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1D Kinetic Energy Density Functionals learned with Symbolic Regression

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Pith's one-line read The paper claims that symbolic regression can rediscover the exact von Weizsäcker kinetic-energy functional for a single 1D electron, but for two or more electrons the best semi-local fits are Thomas-Fermi-like and leave the largest error…

desk verdict First symbolic-regression KEDF search, clean vW rediscovery; the negative semi-local claim is undercut by a narrow operator set and in-sample losses. read the letter →

arxiv 2412.08143 v1 pith:OJ7XDJ4H submitted 2024-12-11 cond-mat.mtrl-sci

Michael A.J. Mitchell , Teresa Del Aguila Ferrandis , Stefano Sanvito This is my paper · ORCID
classification cond-mat.mtrl-sci PACS 71.15.Mb
keywords symbolicregressionkineticenergydensityfunctionalorbital-freeDFTThomas-FermivonWeizsäckerone-dimensionalelectrongasenhancementfactor
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that symbolic regression, a search over closed-form expressions, can rediscover exact kinetic-energy density functionals in one-dimensional systems, but that semi-local functional forms are not enough to represent the non-interacting kinetic energy for few-electron systems. Using exact ground-state densities from non-interacting electrons in random three-Gaussian wells, the search recovers the von Weizsäcker functional for one electron and variants of the Thomas-Fermi functional for every higher electron count. The largest errors appear at two electrons, which the authors attribute to the non-local nodal structure of orbitals that a semi-local expression built from density and its first two derivatives cannot capture. If correct, the result sharpens the known difficulty of orbital-free DFT: accurate kinetic-energy functionals likely need non-local information, not just local density and derivative terms.

What carries the argument

The machinery is symbolic regression adapted to functionals: candidate expressions are built from the density, its first and second derivatives, and, in constrained runs, from the dimensionless reduced variables s(x), q(x), and k(x); each candidate kinetic-energy density is integrated over space and compared to the exact Ts, with complexity penalized. The core identity is the uniform-scaling law Ts[ρλ] = λ³Ts[ρ], which is enforced in the enhancement-factor search by writing Ts[ρ] = (π²/6)∫ρ³ F(s,q,k) dx. The Pareto front of loss versus expression complexity is the selection device that decides which functional wins at each electron number.

What would settle it

Train or test the same symbolic-regression searches on a different ensemble of 1D potentials, such as shallow well potentials, periodic potentials, or potentials producing multi-peaked densities for Ne = 2 to 5, and check whether any semi-local expression built from ρ, ρ′, and ρ″ reaches near-zero loss on a held-out set of exact ground states. A held-out test yielding near-zero error for a semi-local expression at intermediate electron counts would falsify the paper's negative claim, while a demonstration that the reported losses grow substantially out-of-sample would support it.

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Extended reading notes

Core claim

The central claim is that, for one-dimensional non-interacting electron systems, the only accurately learnable semi-local kinetic-energy density functionals are the two exact limits: the von Weizsäcker functional for one electron and Thomas-Fermi-like expressions for many electrons. The search across electron numbers 1 to 20 shows a smooth transition in which the vW functional is recovered exactly at one electron and the TF functional, with a fitted prefactor below the exact 1D value π²/6, dominates the Pareto front at all other electron counts. The authors interpret the elevated losses at intermediate electron counts, especially two electrons, as evidence that the semi-local paradigm is fundamentally limited: orbital orthogonality imposes nodal constraints that cannot be expressed locally through the density and its derivatives.

Load-bearing premise

The conclusion that semi-local functionals are insufficient for few-electron systems rests on treating the random three-Gaussian external potentials used to generate the training densities as representative of all one-dimensional non-interacting electron systems; if that potential ensemble does not sample the density space where semi-local functionals could succeed, the negative result is a property of the dataset rather than of semi-local functionals in general.

