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REVIEW 4 major objections 5 minor 1 cited by

Are Two Hidden Layers Still Enough for the Physics-Informed Neural Networks?

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper argues that one-hidden-layer physics-informed networks, started from a deterministic Euler-like initialization and trained with special loss weighting, match or beat deeper baselines on a range of ODE and PDE benchmarks.

desk verdict A bag of useful tricks for shallow PINNs, but the headline SOTA claim doesn't survive the paper's own Table 7. read the letter →

arxiv 2412.19235 v1 pith:CLLVJRXS submitted 2024-12-26 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 68T0765L0565M70
keywords physics-informedneuralnetworksshallowdeterministicinitializationseparablePINNneuron-by-neurontraininglossweightingLorenzsystemAllen-Cahnequation
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 very shallow physics-informed neural networks—one hidden layer for ordinary differential equations and a separable product of two single-hidden-layer networks for partial differential equations—are enough to solve a range of physical benchmark problems accurately, provided the weights are initialized deterministically from the differential operator and the loss is weighted with care. The central move is a strictly deterministic initialization that mimics the Euler integration step: hidden weights and biases are fixed so that each sigmoid neuron is centered at one grid point, and output weights are set from local derivative matching, so the initial network already behaves like a first-order numerical integrator. Around that starting point the paper develops training modifications—detaching nonlinear right-hand sides, second-derivative weighting, predicted-solution weighting, relative residuals, causal weighting, and gradient normalization—and a gradient-free neuron-by-neuron fitting procedure. On tests including the harmonic oscillator, charged-sphere potential, relativistic slingshot, exponential growth, the Lorenz system, Allen–Cahn, and a fast advection equation, these methods reach relative L2 errors as low as 1e-5 to 1e-6 and, in the Lorenz case, beat the causal-training baseline cited in the paper. If true, this matters because shallow networks avoid vanishing gradients, train quickly, and are far easier to interpret than the deep PINNs usually deployed.

What carries the argument

The load-bearing object is the strictly deterministic initialization (Algorithm 2). It fixes the hidden-layer weights to $W_k^{(1)}=2\Delta\zeta/\Delta x$ and biases to $b_k^{(1)}=-2(k+1)\Delta\zeta$, and sets the output-layer weights from the local derivative-matching identity $W_k^{(2)}=-2\Delta x\,N_l[u(0),x_{k+1}]/\Delta\zeta$, with the output bias equal to the initial value; the parameter $\Delta\zeta=\ln(2+\sqrt{3})/2$ is chosen so that the third derivative of neighboring sigmoids cancels at each neuron center. This construction makes the initial network an Euler-like piecewise-sigmoid integrator, so the subsequent training stages only refine a good starting representation. The separable extension for PDEs, $u_{\theta}(t,x)=\sum_j v_j(t)w_j(x)$, is the second central object: it lets the same one-hidden-layer construction be applied along each coordinate and combined multiplicatively, which is what the paper calls a two-hidden-layer SPINN.

What would settle it

Freeze the hidden layer after Algorithm 2 on an ODE with a known analytic solution, compute the full derivative sum in Eq. (14) at every grid point, and compare it to the one-neuron approximation in Eq. (15). If the relative mismatch is large, say above 10%, on any part of the domain—especially with $\Delta\zeta=\ln(2+\sqrt{3})/2$—then the localization assumption that underlies the output-weight formula is broken, and the claimed source of the method's accuracy is not present.

Watch

Extended reading notes

Core claim

The paper's central claim is that the expressive power of deep PINNs is not needed for the problems considered: a single-hidden-layer network, whose hidden weights and biases are fixed rather than learned and whose output weights are set from the ODE residual at grid points, can be trained to competitive accuracy, and its product with a second single-hidden-layer network (the separable SPINN architecture) extends the same construction to two-dimensional PDEs. The deterministic initialization is derived from the Euler method: with $W_k^{(1)}=2\Delta\zeta/\Delta x$ and $b_k^{(1)}=-2(k+1)\Delta\zeta$, the $k$th sigmoid's steepest response sits at $x_{k+1}$, and assuming that one neuron dominates the derivative at each grid point gives $W_k^{(2)}=-2\Delta x\,N_l[u(0),x_{k+1}]/\Delta\zeta$. The paper chooses $\Delta\zeta=\ln(2+\sqrt{3})/2\approx 0.66$ so that third-derivative interference between neighboring sigmoids cancels, and confirms experimentally that $\Delta\zeta\approx 0.7$ works best. On top of this initialization the paper adds loss-weighting schemes—detaching nonlinear operators, second-derivative weighting, predicted-solution-based weighting, relative residuals, δ-causal training, and gradient normalization—plus a data-driven variant called PIDD initialization and a gradient-free neuron-by-neuron fitting rule. The empirical payoff is reported as competitive accuracy across ODE and PDE benchmarks and, for the Lorenz system, errors below the causal-training baseline quoted in the paper.

