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REVIEW 4 major objections 6 minor 49 references

NestyNet. I. Physics Functions Are Hard to Fit with Neural Networks: A Framework for Accurate Surrogates and Analytic Derivatives

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read On the 120-equation AI Feynman benchmark, NestyNet reports median 2,100x better function values, 1,400x better first derivatives, and 780x better second derivatives than Adam-trained MLPs, with all derivatives analytic.

desk verdict Solid, well-engineered framework with believable analytic-derivative gains; headline factors need a harder baseline and number reconciliation before I'd quote them. read the letter →

arxiv 2608.05862 v1 pith:7ER3CP7R submitted 2026-08-06 astro-ph.IM physics.data-an

Rodrigo Ibata , Wassim Tenachi , Foivos Diakogiannis , Neil Ibata , Anirudh Shankar This is my paper · ORCID
classification astro-ph.IMphysics.data-an
keywords neuralnetworkssurrogatemodelsLevenberg-MarquardtanalyticderivativessoftplussegmentsLaplacianAIFeynmanbenchmarkGalacticdynamics
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

Many smooth functions that matter in physics are exactly the ones standard neural networks fit badly, and this paper's claim is why: the loss landscape induced by stiff, multiscale targets is full of strongly correlated, sloppy parameter directions that first-order optimizers like Adam cannot traverse efficiently. The remedy is a coupled model-and-optimizer, NestyNet, built from a segmented softplus surrogate whose gradients, Hessians, Laplacians, and antiderivatives are closed-form, trained by a tailored Levenberg-Marquardt scheme. On the 120-equation AI Feynman benchmark the framework reports median improvements of 2,100x for function values, 1,400x for first derivatives, and 780x for second derivatives over Adam-trained MLPs, with 540x, 450x, and 250x remaining after L-BFGS refinement, and up to roughly 44x faster than vectorized automatic differentiation. The paper also derives an interpolation law: for fitted errors that are spectrally narrow-band, gradient error is the geometric mean of value and curvature errors, which turns value accuracy plus curvature control into a certificate for derivative accuracy. A sympathetic reader would care because trustworthy derivatives are what make surrogates usable for physics-informed learning, equation discovery, and inverse problems.

What carries the argument

The central object is the segmented softplus network, $f_o(x)=\sum_s a_{os}\sigma(z_{os})$ with $\sigma(z)=\log(1+e^z)$ and each preactivation $z_{os}=K_{os}\cdot x+b_{os}$ affine in the inputs; the softplus is a smooth hinge, so the model is a sum of smooth pieces that can represent multiscale structure while remaining twice differentiable. All differential operators descend from the gate factor $g=\sigma'(z)=1/(1+e^{-z})$: gradients are $\sum a\,g\,K$, Hessians are $\sum a\,g(1-g)\,KK^T$, and the Laplacian collapses to $\sum a\,g(1-g)\,\|K\|^2$, so it costs $O(SWN_x)$ instead of forming a dense Hessian. The same softplus atom sits on the polylogarithm ladder, since $\sigma(z)=-\mathrm{Li}_1(-e^z)$ and $d/dz\,\mathrm{Li}_\nu(-e^z)=\mathrm{Li}_{\nu-1}(-e^z)$, which supplies exact antiderivatives along lines and over boxes. The optimizer is a predictive Levenberg-Marquardt scheme that orders parameters by output-segment, exploits the exact split of $J^T J$ into within-segment blocks and cross-segment coupling, starts from a deterministic canonical initialization built on the identity $\sigma(z)-\sigma(-z)=z$, and adds quadratic path extrapolation, Woodbury fast solves, and an L-BFGS rescue for stagnation.

What would settle it

Train a substantially stronger neural baseline on the same 120 AI Feynman equations and data splits, for example a residual network with several hidden layers of 256 units each, well-tuned initialization, and full-batch L-BFGS; if its median function and derivative errors come within a small factor of NestyNet's, the central improvement factors shrink proportionally and the paper's premise that geometry rather than capacity is the bottleneck would be called into question.

