

Pith reviewed 2026-05-12 00:54 UTC · model grok-4.3

classification 💻 cs.LG
keywords symbolic regressionantiderivative discoveryderivative algebraatomic forestsinterpretable modelsfunction approximationsymbolic function recovery
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The pith

A self-expanding library of atomic functions allows simultaneous symbolic recovery of a function and its antiderivative from data.

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

The paper presents a framework that builds a library of function-derivative pairs using derivative rules starting from seed primitives. Two special primitives make the library efficient by providing trigonometric and exponential-logarithmic atoms early. Additive atomic forests are then used to fit sums of these atoms to data, yielding both a symbolic function and its derivative without needing to integrate symbolically. This produces interpretable models that perform competitively with black-box methods on classification benchmarks.

Core claim

The central discovery is that a derivative algebra generated recursively from elementary seeds via the product and chain rules, augmented by the EML and SOL primitives, creates a complete enough set of atoms for representing many functions. By constructing additive forests from these atoms where each tree's derivative is known by design, the model fits data to recover both F(x) and F'(x) simultaneously through optimization or search.

What carries the argument

Additive atomic forests, defined as finite sums of primitive trees optionally composed via multiplicative nodes, where each primitive carries its derivative from the algebra.

If this is right

  • Both a function and its antiderivative are recovered at once since differentiation is built into the atoms.
  • The library grows dynamically as new functions are added, increasing the range of representable expressions.
  • Sparse combinations of atoms can achieve performance comparable to XGBoost on multiple datasets while remaining interpretable.
  • The method avoids the computational cost of symbolic integration steps.

Where Pith is reading between the lines

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

  • The framework might apply to discovering solutions to differential equations beyond simple antiderivatives.
  • Extending the seeding primitives could further reduce the depth needed for certain function classes.
  • Such atomic libraries could integrate with other symbolic regression techniques to improve search efficiency.

Load-bearing premise

Target functions in practice are well approximated by finite additive combinations of the atomic primitives that the derivative algebra and seeding primitives can generate.

What would settle it

Finding a real dataset where the underlying function cannot be expressed or approximated closely by any finite sum from the grown library, causing the recovery to fail even as the library size increases.

read the original abstract

We present a framework for the simultaneous symbolic recovery of a function and its antiderivative from data. The framework rests on three ideas. First, a derivative algebra: the observation that the product rule $\frac{d}{dx}[f \cdot g] = f'g + fg'$ and the chain rule, applied to a seed set of elementary functions, generate a self-expanding system of function-derivative pairs -- a living library that grows each time a new function is discovered. Second, two complementary primitives -- EML$\,(e^u - \ln v)$, which is theoretically complete for all elementary functions, and SOL$\,(\sin u - \cos v)$, introduced here, which makes trigonometric atoms available at depth~1 instead of depth~$\sim$8 -- that seed the library with core atoms cheaply. Third, additive atomic forests: finite sums of primitive trees, optionally composed via multiplicative nodes, whose derivatives are fitted to data by continuous optimisation or by exhaustive search over the library. Because differentiation of each atom is determined by construction, the forest simultaneously encodes a symbolic expression $F$ and its derivative $F'$; no symbolic integration step is required. The library is not a fixed object: it self-constructs from a small seed set by recursive application of the product rule, chain rule, and the two primitives, and it can grow as newly discovered functions are folded back in. The larger the library, the richer the expressible class of candidate functions. We give conditional completeness, additive-depth, and analytic simultaneous-recovery results for the framework. Empirically, in our reported runs on 17 classification benchmarks, sparse atom combinations match or exceed XGBoost on 13 datasets while producing interpretable formulas.

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 a framework for simultaneous symbolic recovery of a function F and its antiderivative F' from data. It defines a derivative algebra that generates self-expanding libraries of function-derivative pairs from seed elementary functions via the product and chain rules, seeded by the EML (e^u - ln v) and SOL (sin u - cos v) primitives. These libraries support additive atomic forests (finite sums of primitive trees, optionally with multiplicative composition) whose derivatives are known by construction. The paper states conditional completeness, additive-depth, and analytic simultaneous-recovery results, and reports that sparse atom combinations match or exceed XGBoost on 13 of 17 classification benchmarks while yielding interpretable formulas.