Load-bearing premise

The load-bearing premise is that each sigmoid neuron responds mainly at its own grid point, so the network derivative at a point is essentially one neuron's contribution; if neighboring sigmoids overlap heavily, the deterministic start is not a faithful Euler-like approximation and training must repair a bad representation.

Editorial extensions

If this is right

  • For ODEs, the deterministic initialization alone improves the harmonic-oscillator relative L2 error by an order of magnitude over random output-layer initialization, from about 4.7e-3 to 7.7e-4, and the full training pipeline brings it to roughly 2.2e-4.
  • The combination of detaching, δ-causal weighting, second-derivative weighting, and gradient normalization turns the relativistic-slingshot problem from unusable errors into errors around 1e-3 to 1e-2, with relative L2 values such as 1.2e-3 for h and 1.3e-3 for x.
  • Predicted-solution-based weighting or relative-residual weighting extends accurate solutions to multi-scale regimes where plain training fails, e.g., exponential growth on [0,10] improves from 0.93 to 1.72e-3, and the Lorenz system on [0,20] reaches errors around 5e-4 to 8e-4.
  • The gradient-free neuron-by-neuron fitting method reaches Lorenz-system errors below the causal-training baseline reported in the paper (x: 9.8e-4, y: 1.4e-3, z: 6.0e-4) after three sweeps over neurons and without using an optimizer.
  • On the Allen–Cahn PDE, the two-hidden-layer separable SPINN with deterministic initialization and the full weighting stack reaches relative L2 error 4.40e-5, below the δ-causal training baseline of 6.29e-5 listed in the paper.

Reading between the lines

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

  • Editorial inference: the Euler-matching initialization effectively replaces random weight guessing with a discretization prior, so the trained network should inherit the convergence order and error profile of a first-order integrator; one testable consequence is that halving the grid step should roughly halve the error floor before optimization, which the paper's scaling plots hint at but do not a
  • Editorial inference: because only output weights are set analytically and hidden weights stay frozen during the first training stages, the method invites an extreme-learning-machine-style analysis; the paper notes this connection itself, but a rigorous approximation-rate bound for the sigmoid Euler construction is left open.
  • Editorial inference: the generalization measure introduced in the paper, which compares errors on training parameters versus half-shifted parameters, could be applied to any parameterized PINN, not just the separable construction; the low value in the chaotic Lorenz regime suggests the generalization claim should be read as confined to regular parameter regions.
  • Editorial inference: the same deterministic initialization could be extended beyond the tested equations to systems with stiffness or shocks by replacing the uniform grid with adaptive collocation points; nothing in the derivation requires uniformity, so adaptive grid spacing is a natural next experiment.
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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 / 5 minor

Summary. The paper proposes a set of initialization, weighting, and training techniques for physics-informed neural networks with one hidden layer, and for separable PINNs (SPINN) with two such layers, for solving ODEs and PDEs. The contributions include a 'strictly deterministic' initialization (Algorithms 1–2), modifications of causal training and gradient normalization, new loss-weighting schemes (detaching, second-derivative weighting, PSBW, RR), a physics-informed data-driven (PIDD) initialization that fits the network to precomputed reference data (Algorithm 3), a generalization metric for parameterized problems (Appendix B), a gradient-free neuron-by-neuron (NbN) fitting algorithm (Algorithm 5), and an extension to 2D problems using SPINN. The central claim, stated in the abstract and conclusions, is that these shallow networks achieve competitive accuracy and, in some cases, state-of-the-art results.

