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REVIEW 5 major objections 6 minor 1 cited by

A Neural Operator-Based Approach to Symbolic Discovery of PDEs

T0 review · 5 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read NOMTO substitutes pretrained neural operators for exact library operations inside a sparse symbolic graph, letting symbolic regression discover special functions, derivatives, and second-order nonlinear PDEs from data.

desk verdict NOMTO is a genuinely novel architecture, but the evidence is all in-sample, the recovered equations are often structurally wrong, and the abstract overclaims; it deserves serious peer review but needs major revisions before the claims hold. read the letter →

arxiv 2501.08086 v2 pith:GNFI6XZ4 submitted 2025-01-14 cs.AI cs.SC

Sergei Garmaev , Olga Fink This is my paper · ORCID
classification cs.AIcs.SC
keywords symbolicregressionneuraloperatorpartialdifferentialequationdiscoveryFourierconvolutionalsparsecomputationalgraphspecialfunctionsBurgers
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

NOMTO is a symbolic regression framework that turns candidate equations into sparse, differentiable computational graphs whose nodes are neural operators pretrained to approximate library operations such as division, the Gamma function, the Airy function, and derivatives. The paper's central claim is that replacing exact algebraic nodes with these frozen surrogates lets gradient-based search discover symbolic models that Equation Learner (EQL) and Kolmogorov-Arnold Network (KAN) architectures miss, especially expressions with singularities, special functions, and differential operators. The authors support the claim on benchmark expressions and on simulation data, reporting recovery of the two-dimensional heat equation and the coupled two-dimensional Burgers equations with coefficients close to the true values. If correct, NOMTO would extend data-driven governing-equation discovery from local, pointwise operations to nonlocal and special operators without a hand-built library of nonlinear candidate terms.

What carries the argument

The load-bearing object is the neural operator block: a pretrained Fourier Neural Operator or Convolutional Neural Operator that maps a discretized input function to the result of one library operation, wrapped with a cropping layer and an inverse tanh projection so values stay in a bounded, differentiable range. These blocks are wired into a fully connected layered graph; every edge carries a scalar weight, and each node applies its library operation to a weighted sum of the previous layer's outputs, so the graph represents all nonlinear combinations of the library up to a chosen depth. The search itself is sparse weight optimization with RMSProp under a modified $\ell^{1/2}$ penalty, sometimes with periodic pruning of low-energy weights. The surrogates are frozen throughout, which is what supplies gradients for operations such as division, $\Gamma$, and $\mathrm{Ai}$ that are singular, undefined on parts of the domain, or otherwise hard to differentiate exactly.

What would settle it

A direct test is to compute the surrogate approximation error of each frozen neural operator on the actual target data rather than on the Gaussian-mixture training functions and compare it with the target expression's typical magnitude; if the error ratio tracks the near-zero and negative $R^2$ values in Table 3, the surrogates are the bottleneck. Re-running the Burgers rediscovery with exact numerical derivatives replacing the surrogate derivative nodes would settle whether the frozen surrogates, rather than the optimization, prevent NOMTO-CNO from recovering the equation.

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Extended reading notes

Core claim

The central discovery is that a neural operator trained once on random Gaussian-mixture input functions can serve as a fixed, differentiable node inside a symbolic expression graph, standing in for a library operation. Because the surrogate acts on discretized functions rather than on point values, the same block tolerates operations with undefined regions (square root, logarithm), singularities (division, Gamma), special functions (Airy), and derivatives. On the paper's evidence, this design rediscovers the heat equation as $\partial u/\partial t = 1.0\,\partial^2u/\partial x^2 + 1.0\,\partial^2u/\partial y^2$ with NOMTO-CNO and identifies the Burgers equations with FNO surrogates, e.g. $\partial u/\partial t = -0.9u\,\partial u/\partial x - 0.9v\,\partial u/\partial y + 2.0\,\partial^2u/\partial x^2 + 2.0\,\partial^2u/\partial y^2$; the CNO variant fails on Burgers. The paper presents this as evidence that symbolic discovery of second-order nonlinear PDEs is achievable without manual construction of nonlinear combinations.