Watch

Extended reading notes

Core claim

The central claim is that the difficulty in fitting many physics functions with neural networks is not representational power but parameter-space geometry: the right parameters exist, but stiff, strongly correlated loss landscapes make them hard to find with first-order training. NestyNet meets that geometry with a deterministic segmented softplus architecture and a predictive second-order optimizer, and the paper reports that on the AI Feynman suite this combination lowers median function-value error by 2,100x, first-derivative error by 1,400x, and second-derivative error by 780x relative to a fixed two-hidden-layer tanh MLP trained by Adam, with gains of 540x, 450x, and 250x after L-BFGS refinement. Because every derivative is analytic, the improved accuracy does not come at the cost of speed: end-to-end Levenberg-Marquardt steps run up to about 44x faster than a vectorized automatic-differentiation baseline. The same machinery makes gradients, Hessians, Laplacians, and line or box antiderivatives available in closed form, and the paper demonstrates the payoff on PDE residuals and on recovering the vertical acceleration of a model Galactic disk from phase-space densities.

Load-bearing premise

The headline accuracy gains are measured against one fixed baseline, a two-hidden-layer tanh network with 64 units per layer trained in double precision, so the claim assumes that this baseline fairly represents how standard neural networks perform on physics functions.

Editorial extensions

If this is right

  • Derivative-informed tasks inherit the accuracy gains: physics-informed residuals, Hamiltonian dynamics, and inverse-force estimation get closed-form Hessians, Laplacians, and mixed partials, and operator probes show the gain holds for directional derivatives, curvature contractions, and Laplacians.
  • Value accuracy cannot be converted into gradient accuracy without simultaneous curvature control; the interpolation law makes that coupling explicit and gives a capacity-selection rule that needs no derivative data.
  • Surrogates can be integrated exactly: line, axis-parallel, and hyperrectangle integrals of the segmented model are polylogarithm evaluations, so integrating a fitted surrogate adds no numerical error of its own.
  • The framework extends beyond the segmented family: an automatic-differentiation adaptor feeds general neural models through the same predictive LM optimizer, and the matrix-free SPLA preconditioner keeps LM feasible at nearly $10^4$ parameters with near-linear scaling.

Reading between the lines

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

  • Beyond the benchmark, the flat cost of the analytic Laplacian in input dimension suggests high-dimensional PDE workloads such as Fokker-Planck, Schrödinger, or Hamilton-Jacobi equations could see the largest relative speedups, since autograd Laplacians cost one backward pass per dimension.
  • A testable diagnostic follows from the interpolation law: any fitted surrogate can report its spectral tightness ratio on held-out probes, and values well below 1 would signal broadband error where gradient accuracy cannot be inferred from value accuracy alone.
  • The exact polylogarithm antiderivatives could be used to integrate fitted densities without quadrature in downstream normalization or Monte Carlo tasks, a possibility the paper does not develop.
  • If the gains are representative of physics functions more broadly, the same coupled surrogate-optimizer recipe could improve equation-discovery pipelines that depend on stable mixed partials, rather than only pointwise fits.
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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 introduces NestyNet, a deterministic segmented softplus neural-network surrogate whose analytic derivatives (gradients, Hessians, Laplacians) and antiderivatives are available in closed form, coupled with a predictive Levenberg–Marquardt optimizer, a segment-native preconditioning framework (SPLA), and a compositional adaptor ecosystem. The central empirical claims are benchmarked on the 120-equation AI Feynman suite: median improvements over a fixed two-hidden-layer tanh MLP trained with Adam of about 2,100x on function values, 1,400x on first derivatives, and 780x on second derivatives, with smaller but still large factors after L-BFGS refinement; an end-to-end speed advantage over vectorized autograd reaching about 44x; two PINNacle PDE wins plus an expressivity study; and an application to recovering the vertical acceleration of the Galactic disk from the collisionless Boltzmann equation.