Significance. If the claims are substantiated, the paper would make a useful contribution by demonstrating that very shallow PINNs with carefully chosen initialization and training can be a cheaper alternative to deep architectures. The paper's strengths include a large experimental campaign, concrete algorithms that are easy to reimplement, and a deterministic initialization that removes random-seed variability. The NbN method is genuinely gradient-free and the generalization metric is a step toward quantifying interpolation behavior. However, the significance is diminished by the unsupported state-of-the-art claim, the circular evaluation of PIDD, and the lack of error analysis for the heuristic approximations at the core of the initialization; these issues prevent the current manuscript from supporting its strongest conclusions.

major comments (4)
  1. [Abstract, Section 6, Table 7] The claim that NbN training gives 'state-of-the-art results on the Lorentz system' is not supported by the evidence presented. Table 7 reports relative L2 errors for NbN training of (9.8e-4, 1.4e-3, 6.0e-4), which are 3-5 times larger than the PIDD initialization errors (2.0e-4, 6.7e-4, 1.3e-4) listed in the same table, and the only external baseline is 'Causal training [4]'. No comparison with other recent PINN methods for the Lorenz system, no seed or error-bar statistics, and no compute-matched comparison are provided. Since this is the only concrete instance of 'state-of-the-art' offered in the paper, the abstract's central claim rests on an internally contradicted and externally unbenchmarked result.
  2. [Section 4.1, Algorithms 3-4, Examples 15-16 and 22-23] The PIDD initialization is data-driven: the output weights are set using solutions generated by odeint or Chebfun (e.g., Section 4.2.1 states the data were 'generated with odeint'), and the reported relative L2 errors are then computed against the same reference solutions. This makes the PIDD results an interpolation test on the training data rather than an independent solver-free PINN result. The paper should explicitly acknowledge this circularity and distinguish it from the solver-free NbN results. In particular, Table 7 should not be read as a fair comparison between PIDD and NbN, since PIDD has access to the reference data.
  3. [Section 2, Eqs. (14)-(16); Section 4.1, Eqs. (47)-(49)] The deterministic initialization rests on two heuristic assumptions: that the derivative of the network at x_m is dominated by a single neuron's sigma'(0) term, and that neighboring output weights vary slowly. These assumptions are not quantified, no error bounds are derived, and the key hyperparameter delta-zeta is selected empirically (Section 2.1.3 reports 'the best results were achieved at delta-zeta = 0.7'). Additionally, the derivation leading to Eq. (16) is internally inconsistent: the text derives an approximation involving W_{m-1} at x_m but then states 'we have the following equation for the weights W_m' with N evaluated at x_{m+1}. A numerical check of how well Eq. (15) holds as a function of N and delta-zeta, and a sensitivity analysis, would substantially strengthen the paper.
  4. [Section 4.3, Appendix B, Examples 17-18] The 'generalization' properties of the proposed network are evaluated only at the midpoints q_{i+1/2} of the training grid, i.e., as interpolation within the domain [q_min, q_max]. This is not extrapolation to unseen parameter ranges, and the paper even shows in Example 18 that the method fails to generalize in chaotic regimes (mu = 0.023 for rho in [25,30]). The claim of 'pronounced generalizing properties' should be qualified as interpolation between precomputed solution snapshots, and the metric mu should be described as an interpolation-error ratio rather than a measure of genuine generalization.
minor comments (5)
  1. [Throughout] The spelling 'Lorentz system' appears in Examples 16 and 20, Table 7, and the Conclusions; the correct name is 'Lorenz system' (as used in the equations and in the reference to Lorenz 1963).
  2. [Section 3.5.1] The paragraph 'We noticed that changing the weights lambda_ic and lambda_r can lead to better accuracy...' indicates that key hyperparameters were tuned after seeing the test results. The paper would benefit from a description of a validation or cross-validation protocol that separates hyperparameter selection from final performance evaluation.
  3. [Section 4.2.2, Example 16] The phrase 'under which the calculations in [4] were carried out' would be clearer if the exact reference values of sigma, rho, and beta were stated with the units or conventions used; currently the reader must infer them from the equation.
  4. [Appendix B, Eq. (65)] The definition of the generalization measure mu is correct but the textual description 'the ratio of the relative total L2 error of prediction of set {q_{i+1/2}} to the ratio of the relative total L2 error of prediction of set {q_i}' is garbled; it should say 'the ratio of the error on the training grid to the error on the midpoints'.
  5. [Figure 7 caption] The caption 'Contribution to the solution u_{theta,1} of the first 1 (a), 11 (b), 21 (c)...' is missing commas and the first item should read 'first 1 neuron (a), first 11 neurons (b), first 21 neurons (c), ...'.