Significance. If the simultaneous F/F' recovery and derivative-algebra construction hold under proper validation, the approach would provide a distinctive contribution to symbolic regression by guaranteeing derivative information without post-hoc integration. The self-expanding library mechanism and the two seeding primitives (particularly SOL for reducing trigonometric depth) are technically interesting and could enable richer expressivity than fixed-basis methods. The manuscript does not, however, supply machine-checked proofs or reproducible code artifacts that would strengthen the theoretical claims.

major comments (2)
  1. [Abstract] Abstract and experimental evaluation: The central claim is simultaneous symbolic recovery of F and its antiderivative F' from data, with derivatives guaranteed by the derivative algebra. The reported results use 17 classification benchmarks that supply discrete labels rather than continuous observations of F or F'. No description is given of held-out derivative validation, numerical quadrature checks (integrating the discovered F' to recover F), or synthetic regression tasks with known antiderivatives. This leaves the distinguishing feature of the framework without direct empirical support.
  2. [Abstract] Abstract: The performance claim that 'sparse atom combinations match or exceed XGBoost on 13 datasets' is presented without details on the optimization procedure for fitting coefficients, the exact construction of the atomic library at each run, statistical significance testing, or data preprocessing. Because the library is generative and the fitting involves continuous optimisation or exhaustive search, the absence of these specifics makes it impossible to assess whether the reported wins are attributable to the derivative-algebra construction or to standard additive-model advantages.
minor comments (1)
  1. [Abstract] Notation for the two seeding primitives (EML and SOL) is introduced without an explicit table or equation defining their precise functional forms and derivative pairs at the point of first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which correctly identify gaps in empirical validation of the core simultaneous-recovery claim and in experimental transparency. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract and experimental evaluation: The central claim is simultaneous symbolic recovery of a function F and its antiderivative F' from data, with derivatives guaranteed by the derivative algebra. The reported results use 17 classification benchmarks that supply discrete labels rather than continuous observations of F or F'. No description is given of held-out derivative validation, numerical quadrature checks (integrating the discovered F' to recover F), or synthetic regression tasks with known antiderivatives. This leaves the distinguishing feature of the framework without direct empirical support.

    Authors: We agree that the reported experiments evaluate predictive performance on classification tasks using the discovered symbolic expressions, without explicit held-out validation of derivative accuracy or synthetic tasks with known antiderivatives. While the derivative algebra guarantees F' by construction, this does not substitute for direct empirical checks of the simultaneous-recovery property. In the revised manuscript we will add a dedicated experimental subsection containing (i) synthetic regression benchmarks with analytically known F and F', (ii) numerical quadrature verification that integrating the recovered F' recovers F within tolerance, and (iii) held-out derivative error metrics. These additions will directly support the central claim. revision: yes

  2. Referee: [Abstract] Abstract: The performance claim that 'sparse atom combinations match or exceed XGBoost on 13 datasets' is presented without details on the optimization procedure for fitting coefficients, the exact construction of the atomic library at each run, statistical significance testing, or data preprocessing. Because the library is generative and the fitting involves continuous optimisation or exhaustive search, the absence of these specifics makes it impossible to assess whether the reported wins are attributable to the derivative-algebra construction or to standard additive-model advantages.

    Authors: The referee is correct that the current manuscript omits key methodological details required to interpret the performance numbers. We will expand the experimental section to specify: the coefficient-fitting procedure (continuous optimisation versus exhaustive sparse search), the precise library-construction protocol and its size per run, the statistical tests employed (including p-values and multiple-run variance), and all preprocessing steps. These clarifications will allow readers to isolate the contribution of the derivative-algebra mechanism from generic additive-model benefits. revision: yes

Circularity Check

0 steps flagged

No circularity: generative library and by-construction derivative are independent of fitted recovery claims

full rationale

The derivation chain begins with a seed set of primitives and applies the product/chain rules plus two explicit seeding functions (EML, SOL) to generate an expanding library of atom/derivative pairs. This process is strictly generative from the stated algebraic rules and does not presuppose the target functions or their antiderivatives. The additive atomic forests are then selected and fitted to data via optimization or exhaustive search; the fact that each atom carries its derivative by construction means the symbolic F and F' are recovered together, but this is an explicit design property rather than a reduction of the fitting result to its own inputs. Conditional completeness and analytic recovery statements are derived from the same generative rules under stated assumptions and do not collapse into self-definition. No load-bearing self-citations or fitted parameters renamed as predictions appear in the provided description. The classification-benchmark results therefore constitute an external empirical test of the fitting procedure, not a tautology.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 3 invented entities

The framework rests on the derivative algebra as a domain assumption, the two introduced primitives as invented entities, and free parameters in the choice of seed atoms and fitting coefficients.