Significance. If the headline comparisons survive a more rigorous baseline protocol, NestyNet would be a practically valuable substrate for scientific machine learning: the analytic derivative and antiderivative machinery is correct and cheap, the canonical initialization gives reproducible fits, and the benchmark protocol is mostly disciplined (shared splits and transforms, a single declared suite, no per-equation refits, and row-level provenance with the code release). The paper is also commendably explicit about what it does not claim: SPLA is not needed for the main results, the supervised PDE fits are not PINN solutions, and the interpolation law is an empirical regularity rather than a theorem. The main weakness is that the quantitative magnitude of the claimed gains rests on a single unsearched MLP baseline and on single-run medians, so the improvement factors should be treated as upper-bound estimates until the baseline is strengthened or the claims are narrowed.

major comments (4)
  1. [§8.3, Figure 2] The headline improvement factors are ratios against a single fixed baseline: a two-hidden-layer tanh MLP with 64 units per layer, trained in double precision. No hyperparameter search, learning-rate schedule, architecture variants, restarts, or training budgets are reported for this baseline, and its parameter count is not given. Because the abstract's 2,100x/1,400x/780x and 540x/450x/250x claims are measured against this one network, the representativeness of the baseline is load-bearing. Please either add a small, declared baseline-selection protocol on the same 120 equations (for example, a random search over width, depth, activation, and learning rate with the same splits and transforms) and report the resulting distribution of improvement factors, or explicitly restrict all headline claims to 'the fixed 64x64 tanh MLP baseline used here.' Without this, the quantitative magnitude of the claimed gains is not established.
  2. [Abstract; §8.3–§8.4, Figures 2–3] The numbers in the abstract do not match the numbers in the figures. The abstract quotes median improvements of 2,100x/1,400x/780x for Adam and 540x/450x/250x for L-BFGS, while Figure 2 reports 2,016x/1,572x/872x and 593x/475x/271x, and §8.4 reports yet another set (2,016x/960x/1,466x) for its RMS-norm analysis. Similarly, the integral gain is 5,260x in Figure 2 but roughly 4,800x in the conclusions. The source of the abstract values is not explained. Since these medians are the paper's central quantitative claims, please harmonize the values and state exactly which per-axis metrics, panels, and norm definitions produce each quoted factor.
  3. [§8.3, Figures 2–3] All improvement factors are single-run medians with no repeated-seed variability. The MLP baseline is randomly initialized, and no number of seeds, standard deviations, or per-equation variability is reported; the deterministic canonical initialization of NestyNet does not by itself determine the baseline's noise. Please report multiple seeds for the baseline (and ideally for NestyNet too) or provide a clear justification for why the medians are insensitive to seed choice, so the reader can judge whether the 98–100% win rates and the factor magnitudes are stable.
  4. [§10; §8.2–§8.3] The conclusion states that 'all of these results were obtained with a single, fixed hyperparameter configuration applied across the entire benchmark,' but the numerical sections use different model sizes and recipes: §8.2 crosses two model families and four training recipes, §8.3 uses dual-layer 20-segment models, and §8.7 sweeps segment counts up to 8,960 parameters. Please specify precisely which hyperparameters were fixed across which benchmark suite and which ones varied by design; as written, the reproducibility claim is ambiguous.
minor comments (6)
  1. [Figures 2–3] The caption says color encodes the benchmark equation index, but no color bar is shown; please add a color bar or remove the color encoding.
  2. [Table 2] The column headings appear garbled ('W all (s)', 'V al.'); please typeset them properly and define each column in the caption.
  3. [§8.5, footnote 8] The statement that torch.compile 'silently miscompiles the forward-mode autograd Jacobian to zero' is surprising and affects the choice of eager autograd as the baseline; please cite a PyTorch issue or include a minimal reproducer in the released code.
  4. [§8.2 vs §8.3] Section 8.2 retains 88 problems after excluding targets with dynamic range above four decades, while §8.3 reports results for all 120 problems with an asinh conditioning policy; please add one sentence explaining how the two protocols relate.
  5. [§5.2] The Armijo backtracking along the LM direction is mentioned, but no default constants or safeguarding details are given; please provide the Armijo parameters or a citation.
  6. [§8.4, Eq. (41)] The curvature-budget inequality is stated without derivation; a one-line derivation from Eq. (8) and g(1−g)≤1/4 would help readers verify the bound.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are external benchmark comparisons and independent empirical evaluations, and self-citations are peripheral.