Load-bearing premise

The load-bearing premise is that the frozen neural operator surrogates, trained only on random Gaussian-mixture functions, approximate every library operation accurately enough on the target data for the sparse graph optimization to recover the correct operator structure and coefficients; the paper's own f2 and f4 results and the NOMTO-CNO Burgers failure show this premise can fail.

Editorial extensions

If this is right

  • NOMTO can discover compact equations containing nonlocal, singular, or special operator terms directly from discretized field data, without a manually constructed library of candidate nonlinear terms.
  • Once a library operation is learned on random Gaussian-mixture functions, the same pretrained surrogate block can be frozen into any new NOMTO graph, so discovery tasks share the same operator library.
  • The redefined versions of functions such as $\sqrt{x}$ and $\ln x$ make the search robust to undefined regions, which the paper argues is acceptable because experimental measurements are well defined on the domain of interest.
  • With FNO surrogates the method recovers the structure and coefficients of the two-dimensional heat and Burgers equations; with CNO surrogates it recovers the heat equation but not Burgers, indicating that surrogate choice is decisive.
  • A two-layer computational graph, together with the $\ell^{1/2}$ sparsity penalty, is sufficient to rediscover second-order nonlinear PDEs on a $50 \times 50 \times 50$ spatio-temporal grid from 1,000 simulations.

Reading between the lines

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

  • Beyond the paper: because neural operators can approximate operators rather than only pointwise functions, the same graph construction should extend to integro-differential or delayed equations, a regime the paper does not test.
  • Beyond the paper: the contrast between FNO and CNO on Burgers suggests surrogate fidelity is the dominant bottleneck; an immediate testable extension is to retrain the surrogates on the target data distribution before running the sparse optimization.
  • Beyond the paper: the recovered coefficients can be no more accurate than the surrogate's approximation error at the target input magnitudes, which the clipping and tanh projection in the paper partially mitigate but do not remove.
  • Beyond the paper: adding temporal integral nodes to the library would let the framework attempt memory-effect equations of the sort the abstract motivates, since the derivative nodes already demonstrate that differential operators are discoverable.
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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

5 major / 6 minor

Summary. The manuscript introduces NOMTO, a symbolic-regression architecture in which pretrained neural operators (FNO or CNO) replace the exact operations of an Equation Learner-type graph. Each library operation (identity, multiplication, division, derivatives, sin, cos, sqrt, ln, Airy, Gamma) is approximated by a frozen neural operator trained on random Gaussian-mixture input functions, and the graph weights are optimized under an l1/2 sparsity penalty. The authors evaluate NOMTO on selected Nguyen/Keijzer benchmarks, on expressions containing derivatives and special functions, and on rediscovery of the 2-D heat and Burgers equations from simulations. The central claim is that NOMTO extends symbolic model discovery to nonlocal operators, singularities, special functions, and second-order nonlinear PDEs.

Significance. The idea of using neural operators as differentiable surrogate nodes for symbolic operations is novel and potentially useful, and it is clearly specified in the methodology. The paper is generally well written, the architecture is described in detail, and the PDE simulation protocols are documented in the appendices. If validated out-of-sample, NOMTO could extend symbolic regression to derivatives, special functions, and operator-level terms that existing EQL/KAN approaches cannot easily represent. However, the current evaluation is entirely in-sample, the CNO variant fails on Burgers rediscovery, and the reported failures for f2/f4 are incompatible with the abstract's unconditional claims. The nonlocal-operator and memory-effect promises in the abstract are not tested in any experiment.

