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REVIEW 3 major objections 5 minor 28 references

About rectified sigmoid function for enhancing the accuracy of Physics-Informed Neural Networks

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A rectified sigmoid makes single-hidden-layer PINNs an order of magnitude more accurate on ODEs.

desk verdict A correct and neat derivation of a piecewise-linear activation trick for small PINNs, but the headline accuracy claim rests on interpolation with undisclosed reference data and is broader than the Lorenz numbers support. read the letter →

arxiv 2412.20851 v1 pith:LKKLNPAV submitted 2024-12-30 math.NA cs.AIcs.LGcs.NAphysics.comp-ph

Vasiliy A. Es'kin , Alexey O. Malkhanov , Mikhail E. Smorkalov This is my paper · ORCID
classification math.NAcs.AIcs.LGcs.NAphysics.comp-ph MSC 68T0765L05
keywords rectifiedsigmoidphysics-informedneuralnetworkssinglehiddenlayerordinarydifferentialequationsPIDDinitializationneuron-by-neurontrainingharmonicoscillatorLorenzsystem
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 a small, explicit change to the activation function — replacing the smooth sigmoid with a piecewise-linear "rectified sigmoid" — improves the accuracy of single-hidden-layer physics-informed neural networks on ordinary differential equations by at least an order of magnitude. The reported gains appear in three very different test systems: a linear harmonic oscillator, a relativistic electron slingshot, and the chaotic Lorenz system. If the claim holds, accurate physics-informed solving no longer needs deep networks or gradient-based training; a closed-form initialization followed by a few gradient-free fitting passes would suffice. The proposed initialization is "physics-informed data-driven" (PIDD), and the fitting method is "neuron-by-neuron" (NbN), both adapted from the sigmoid version of the method.

What carries the argument

The load-bearing object is the rectified sigmoid activation, defined piecewise as $\mathrm{Re}\text{-}\sigma(x)=0$ for $x<-1$, $\tfrac12(x+1)$ for $-1\le x\le 1$, and $1$ for $x>1$, which is a sum of two ReLU functions: $\mathrm{Re}\text{-}\sigma(x)=\tfrac12[\mathrm{ReLU}(x+1)-\mathrm{ReLU}(x-1)]$. Because its derivative is the box function with $\mathrm{Re}\text{-}\sigma'(0)=1/2$ and with the chosen spacing $\Delta\zeta=1$ the derivative of one neuron does not overlap the neighbouring neuron, the initialization formula collapses to $W^{(2)}_k = -\Delta x\, N_l[u_k,x_k]/\Delta\zeta$, a direct closed-form expression for the output weights from the ODE residual at grid points. This converts the physics equations into an explicit, gradient-free construction of the network.

What would settle it

Take a new ODE with a known exact solution, initialize the Re-$\sigma$ network without supplying any grid values $\{u_k\}$ — setting $W^{(2)}_k$ only from the ODE residual and the initial condition — and compare against the sigmoid network. If the Re-$\sigma$ network no longer beats sigmoid by an order of magnitude, the reported gain depends on the precomputed solution points.

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Extended reading notes

Core claim

The central claim is that substituting the sigmoid activation with the rectified sigmoid $\mathrm{Re}\text{-}\sigma(x) = 0$ for $x<-1$, $\tfrac12(x+1)$ for $-1\le x\le 1$, and $1$ for $x>1$ reduces the relative $L^2$ error of the predicted ODE solution by at least an order of magnitude for any given value, after the same initialization and training procedures. The paper reports this for PIDD initialization and for NbN training (three epochs), on the harmonic oscillator, the relativistic slingshot, and the Lorenz system. In the harmonic-oscillator case the Re-$\sigma$ network's error is close to the error of the standard high-precision reference solver itself.

Load-bearing premise

The initialization algorithm needs the solution values $\{u_k\}$ at the grid points before training, and the paper never states where they come from; if they come from the same reference solver used to measure error, the reported accuracy may reflect interpolation of that solver's output rather than physics-informed prediction.