full rationale

The paper's headline results (AI Feynman improvement factors, PINNacle transfers, derivative-speed comparisons, and the galactic-acceleration demonstration) are empirical evaluations against external benchmarks and independent baselines, not derivations that assume their conclusions. The derivative-gap 'interpolation law' is a classical Landau–Kolmogorov inequality (acknowledged as such, with citations to Hardy et al. 1952) followed by an empirical near-equality check; the tightness ratio is measured, not fitted, and the prediction of gradient gains from value and curvature gains is an algebraic consequence of the inequality, not a fitted parameter renamed as a prediction. Self-citations to the SPLA theory paper and to Papers II-IV are not load-bearing: the SPLA speedup in Figure 6 is a measured wall-clock comparison, and the cited theory paper supplies explanation rather than the evidence for the empirical result; Papers II-IV are downstream applications. Potential concerns about the fixed MLP baseline are fairness and representativeness issues, which are correctness risks, not circularity: no equation, parameter, or prediction in the paper is defined in terms of the target conclusion it is used to support. The paper is self-contained against external benchmarks, so the appropriate circularity score is 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central results rest on standard facts (Cauchy-Schwarz, polylogarithm identities) plus the empirical narrow-band error premise and the domain representativeness of the benchmarks. No target result is assumed in the derivation; the main external assumption is that the chosen baselines represent standard neural network surrogates.

free parameters (4)
  • Segment count S = 20 (dual-layer AI Feynman), 32 (PDE), 4 (Galactic disk)
    Architecture capacity chosen by the authors; central benchmark uses 20-segment dual-layer and 32-segment single-stage for PDE without per-equation tuning, while the Galactic disk count is selected by the derivative-space capacity rule.
  • N_history (quadratic extrapolation history) = 10
    Fixed default in Sec 5.4; not tuned per problem.
  • L-BFGS rescue parameters = 5 rejected steps trigger; 10 curvature pairs; 20 max iterations; 10 max attempts
    Fixed defaults in Sec 5.6; algorithmic constants, not data-fitted.
  • Patch family set F = {mean, slope, quad}
    Regularization design choice in Sec 5.3; the evidence loop sets the strengths, but the family set is fixed.
assumptions (5)
  • standard math Proposition 1 (interpolation inequality between L2 norms of value, gradient, and Hessian)
    Invoked in Sec 8.4 to explain the derivative gap; the inequality itself is classical Cauchy-Schwarz, but its use as a prediction assumes narrow-band error.
  • standard math Polylogarithm identities d/dz Li_nu(-e^z) = Li_{nu-1}(-e^z)
    Used in Sec 3.2 to obtain antiderivatives.
  • domain assumption Narrow-band error premise (rho_e ~ 1)
    Empirical claim in Sec 8.4 that converged fits have error spectra concentrated near one wavenumber; required for the gradient-error bound to be a prediction.
  • domain assumption AI Feynman benchmark equations are representative of physics functions
    Used to generalize benchmark results to 'physics functions are hard to fit'.
  • domain assumption Collisionless Boltzmann equation reduces to v d_z ln f + K_z d_v ln f = 0 for stationary phase-mixed populations
    Sec 9, used for the vertical acceleration estimator.