Circularity Check

3 steps flagged · score 6.0 of 10

PIDD accuracy is a fit residual, the 'generalization' measure tests interpolation, and the loss-weighting rule is a load-bearing self-citation.

  1. fitted input called prediction [Section 4.2.1 (Example 15) and Section 4.2.2 (Example 16), with Algorithm 3]
    "The reference solutions are obtained by using the odeint solver of scipy.integrate library. The data {uk}N k=0 for uniform distribution points {xk}N k=0 were generated with odeint for N = 20000 in 0.004 seconds. Results of PIDD initialization for neural network with N = 20000 are following: the relative L2 errors are ϵ[uθ;1, uref 1] = 5.67×10−5 and ϵ[uθ;2, uref 2] = 6.82×10−4."

    Algorithm 3 sets each output weight to W(2)_k = −(Δx/(2Δζ)) N[u_k, x_k]/κ_k, so the output-layer parameters are algebraic functions of the reference solution u_k at the same grid points x_k. The reported relative L2 error (Appendix A, Eq. (64)) is then computed against that same reference solution u. The number therefore measures interpolation or fitting residual on the training grid, not an independent prediction; a network whose output layer is computed from the reference data is forced to match those values up to the κ localization correction. Presenting this as competitive accuracy treats a fitted input as a predicted result.

  2. self definitional [Section 4.3 (Suggestion 9, Algorithm 4) and Appendix B]
    "The second set of solution values is calculated for the calculated grid of the parameter q in steps of ∆q, offset by ∆q/2, i.e. at points as far away as possible from the nearest points of q where the training was carried out, but not out of the training domain."

    The Q network is constructed by least squares W(2)a = I with bump activations chosen so that Q_j(q_n) ≈ 1 at the training parameters and values close to zero for other q; hence every q in [q_min, q_max], including the midpoints q_{j+1/2}, lies inside the interpolation span of the fitted bumps. The generalization measure µ = ϵ1/ϵ1/2 therefore compares fit on the training parameter grid with interpolation at midpoints of that same grid, both strictly inside the training domain. The claimed 'pronounced generalizing properties' are, by construction, properties of an interpolant rather than extrapolation to unseen parameter regions.

1 more flagged steps
  1. self citation load bearing [Section 3.2 and Section 6, citing reference [10]]
    "The original problem described by a differential equation (1) accompanied with the initial condition (2) can be reformulated to the problem described by the differential equation only. The procedure of reformulation is given in the [10]. According to this method the weights λk ic and λk r are related to each other by means of the relation λk ic = βkλk r."

    Reference [10] is by the same group (Es'kin, Davydov, Egorova, Malkhanov, Akhukov, Smorkalov), and the conclusion states that 'in all numerical experiments ... the relations between λr and λic loss functions were taken based on the method proposed in [10]'. This λic = βλr relation is not re-derived or independently verified in the present paper, yet it is load-bearing for every reported training result that uses the composite loss (4). The experimental accuracy numbers therefore rest on a self-citation chain, even though the relation is not the sole determinant of the accuracy values.

full rationale

The strictly deterministic initialization (Algorithms 1–2) and the neuron-by-neuron training (Algorithm 5) are largely self-contained: the output weights are set from the ODE residual at grid points and then refined by the network's own predictions, without importing a reference solution. The central architectural claim that one or two hidden layers suffice also has independent content. However, the PIDD accuracy claims (Examples 15, 16, 22, 23) import the reference solution as the input to Algorithm 3 and then evaluate against the same reference, so those error numbers are interpolation residuals rather than independent predictions. The generalization section similarly constructs Q from the training-parameter grid and then measures at midpoints of that grid, so µ quantifies interpolation, not extrapolation. The λic = βλr weighting used throughout is taken from the authors' own prior work [10] without independent verification, making it a load-bearing self-citation. Separately, the Lorenz 'state-of-the-art' statement is weakened by the paper's own Table 7, where PIDD is 3–5x more accurate than NbN, and no external SOTA baseline is supplied; that is a correctness/support problem rather than circularity. Overall, the thesis retains independent content, but two of the headline empirical claims reduce by construction, giving a partial circularity score.