Editorial extensions

If this is right

  • Single-hidden-layer networks with the rectified sigmoid reach errors near $10^{-6}$ on the harmonic oscillator, close to the reference solver's own error in the reported setup.
  • The reported order-of-magnitude gap over the sigmoid appears both after initialization alone (PIDD) and after the gradient-free NbN training, and across linear, nonlinear, and chaotic systems.
  • Training stays cheap: the PIDD runs in milliseconds and the NbN fitting in a few minutes in the reported experiments, with three epochs sufficient.
  • Because the same algorithms transfer from sigmoid to Re-$\sigma$ with only the spacing constant $\Delta\zeta$ changed, the method applies directly to the earlier single-hidden-layer toolbox.

Reading between the lines

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

  • The paper compares Re-$\sigma$ only against the sigmoid; a direct comparison with plain ReLU under the same PIDD initialization would isolate whether the benefit comes from the rectified shape or from the smooth central slope, an experiment the paper does not run.
  • The 'any given value' strength of the claim rests on three systems at two widths; a width and parameter sweep would show whether the gap persists for very small networks and for PDEs.
  • Because the output weights are set by the residual formula, the scheme could be used for fast parameter sweeps or surrogate modeling of a family of ODEs by re-initializing with new residuals, an application left implicit.
  • If the solution values consumed by the PIDD algorithm come from a reference solver, the reported errors measure interpolation of that solver's output; a natural test is to run the same initialization with the grid values withheld and watch whether the gap survives.
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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

3 major / 5 minor

Summary. The paper proposes a rectified sigmoid activation function Re-sigma (Eq. 4) for single-hidden-layer physics-informed neural networks solving ODEs, together with a physics-informed data-driven (PIDD) initialization (Algorithm 1) and a neuron-by-neuron (NbN) training procedure (Algorithm 2) adapted from previous work. The main technical claim is that, because Re-sigma has compact support and Delta-zeta = 1, each neuron affects the derivative only at its own collocation point, so the output weights are determined exactly by the ODE residual, as shown in Eqs. (6)-(8). The numerical experiments in Section 3 compare relative L2 errors of networks with sigmoid and Re-sigma activations on a harmonic oscillator, a relativistic slingshot, and a Lorenz system, and Section 3.2 claims an order-of-magnitude error reduction for any given value. The central unresolved issue is the provenance of the solution values {u_k} consumed by Algorithm 1, which the paper never states.

Significance. If the central claim were fully established, the paper would offer a practically useful, almost parameter-free way to obtain accurate ODE solutions from tiny single-hidden-layer networks, with a simple and elegant derivative construction. The Section 2 derivation is correct as far as it goes: the compact-support argument makes Eq. (8) exact at the grid points, and the explicit algorithms make the method easy to reproduce. However, the claimed superiority over the sigmoid activation is only as strong as the source of the input data in Algorithm 1; if {u_k} are the reference solver values, the reported errors are interpolation errors, not physics-informed solution errors. The paper also does not quantify uncertainty, but because the algorithms are deterministic this is less important than the data-source issue.