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Cite this review

Pith. "Pith review of NestyNet. I. Physics Functions Are Hard to Fit with Neural Networks: A Framework for Accurate Surrogates and Analytic Derivatives." pith.science (2026). https://pith.science/paper/7ER3CP7R

@misc{pith2026260805862,
  author       = {Pith},
  title        = {Pith review of: NestyNet. I. Physics Functions Are Hard to Fit with Neural Networks: A Framework for Accurate Surrogates and Analytic Derivatives},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ER3CP7R}},
  note         = {Machine review of arXiv:2608.05862}
}
abstract

Many of the smooth functions that matter most in physics are precisely the ones that standard neural network methods struggle to fit accurately. Here we present NestyNet, a coupled model-and-optimizer framework capable of fitting such targets to high accuracy while also delivering their gradients, Hessians, Laplacians, and antiderivatives analytically and at low cost. This makes it a natural substrate for scientific machine learning tasks. The model is a deterministic segmented analytic surrogate, and its optimizer is a second-order Levenberg--Marquardt scheme whose damping and linear solves are tailored to the stiff, strongly correlated parameter geometries induced by multiscale and sharply structured targets typical in physics and other scientific applications. On the AI Feynman benchmark of 120 physics equations, NestyNet achieves median improvement factors of $2\,100\times$ for function values, $1\,400\times$ for first derivatives, and $780\times$ for second derivatives relative to standard neural networks trained with first-order optimization (Adam). Even after refining those fits with quasi-Newton (L-BFGS) optimization, the corresponding improvements are $540\times$, $450\times$, and $250\times$. Owing to the analytic design it is up to $\approx 44\times$ faster than vectorized automatic-differentiation (autograd) baselines, with a margin growing with model size. The same analytic-derivative framework also supports vector- and complex-valued targets, measurement uncertainties in both inputs and outputs, and constraints, and its modules can be composed flexibly to build scientifically useful model architectures, all without reverting to autograd. Together, these components provide a practical modular framework for fitting difficult scientific surrogates while delivering accurate differential operators for subsequent analysis.

Figures

Figures reproduced from arXiv: 2608.05862 by the authors.

Figure 1
Figure 1. Optimizer comparison on the same 960-param￾eter (dual-layer) segmented model fit to AI Feynman #29, from identical initialization and with the same full-batch for￾ward-MSE objective. Dotted lines show training MSE and solid lines show validation MSE, and the legend lists each method’s total optimizer wall-clock time. Predictive LM reaches the lowest error by a wide margin (2.27 × 10−6 best validation MSE in 688 s, v… view at source ↗
Figure 2
Figure 2. Direct MLP-vs-NestyNet error comparison across the benchmark suite. Circles show Adam-trained MLPs, while triangles show full-batch L-BFGS runs initialized from the corresponding Adam solution. The dashed line marks parity, so points below the line favor NestyNet. For the derivative and integral panels, each plotted point is the per-equation median over input axes. The panels show relative L 2 errors for values, fir… view at source ↗
Figure 3
Figure 3. Operator-focused MLP-vs-NestyNet comparison across the benchmark suite. Symbols as in [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The derivative-gap interpolation law across the AI Feynman suite. Panel (a): measured gradient error ∥∇e∥L2 of the fit error e = ˆf − f against the interpolation prediction (∥e∥L2 |e|H2 ) 1/2 of Proposition 1, for all 120 equations and all three model families (Adam ML…
Figure 5
Figure 5. Figure 5: Analytic versus autograd cost, end to end. Left: LM-step runtime versus segment count for one-layer segmented and two-layer dual-segmented models fit to AI Feynman problem #29 (top), and the speedup relative to a vectorized functorch jacfwd autograd baseline (bottom). …
Figure 6
Figure 6. Figure 6: End-to-end runtime comparison of three inner LM solvers on AI Feynman problem #11, the five-input tar￾get x0 [x1 + x2x3 sin x4], as the single-layer segment count grows. Each solver is run from identical initializations to the same training loss of 10−3 . The top panel…
Figure 7
Figure 7. Figure 7: Statistical recovery of the Galactic vertical acceleration and its derivative tower on a mock isothermal slab. (a) The certified-gradient estimator of Eq. (43) in 10 bins across the core |z| ≤ a, from 5 × 105 stars. The inner bars are the split-half statistical uncerta…
Figure 8
Figure 8. Figure 8: The population hierarchy. Top: three mock tracer populations of dispersion σp = 0.7, 1, 1.4 (in units of the potential’s σ) in one shared potential, each recovering Kz(z), with the stacked shared-force solution (squares, er￾ror bars showing the cross-population scatter…

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