major comments (5)
  1. [§2.1, Tables 1–2; §2.3, Fig. 2 and Appendix D] All reported MSE and R2 values are computed on the same data used to optimize the computational graph; no held-out split, random seed, or repeated-run statistics are provided. In Table 2, NOMTO-FNO 'recovers' Nguyen-7 as a mixture of sqrt, x, and cosine terms with R2=0.9604, and Keijzer-14 as a structurally different expression with R2=0.8151, yet both are described as successful. In Table 3, f2 and f4 have R2=-0.05 and 0.038, respectively, while the text states that the structure was captured. Since structurally wrong equations can achieve high in-sample R2, the reported metrics do not establish symbolic rediscovery; structural-equivalence rates and out-of-sample residual evaluation are needed.
  2. [§2.3.2, Eqs. (4)–(7)] For the Burgers rediscovery experiment, NOMTO-CNO produces Eq. (6), which includes spurious terms such as 0.5u^2 du/dx, -6.0(du/dx)^2, and 20 du/dx, and omits the u du/dx term with coefficient close to -1 as well as the du/dy diffusion term. This is not a rediscovery of Eq. (2). The abstract's unconditional claim that NOMTO 'accurately rediscover[s] second-order non-linear partial differential equations' is therefore supported only by the FNO variant, and even that variant's convective coefficients are 0.84–0.93 rather than 1. The claims should either be restricted to the successful conditions or the CNO failure should be analyzed and addressed.
  3. [Abstract and §1; §2.3] The abstract and introduction motivate NOMTO with 'nonlocal differential operators' and 'temporal memory effects', but none of the experiments involve a nonlocal operator or a memory/integral term. The heat and Burgers rediscovery tasks in Section 2.3 are local PDEs, and the symbolic-expression benchmarks in Sections 2.1 and 2.2 use only pointwise algebraic operations, derivatives, and special functions. The nonlocal and memory-effect claims should be either removed or supported by dedicated experiments with operators such as fractional derivatives, integral terms, or auxiliary-field couplings.
  4. [§4.2, Eq. (8)] The surrogate library uses redefined versions of sqrt and ln: sqrt(x)=0 for x<=0 and ln(x)=ln|x| for x!=0. The extracted expressions in Tables 2 and 3, however, are written with the standard mathematical symbols sqrt and ln. For negative arguments, the standard and redefined functions differ, so it is not justified to claim that NOMTO 'successfully identify[ies] symbolic expressions containing elementary functions with singularities'. At best, the method identifies the redefined surrogate operations, which is a different target. This issue also affects the singularity-handling claims in Sections 2.1 and 3.
  5. [§4.2 vs. §2.1] The pretraining distribution for the neural operator surrogates coincides with the input distribution used for the benchmarks: both use random mixtures of Gaussians on [-10,10] with 100 discretization points. Consequently, the reported successes may reflect interpolation within the surrogate's training distribution rather than general operator approximation. At least one evaluation with input functions drawn from a different family, or with analytic functions outside the Gaussian-mixture class, is needed to establish that the surrogate nodes are not simply memorizing the training distribution.
minor comments (6)
  1. [§4.4, Eq. (9)] The two cases in Eq. (9) both read 'for |w| >= a'; the second branch should be 'for |w| < a'.
  2. [§3] The Discussion section refers to 'NOMAD' instead of 'NOMTO'.
  3. [§4.2] There are typos: 'tahn' should be 'tanh', and the dataset size '10 5 samples' should be written as 10^5 samples.
  4. [§2.3.1] The phrase 'rediscover the patial differential equation' contains a typo: 'partial'.
  5. [§2.1] The text refers to 'test expressions', but no held-out test set is described; 'benchmark expressions' would be more accurate.
  6. [§5] The data-availability statement says code and data will be released upon acceptance; providing a public artifact or an anonymous repository with the submission would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found: NOMTO's symbolic search is an empirical fitting procedure validated against external benchmark expressions and independently simulated PDE targets.

full rationale

I walked the claimed derivation chain—Sections 2.1–2.3 as empirical demonstrations, Section 4.2 as the surrogate-training step, and Section 4.3–4.4 as the graph optimization and expression extraction—and found no step that reduces to its own inputs by construction. The neural-operator surrogates are pre-trained on random Gaussian-mixture inputs with outputs computed from exact library operations (Section 4.2); this is supervised pretraining of subroutines, not a derivation of the target expressions, and the target expressions are not used to train the surrogates. The benchmark experiments compare against external Nguyen and Keijzer ground truths, and the PDE experiments use independently simulated heat and Burgers solutions with random initial conditions; the target PDEs are not inserted into the library or used to set the graph coefficients. The only author-overlapping citation is [23] for convolutional neural operators, but it is used as a standard architecture component and not as a load-bearing uniqueness theorem or as the justification that the discovered equation must hold. The main weaknesses are validity issues rather than circularity: all reported metrics are in-sample (e.g., Figure 2 and Appendix D report MSE on the training simulations), some recovered expressions have high R2 while being structurally wrong (Table 2: Nguyen-7, Nguyen-2), and NOMTO-CNO fails on Burgers (Eqs. 6–7). These concerns bear on generalization and reproducibility, but they do not make any derivation equivalent to its inputs. Therefore the appropriate circularity score is 0.