Assumptions & free parameters 8 free parameters · 7 assumptions · 0 invented entities

The central claims rest on several hand-chosen constants, a localization approximation that is not rigorously justified, and self-cited weighting relations. No new physical entities are introduced.

free parameters (8)
  • delta-zeta (Algorithm 2) = 0.7
    Selected per problem after scanning values; Section 2.1.3: 'the best results were achieved at delta-zeta = 0.7'. Later derived value ln(2+sqrt(3))/2 approx 0.66 is used only for PIDD.
  • Loss weights lambda_ic, lambda_r = varies per example (e.g. 1/10, 10^3/10^4, 10^5/10^6)
    Chosen by hand per experiment and sometimes changed after observing results; Section 3.5.1 reports the best after noticing that changing weights improves accuracy.
  • Causality parameter epsilon initial = 10^-8, 10^-24, doubled with threshold delta_w = 0.99
    Free parameter in causal weighting (Eq. 29); initial value varies per experiment.
  • PSBW/RR decay gamma = 0.9 (PSBW), 0.99 (RR)
    Chosen in Section 3.6; controls exponential moving average of solution magnitudes.
  • Gradient normalization moving average alpha = 0.9
    Eq. (32), chosen in Section 3.3.
  • PIDD truncation window L = 10
    Set after Eq. (49): the authors state 'we limited L to 10'; affects kappa_m normalization.
  • beta_k relation for loss weights = varies with problem
    Taken from self-cited [10] as beta_k = max(N_t/X, 1/sigma[0.5], 1/sigma[2]); sets lambda_ic = beta lambda_r and is not independently derived here.
  • Training schedules (stages, epochs, learning rates) = varies, e.g. Adam 1e-3, LBFGS, Adam 1e-5, up to 300000 epochs
    Every example uses a different multi-stage schedule chosen by the authors; no justification or sensitivity analysis is given.
assumptions (7)
  • standard math Universal approximation theorem for single-hidden-layer sigmoid networks
    Invoked in Section 1 via [24-26] to justify using one hidden layer.
  • ad hoc to paper A sigmoid neuron's response is localized and approximately linear on [-delta-zeta, delta-zeta]
    Section 2.1.3 uses this to justify weight scaling W^(1) = 2 delta-zeta / delta-x and bias placement.
  • ad hoc to paper Derivative of the network at x_m is dominated by a single neuron's sigma'(0) term
    Eq. (15): partial u / partial x at x_m approx (delta-zeta / (2 delta-x)) W^(2)_{m-1}; basis for Eq. (16) and Algorithm 2.
  • ad hoc to paper Output weights W^(2)_k vary slowly near index m, allowing truncation to L neighbors
    Eq. (47)-(49): used to derive the PIDD weight formula (49); no error bound is given.
  • domain assumption delta-causal weighting relation beta from ref [10] is valid
    Used throughout to set lambda_ic = beta lambda_r; the cited paper is by the same authors.
  • domain assumption Euler method global error O(X/N) is a guide for initialization error
    Section 2 states 'we can expect the same order of prediction error' from the initialization.
  • domain assumption Reference solutions from odeint/Chebfun are accurate ground truth
    PIDD initialization and all error evaluations rely on these solvers; no tolerance or convergence details are given.

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Cite this review

Pith. "Pith review of Are Two Hidden Layers Still Enough for the Physics-Informed Neural Networks?." pith.science (2026). https://pith.science/paper/CLLVJRXS

@misc{pith2026241219235,
  author       = {Pith},
  title        = {Pith review of: Are Two Hidden Layers Still Enough for the Physics-Informed Neural Networks?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CLLVJRXS}},
  note         = {Machine review of arXiv:2412.19235}
}
read the original abstract

The article discusses the development of various methods and techniques for initializing and training neural networks with a single hidden layer, as well as training a separable physics-informed neural network consisting of neural networks with a single hidden layer to solve physical problems described by ordinary differential equations (ODEs) and partial differential equations (PDEs). A method for strictly deterministic initialization of a neural network with one hidden layer for solving physical problems described by an ODE is proposed. Modifications to existing methods for weighting the loss function are given, as well as new methods developed for training strictly deterministic-initialized neural networks to solve ODEs (detaching, additional weighting based on the second derivative, predicted solution-based weighting, relative residuals). An algorithm for physics-informed data-driven initialization of a neural network with one hidden layer is proposed. A neural network with pronounced generalizing properties is presented, whose generalizing abilities of which can be precisely controlled by adjusting network parameters. A metric for measuring the generalization of such neural network has been introduced. A gradient-free neuron-by-neuron fitting method has been developed for adjusting the parameters of a single-hidden-layer neural network, which does not require the use of an optimizer or solver for its implementation. The proposed methods have been extended to 2D problems using the separable physics-informed neural networks approach. Numerous experiments have been carried out to develop the above methods and approaches. Experiments on physical problems, such as solving various ODEs and PDEs, have demonstrated that these methods for initializing and training neural networks with one or two hidden layers (SPINN) achieve competitive accuracy and, in some cases, state-of-the-art results.