Editorial extensions

If this is right

  • At one electron, symbolic regression returns the exact von Weizsäcker functional with a percentage loss of about 0.00027%, so the method can rediscover a known exact limit without being told it.
  • At every other electron count, the best semi-local form is a Thomas-Fermi-like functional whose fitted prefactor stays below π²/6 and approaches about 1.635 for large Ne, so the 1D TF functional systematically overestimates the kinetic energy of confined non-uniform systems.
  • The largest percentage loss occurs at Ne = 2, identifying the few-electron regime as the hardest case for semi-local kinetic-energy density functionals.
  • When the search is seeded with vW and TF, a 'TF − λvW' linear combination becomes the second-best expression for intermediate Ne with λ ≈ 0.265 ± 0.008 in 1D, while in 3D the optimal combination reported in the literature is TF + λvW with λ ≈ 0.2.
  • Constraining the search to an enhancement factor that respects uniform scaling does not change the picture: the winning forms are still vW for one electron and TF-like variants otherwise, with the extra dimensionless terms integrating to essentially zero.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension the paper leaves implicit is to split the functional into a fixed non-local kernel plus a learned semi-local correction; the authors mention this avenue but do not test it, and the negative result here is what makes that split worth trying.
  • Since the losses are reported in-sample, the gap between semi-local and exact might grow when measured on densities outside the training set; a train/test split could make the few-electron failure more severe than the figures suggest.
  • The paper does not evaluate the variational accuracy of the functional derivatives of the discovered expressions; if those derivatives are inaccurate, even the recovered vW and TF forms may not be directly usable in an orbital-free minimization.
  • The same symbolic-regression setup could be applied to 3D confined systems or to electrons with spin, where the density has nodal surfaces; a failure there would extend the paper's conclusion beyond 1D.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper uses symbolic regression (SR), with a modified PySR implementation, to search for one-dimensional kinetic-energy density functionals (KEDFs) from exact-diagonalization ground states of non-interacting spinless electrons in random three-Gaussian external potentials. Three search strategies are presented: an unconstrained search over densities and their first two derivatives, a search seeded with the von Weizsäcker (vW) and Thomas-Fermi (TF) functionals as input features, and a search constrained to the form of a TF enhancement factor. For Ne=1 the SR recovers the vW functional with a fitted prefactor 0.12499846, very close to the exact 1/8; for Ne>=2 the best-scoring expressions are variants of the TF functional, with in-sample percentage losses decreasing with Ne and largest around 2% for Ne=2. The authors conclude that semi-local expressions are insufficient for the few-electron KEDF in 1D.

Significance. If the conclusions are robust, the paper provides a useful benchmark showing that symbolic regression can rediscover exact KEDF limits (vW for one electron, TF for many electrons) and it highlights the difficulty of constructing accurate semi-local KEDFs for few-electron systems. The exact-diagonalization data generation is simple and the Ne=1 recovery is a clean sanity check. However, the central negative claim rests on a search with a narrow operator set and on in-sample loss only; the manuscript does not report held-out tests, dataset sizes, or parameter ranges, and it does not validate functional derivatives. These omissions currently prevent the negative conclusion from being established as a general statement about semi-local KEDFs.