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

The central claim rests on the surrogate fidelity of pretrained neural operators, on the user choosing a library and graph depth that contain the target equation's operations, and on numerical derivatives being accurate. The method itself is a data-fitting procedure, so the recovered coefficients are fitted values, not derived predictions; the main circularity risk is milder: surrogates and target data share a Gaussian-mixture structure and PDE metrics are computed on the training simulations.

free parameters (5)
  • tanh projection clipping bound = [-tanh(1000), tanh(1000)]
    Chosen by hand to prevent gradient explosion; directly shapes surrogate predictions and final coefficient accuracy (Section 4.2).
  • l1/2 penalty smoothing constant a = 0.01
    Defines modified sparsity norm in Eq. (9); threshold for treating weights as nonzero, affects final expression compactness.
  • pruning schedule and fraction = prune 10% lowest-energy weights every 50 epochs after epoch 100
    Heuristic that determines which terms survive; minimal nonzero weight count is a tunable parameter left unspecified.
  • sparsity regularization coefficient lambda = not reported
    Weight on l1/2 penalty in Eq. (10) is not specified, yet it controls compactness of discovered expressions.
  • output clipping range = [-100, 100] for labels
    Applied to all target values; authors state it prevents precise coefficient recovery in high-magnitude expressions (Section 2.2).
assumptions (5)
  • domain assumption Neural operator surrogates trained on random Gaussian mixtures provide sufficiently accurate approximations of library operations on target data.
    Section 4.2 and 4.3; failures in f2/f4 and NOMTO-CNO Burgers indicate this assumption can fail.
  • domain assumption The library of operations and graph depth are chosen to contain the operations needed to express the target equation.
    Discovered models are restricted to user-provided library; Section 2.3 uses minimal libraries tailored to each PDE.
  • domain assumption Second-order central finite differences provide accurate reference derivatives.
    Used to compute the reference time derivatives and to train derivative surrogates; accuracy depends on grid resolution and solution smoothness.
  • ad hoc to paper Redefined versions of sqrt and ln in Eq. (8) preserve symbolic discovery capability for functions with undefined regions.
    Surrogates learn modified functions, and final symbolic expressions are validated against original benchmarks while extraction uses exact operations at defined points.
  • ad hoc to paper For Burgers, replacing surrogate multiplication with exact PyTorch multiplication does not change the discovered structure.
    Section 2.3.2 states multiplication operations were replaced with exact counterparts to accelerate optimization, but the effect on results is not validated.

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Cite this review

Pith. "Pith review of A Neural Operator-Based Approach to Symbolic Discovery of PDEs." pith.science (2026). https://pith.science/paper/GNFI6XZ4

@misc{pith2026250108086,
  author       = {Pith},
  title        = {Pith review of: A Neural Operator-Based Approach to Symbolic Discovery of PDEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GNFI6XZ4}},
  note         = {Machine review of arXiv:2501.08086}
}
read the original abstract

Discovering governing equations from data remains challenging when the underlying dynamics involve nonlocal differential operators, field interactions governed by auxiliary equations, or temporal memory effects. We propose Neural Operator-based symbolic Model approximaTion and discOvery (NOMTO), a framework that extends Equation Learner-type symbolic architectures by incorporating pretrained neural operators as nodes in the symbolic network. NOMTO represents candidate equations as sparse differentiable computational graphs that combine algebraic operations with fixed neural operator surrogates pretrained to approximate nonlinear operators. We evaluate the method on model-discovery problems involving nonlocal spatial operators, couplings mediated by auxiliary field equations, and temporal integral terms representing memory effects. The results show that NOMTO can recover compact governing equations containing nonlocal operator terms, thereby extending symbolic model discovery beyond libraries restricted to local derivatives and point-wise algebraic combinations.

Figures

Figures reproduced from arXiv: 2501.08086 by the authors.

Figure 1
Figure 1. The general implementation steps and architecture of the Neural Operator-based symbolic Model [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Heat equation rediscovery results: The top row presents the outcomes from NOMTO-FNO; while [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Construction of the Neural Operator Block. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Prediction MSE distributions within the simulation domain: NOMTO-FNO for [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

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