free parameters (2)
  • seed elementary functions
    Initial set chosen to start the self-expanding library.
  • optimization coefficients
    Parameters adjusted when fitting forests to data.
axioms (2)
  • domain assumption Product and chain rules applied to a seed set generate a self-expanding system of function-derivative pairs.
    Foundation of the derivative algebra stated in the abstract.
  • domain assumption EML and SOL primitives are theoretically complete for elementary functions.
    Claimed completeness for seeding the library.
invented entities (3)
  • EML primitive (e^u - ln v) no independent evidence
    purpose: Seed library with exponentials and logs at low depth.
    New primitive introduced in the framework.
  • SOL primitive (sin u - cos v) no independent evidence
    purpose: Provide trigonometric atoms at depth 1.
    New primitive introduced here.
  • Additive atomic forests no independent evidence
    purpose: Represent candidate functions as sums of primitive trees.
    Core new representation for joint F and F' encoding.

pith-pipeline@v0.9.0 · 5606 in / 1563 out tokens · 60618 ms · 2026-05-12T00:54:20.582023+00:00 · methodology

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 3 internal anchors

  1. [1]

    S. L. Brunton, J. L. Proctor, and J. N. Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15):3932--3937, 2016

    work page 2016

  2. [2]

    Chen and C

    T. Chen and C. Guestrin. XGBoost : A scalable tree boosting system. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 785--794, 2016

    work page 2016

  3. [3]

    R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. Duvenaud. Neural ordinary differential equations. In Advances in Neural Information Processing Systems 31 (NeurIPS), 2018

    work page 2018

  4. [4]

    M. Cranmer. Interpretable machine learning for science with PySR and SymbolicRegression.jl . arXiv preprint arXiv:2305.01582, 2023

    work page internal anchor Pith review arXiv 2023

  5. [5]

    Greydanus, M

    S. Greydanus, M. Dzamba, and J. Yosinski. Hamiltonian neural networks. In Advances in Neural Information Processing Systems 32 (NeurIPS), 2019

    work page 2019

  6. [6]

    D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In Proceedings of the 3rd International Conference on Learning Representations (ICLR), 2015

    work page 2015

  7. [7]

    Lelli, S

    F. Lelli, S. S. McGaugh, and J. M. Schombert. SPARC : Mass models for 175 disk galaxies with Spitzer photometry and accurate rotation curves. The Astronomical Journal, 153(6):240, 2017

    work page 2017

  8. [8]

    Liouville

    J. Liouville. Premier m\'emoire sur la d\'etermination des int\'egrales dont la valeur est alg\'ebrique. Journal de l'\'Ecole Polytechnique, 14:124--148, 1833

    work page

  9. [9]

    S. S. McGaugh, F. Lelli, and J. M. Schombert. Radial acceleration relation in rotationally supported galaxies. Physical Review Letters, 117(20):201101, 2016

    work page 2016

  10. [10]

    M. Milgrom. A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. The Astrophysical Journal, 270:365--370, 1983

    work page 1983

  11. [11]

    All elementary functions from a single binary operator

    A. Odrzywo ek. All elementary functions from a single binary operator. arXiv preprint arXiv:2603.21852, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  12. [12]

    Raissi, P

    M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686--707, 2019

    work page 2019

  13. [13]

    R. H. Risch. The problem of integration in finite terms. Transactions of the American Mathematical Society, 139:167--189, 1969

    work page 1969

  14. [14]

    R. H. Risch. The solution of the problem of integration in finite terms. Bulletin of the American Mathematical Society, 76(3):605--608, 1970

    work page 1970

  15. [15]

    J. F. Ritt. Integration in Finite Terms: Liouville's Theory of Elementary Methods . Columbia University Press, 1948

    work page 1948

  16. [16]

    Schmidt and H

    M. Schmidt and H. Lipson. Distilling free-form natural laws from experimental data. Science, 324(5923):81--85, 2009

    work page 2009

  17. [17]

    Algebraic structure behind Odrzywo{\l}ek's EML operator

    T. Stachowiak. Algebraic structure behind Odrzywo ek's EML operator. arXiv preprint arXiv:2604.23893, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  18. [18]

    Tibshirani

    R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society: Series B, 58(1):267--288, 1996

    work page 1996

  19. [19]

    Udrescu and M

    S.-M. Udrescu and M. Tegmark. AI Feynman : A physics-inspired method for symbolic regression. Science Advances, 6(16):eaay2631, 2020

    work page 2020