major comments (4)
  1. [II.D, Figs. 4-8] The dataset description in Section II.D gives no number of random potentials per Ne, no parameter ranges for Ai, bi, ci (aside from a vague statement that the Ai range is chosen according to Ne), and no train/test split. All reported percentage losses in Figs. 4, 5, 6, and 8 are therefore in-sample fit errors. This is load-bearing because the central negative claim, that semi-local expressions are insufficient for few-electron systems, depends on the losses not being an artifact of fitting a small or unrepresentative set of potentials. The authors should report the dataset size, the exact parameter sampling ranges, and a held-out test (e.g., cross-validation over potentials) before drawing conclusions about generalizability.
  2. [III.A] The unconstrained search allows only the operators +, ×, ÷, and squaring, with inputs restricted to ρ, ρ', and ρ''. Many plausible semi-local KEDF forms, including fractional powers of the density such as ρ^α with non-integer α, cannot be represented exactly in this search space. The claim that 'a semi-local expression of the functional appears not sufficient' (Introduction) is therefore only supported with respect to this restricted operator set. Moreover, Fig. 6 shows that second-best functionals often have substantially lower loss than the Pareto-front choice but are discarded by the complexity penalty; this indicates that the parsimony criterion, not an intrinsic limitation of semi-local forms, may be responsible for the reported losses. The conclusions should be rephrased as applying to the specific SR search space and complexity trade-off used here.
  3. [II.A and IV] The Introduction and Abstract motivate the work by the need for KEDFs with accurate functional derivatives for OFDFT minimization, and the paper states that analytical SR expressions make derivatives easy to compute. However, no functional derivative is ever computed or tested. Given that the manuscript explicitly frames derivative accuracy as a central advantage of SR, the authors should at least verify δTs/δρ for the Ne=1 and Ne=2 fitted functionals against the exact Kohn-Sham potential, or otherwise clearly delimit derivative accuracy as future work. As written, the claimed advantage for OFDFT is unsubstantiated.
  4. [III.A-III.C] Symbolic regression is a stochastic global optimization procedure, yet the paper reports only a single Pareto front for each Ne and gives no statistics over independent SR runs or random seeds. The reported losses and functional forms could be non-reproducible if the genetic search is sensitive to initialization. The authors should provide repeated-run statistics (e.g., median and spread of losses across seeds) or justify why a single run is representative. This concern applies to all three searches and to the specific functional forms quoted in Figs. 4, 5, and 8.
minor comments (6)
  1. [II.A, Eq. (5)] Equation (5) defines ρ(r) = Σ_i |φ_i(r)|, but the square is missing; it should be ρ(r) = Σ_i |φ_i(r)|^2.
  2. [II.D] The text refers to 'Born-von Karman boundary conditions' but states ψ(x0)=ψ(x_N−1)=0, which are hard-wall Dirichlet boundary conditions, not periodic Born-von Karman conditions.
  3. [Abstract and I] The abstract and introduction contain the typo 'von von Weizsäcker' and later 'Plack constant' should be 'Planck constant'.
  4. [III.C, Fig. 8] The caption of Fig. 8 has 'grey dashed line os to guide the eye'; 'os' should be 'is'.
  5. [III.B] The discussion of the λ prefactor in 'TF - λvW' says the standard deviation is small but 'statistically insignificant'; this phrasing is confusing because the standard deviation is a descriptive statistic, not a significance test. Clarify what is meant.
  6. [References] Reference [14] has the same title as reference [13] but appears to be a different paper (by del Mazo-Sevillano and Hermann); please verify the title and citation details.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SR fits and constrained searches are disclosed as such, and the central vW/TF findings are data-driven rather than definitionally forced.

full rationale

The paper's derivation chain is a symbolic-regression fit of candidate kinetic-energy density functionals to exact Ts values obtained by diagonalizing non-interacting 1D systems. The one-electron result (vW) and the multi-electron results (TF-like forms) are outputs of a search over a stated operator set, not identities imported from the fitting target. In the unconstrained search (Section III.A), vW and TF are not given as inputs; their recovery is a genuine data-driven finding. In Section III.B the authors explicitly seed vW and TF as input features and describe the outcome as expected, which is an acknowledged inductive bias rather than a disguised prediction; moreover, the second-best functionals show that the Pareto-front choice is a complexity trade-off, not a forced identity. In Section III.C, Eq. (13) imposes the TF prefactor through an enhancement-factor ansatz, but the search still had to find F; the fact that F's gradient terms integrate to roughly zero is an empirical result within that ansatz. No self-citation chain or imported uniqueness theorem is load-bearing. The main weakness is external validity: no train/test split or sample counts are reported, and the restricted operator set limits the space of semi-local forms explored, so the negative conclusion about semi-local KEDFs in the few-electron case is an absence-of-evidence claim. That is a correctness and generalization concern, not circularity.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

All quantitative results are obtained by fitting analytic expressions to exact Ts values computed by diagonalization. The only structural input beyond the standard DFT framework is the scaling-inspired enhancement-factor ansatz of Eq. (13). No new physical entities are introduced, but every numerical prefactor in the reported functionals is a fitted parameter.