major comments (3)
  1. [Algorithm 1; Section 3.2; Eq. (8)] The PIDD initialization in Algorithm 1 requires the solution values {u_k} at every collocation point, and Eq. (8) sets W_m^(2) = -Delta x N_l[u_m, x_m]/Delta zeta from those values. The paper never states where these values come from; in Section 3 the only numerical reference mentioned is scipy's odeint solver, used for the 'reference solutions'. If {u_k} are the odeint outputs, then the networks in Table 1 are constructed by interpolation of the reference trajectory, not by solving the ODE from the initial condition alone, and the comparison with the sigmoid activation is a comparison of two interpolation bases. Because Section 3.2's order-of-magnitude claim rests entirely on Table 1, this missing provenance is load-bearing. Please state the source of {u_k} explicitly, or re-run the PIDD experiments using only the initial condition and the ODE residual.
  2. [Section 3.2 vs. Section 3.3, Tables 1 and 2] The claim in Section 3.2 that replacing sigmoid with rectified sigmoid 'results in a reduction of the relative error by at least an order of magnitude for any given value' is not supported by the NbN results: in Table 2, for the Lorenz system the Re-sigma relative errors are only about 1.3 times smaller than the sigma errors (e.g., 2.63e-3 vs 3.47e-3 for x). If the claim is meant to cover only the PIDD initialization, the text should say so; if it is meant to cover the proposed method generally, the statement needs to be revised to match Table 2.
  3. [Section 3.2, harmonic oscillator paragraph] The observation that Re-sigma's relative L2 errors for the harmonic oscillator (3.88e-6 and 2.63e-6) are close to the odeint-vs-exact errors (1.03e-6 and 1.04e-6) is consistent with the interpretation that the Re-sigma network is reproducing the odeint reference solution to the accuracy of that solver. The paper should state whether the reported errors therefore measure the network's fit to the reference data, and should discuss how this relates to the claim that the network 'solves' the ODE.
minor comments (5)
  1. [Throughout] The system in Section 3.1.3 and in Tables 1 and 2 is called 'Lorentz system'; the standard name is 'Lorenz system' (also used in Eq. (13)).
  2. [Section 2, Eq. (6)] There is a typo: 'derivate' should be 'derivative', and the phrase 'Pay attention Re-sigma'(0) = 1/2' needs a comma or semicolon.
  3. [Algorithm 1 vs Eq. (8)] The subscript l is used in Eq. (8) (N_l[u_m, x_m]) but omitted in Algorithm 1's W_k^(2) update; make the component index consistent.
  4. [Algorithm 2] The initialization in Algorithm 2 uses N_l[u(0), x_k] but the notation u(0) is ambiguous for a system: it should be stated whether this means the constant initial vector evaluated at every collocation point, and how this differs from the correction term b0 used in Algorithm 1.
  5. [Section 3.3, Figure 1] The text says the absolute errors are 'significantly lower' than those of vanilla PINN, but no definition, settings, or results for vanilla PINN are given; either cite the specific comparison in [21] or add the missing details.

Circularity Check

1 steps flagged · score 6.0 of 10

PIDD initialization consumes the reference solution as input, so the Section 3.2 'order-of-magnitude' comparison measures in-sample quadrature accuracy, not independent PINN prediction.

  1. fitted input called prediction [Algorithm 1 (Data line, Eq. 8) and Section 3.2 (Table 1, reference-solution sentence)]
    "Data: {u_k}_{k=0}^{N-1} for {x_k}_{k=0}^{N-1} ... W_k^(2) ← −Δx/Δζ N [u_k, x_k]; ... The reference solutions were obtained using the odeint solver of scipy.integrate library. ... Column 3 (relative L2 errors) shows the relative errors between solutions, provided by neural networks and solutions obtained using odeint solver of scipy.integrate library."

    Algorithm 1 sets every second-layer weight from N[u_k, x_k] at every collocation point, so the solution values u_k must be known before the network is constructed. The only solution values named in Section 3 are the odeint reference values. With the Re-σ tent basis (W_k^(1)=2/Δx, b_k^(1)=−2k), the initialized network output at x_m is a trapezoidal quadrature of the derivative field built from those same u_k values. Table 1 then reports error against the odeint values that supplied Eq. 8. The network is thus constructed from the reference trajectory; the error measures quadrature accuracy of a data-driven fit, not independent PINN solving. The σ-vs-Re-σ comparison compares two interpolation bases, so the 'order of magnitude' sentence is an in-sample interpolation claim.