free parameters (5)
  • vW prefactor (Ne=1, unconstrained) = 0.124998460964301
    SR-fitted prefactor of (rho')^2/rho in the one-electron case; compares to the exact value 1/8. The tiny deviation likely reflects numerical discretization.
  • TF prefactor for Ne>=2 = Values below pi^2/6, asymptoting near 1.635
    Fitted coefficient of rho^3 in each electron-number run; the paper notes it remains below the exact 1D Thomas-Fermi value of pi^2/6.
  • gamma in gammaTF expressions = Consistently less than one
    Multiplicative scaling factor found in the second-best and constrained searches, fitted to minimize the loss.
  • lambda in TF - lambda vW = 0.26545 +/- 0.007658 (average)
    Fitted weight of the vW correction in linear combinations; the standard deviation is reported across systems.
  • Parsimony/complexity weighting = Chosen to favor more complex expressions
    Hand-chosen hyperparameter in SymbolicRegression.jl that determines which expression sits on the Pareto front (Section III.B).
assumptions (5)
  • standard math The non-interacting KEDF obeys uniform scaling Ts[rho_lambda] = lambda^3 Ts[rho] (Eq. 11).
    Used to define the enhancement-factor ansatz in Section III.C; accepted exact constraint from Levy and Perdew.
  • domain assumption The search space restricted to local and semi-local functionals of rho, rho', and rho'' is capable of representing the target KEDF.
    The central modeling choice. The paper's negative conclusion is that this is false for few-electron systems, but the search still assumes it as the hypothesis space.
  • ad hoc to paper Ground states from random three-Gaussian potentials with unspecified parameter ranges are representative of 1D non-interacting electron systems.
    Dataset construction in Section II.D. No justification of coverage, no train/test split, so generalization of the fitted functionals is assumed.
  • domain assumption Exact diagonalization on a uniform 2000-point grid gives converged ground-state densities and kinetic energies.
    No convergence checks or discretization error estimates are reported (Section II.D).
  • standard math The Hohenberg-Kohn and Kohn-Sham theorems provide the variational DFT framework.
    Background of Section I; standard and not controversial.

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Cite this review

Pith. "Pith review of 1D Kinetic Energy Density Functionals learned with Symbolic Regression." pith.science (2026). https://pith.science/paper/OJ7XDJ4H

@misc{pith2026241208143,
  author       = {Pith},
  title        = {Pith review of: 1D Kinetic Energy Density Functionals learned with Symbolic Regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OJ7XDJ4H}},
  note         = {Machine review of arXiv:2412.08143}
}
read the original abstract

Orbital-free density functional theory promises to deliver linear-scaling electronic structure calculations. This requires the knowledge of the non-interacting kinetic-energy density functional (KEDF), which should be accurate and must admit accurate functional derivatives, so that a minimization procedure can be designed. In this work, symbolic regression is explored as an alternative means to machine-learn the KEDF, which results into analytical expressions, whose functional derivatives are easy to compute. The so-determined semi-local functional forms are investigated as a function of the electron number, and we are able to track the transition from the von Weizs\"acker functional, exact for the one-electron case, to the Thomas-Fermi functional, exact in the homogeneous electron gas limit. A number of separate searches are performed, ranging from totally unconstrained to constrained in the form of an enhancement factor. This work highlights the complexity in constructing semi-local approximations of the KEDF and the potential of symbolic regression to advance the search.

Figures

Figures reproduced from arXiv: 2412.08143 by the authors.

Figure 2
Figure 2. FIG. 2. A diagram depicting the modification required by the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. An example of the symbolic regression procedure for the case of one electron in its ground state. In panel (a) we show [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Percentage error of the best-performing functionals [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: As expected, the vW functional tops the search for the one-electron case, this time with essentially zero per￾centage loss. Then, at any other electron count the TF expression appears to be at the Pareto front, namely it is the best trade-off between accuracy and compl…
Figure 5
Figure 5. Figure 5: FIG. 5. Percentage error of the best-performing functionals [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Percentage loss error of the best performing func [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Demonstration of the ambiguity in the identification [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

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