full rationale

The main circularity is concentrated in the PIDD experiments. The algorithm's input is the solution grid {u_k}; the experiments' only named source of such values is the odeint reference solution, and the accuracy metric in Table 1 is the deviation from that same reference. Because the Re-sigmoid basis has disjoint local support on the grid (each neuron activates only on its own cell and the neighboring cell), Eq. 8 turns the initialization into a quadrature formula for the reference derivative, so the 'prediction' is largely determined by the reference values themselves. This does not make the comparison between σ and Re-σ meaningless as a study of basis functions, but it means the strong claim in Section 3.2 ('reduction of the relative error by at least an order of magnitude for any given value') is not a demonstration that Re-sigmoid PINNs solve ODEs better from first principles. The NbN results (Table 2, Algorithm 2) are not circular in the same way: Algorithm 2 states 'Data: —' and updates weights from the network's own current solution, so those experiments provide partially independent support, although for the chaotic Lorenz case the Re-σ advantage is only about 1.3×, not an order of magnitude. No load-bearing self-citation or uniqueness-import pattern was found; Algorithms 3 and 4 are reproduced in the appendices, so the comparison rests on the paper's own equations rather than on [21] as an unverified authority. Overall, because the central quantitative claim rests on a data-driven initialization evaluated on the same data, the circularity score is 6.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central derivation has no fitted physical parameters, but it depends on several unstated assumptions. The most serious is the availability of the initialization data {u_k}; without knowing their provenance, the error tables may measure interpolation rather than solution. The other entries are standard numerical-analysis assumptions (reference solver accuracy) and hand-chosen hyperparameters (N, E, time windows). The rectified sigmoid is treated by the paper as a new proposed entity, but it is the standard hard sigmoid function, so it is not counted as an invented entity here.

free parameters (3)
  • Number of hidden neurons / collocation points N = 20000
    Chosen by the authors without sensitivity analysis; errors likely improve with N, so the reported gain may depend on this specific value.
  • Number of training epochs E (NbN) = 3
    Stated in Section 3.3 ('it was enough to use 3 epochs'); no study showing convergence or independence.
  • Number of time windows for NbN = 20
    The time domain is split into 20 disjoint windows of size T/20, chosen by hand.
assumptions (4)
  • domain assumption The ODE initial value problems are well-posed and have unique solutions on [0,T].
    The three test problems (harmonic oscillator, relativistic slingshot, Lorenz system) are assumed to have unique solutions that the networks approximate.
  • domain assumption The reference solutions from scipy.integrate.odeint are accurate enough to serve as ground truth.
    All relative L2 errors are computed against odeint; for the chaotic Lorenz system, reference errors are never quantified.
  • ad hoc to paper The input data {u_k} for the PIDD initialization are available and consistent with the ODE.
    Algorithm 1 requires the solution values at all grid points as input, but the paper never specifies their origin; this is the main circularity risk.
  • ad hoc to paper Three epochs of neuron-by-neuron updates suffice for convergence.
    No convergence analysis or epoch-scaling experiment is provided; the claim that E=3 gives 'high-precision results' is an empirical assertion.

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Cite this review

Pith. "Pith review of About rectified sigmoid function for enhancing the accuracy of Physics-Informed Neural Networks." pith.science (2026). https://pith.science/paper/LKKLNPAV

@misc{pith2026241220851,
  author       = {Pith},
  title        = {Pith review of: About rectified sigmoid function for enhancing the accuracy of Physics-Informed Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LKKLNPAV}},
  note         = {Machine review of arXiv:2412.20851}
}
read the original abstract

The article is devoted to the study of neural networks with one hidden layer and a modified activation function for solving physical problems. A rectified sigmoid activation function has been proposed to solve physical problems described by the ODE with neural networks. Algorithms for physics-informed data-driven initialization of a neural network and a neuron-by-neuron gradient-free fitting method have been presented for the neural network with this activation function. Numerical experiments demonstrate the superiority of neural networks with a rectified sigmoid function over neural networks with a sigmoid function in the accuracy of solving physical problems (harmonic oscillator, relativistic slingshot, and Lorentz system).

Figures

Figures reproduced from arXiv: 2412.20851 by the authors.

Figure 1
Figure 1. Over the all-time interval, the absolute errors for all values obtained by such neural networks by NbN training [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 1
Figure 1. (a), (b), (c) and (d) are comparison of the predicted (red dash lines) and reference solutions (blue solid lines) [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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