Figures

Figures reproduced from arXiv: 2412.19235 by the authors.

Figure 1
Figure 1. Approximations of the graph of a given function [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Explanatory figure for Algorithm 1. Algorithm 1: Initialization of physics-informed neural networks Data: — Result: Initialized neural network uθ;l(x) of PINN uθ, which consists of N neurons on hidden layer ∆x ← X/N; for k = 0, . . . , N − 1 do W (1) k ← 1; b (1) k ← −(k + 1)∆x; W (2) k are initialized with the Glorot scheme [38]; b (2) 0 ← ul(0). 2.1 Numerical experiments For our experiments we used Pytorch [39] ve… view at source ↗
Figure 3
Figure 3. Harmonic Oscillator. (a) and (b) are dependences of relative [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (33 more)
Figure 4
Figure 4. Figure 4: Harmonic Oscillator. (a) and (b) are dependences of relative [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Harmonic Oscillator. (a) and (b) are dependences of relative [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Harmonic Oscillator. (a) and (b) are comparisons of the predicted (red dash lines) and reference solutions [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Harmonic Oscillator. Contribution to the solution [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Electric Potential. (a) and (b) are dependences of relative [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: Electric Potential. (a) is a comparison of the predicted (red dash lines) and reference solutions (blue solid [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: Explanatory figure for strictly deterministic initialization of PINN (Algorithm 2). [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: Harmonic Oscillator. (a) and (b) are dependences of relative [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 12
Figure 12. Figure 12: Harmonic Oscillator. (a) and (b) are dependences of relative [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]
Figure 13
Figure 13. Figure 13: Harmonic Oscillator. (a) and (d) are comparisons of the predicted (red dash lines) and reference solutions [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 14
Figure 14. Figure 14: Electric Potential. (a) and (b) are dependences of relative [PITH_FULL_IMAGE:figures/full_fig_p012_14.png]
Figure 15
Figure 15. Figure 15: Electric Potential. (a) is a comparison of the predicted (red dash lines) and reference solutions (blue solid [PITH_FULL_IMAGE:figures/full_fig_p013_15.png]
Figure 16
Figure 16. Figure 16: Relativistic slingshot. (a), (b), (c) and (d) are comparisons of the predicted (red dash lines) and reference [PITH_FULL_IMAGE:figures/full_fig_p014_16.png]
Figure 17
Figure 17. Figure 17: Electric potential of the charged plane. (a) is a comparison of the predicted (red dash lines) and reference [PITH_FULL_IMAGE:figures/full_fig_p018_17.png]
Figure 18
Figure 18. Figure 18: Relativistic slingshot. (a), (b), (c) and (d) are comparisons of the predicted (red dash lines) and reference [PITH_FULL_IMAGE:figures/full_fig_p020_18.png]
Figure 19
Figure 19. Figure 19: Relativistic slingshot. (a), (b), (c) and (d) are comparisons of the predicted (red dash lines) and reference [PITH_FULL_IMAGE:figures/full_fig_p020_19.png]
Figure 20
Figure 20. Figure 20: The exponential growth. (a) and (b) are comparisons of the predicted (red dash lines) and reference solutions [PITH_FULL_IMAGE:figures/full_fig_p021_20.png]
Figure 21
Figure 21. Figure 21: The exponential growth. (a) is a comparison of the predicted (red dash lines) and reference solutions [PITH_FULL_IMAGE:figures/full_fig_p022_21.png]
Figure 22
Figure 22. Figure 22: Relativistic slingshot. (a), (b), (c) and (d) are comparisons of the predicted (red dash lines) and reference [PITH_FULL_IMAGE:figures/full_fig_p023_22.png]
Figure 23
Figure 23. Figure 23: Relativistic slingshot. (a), (b), (c) and (d) are comparisons of the predicted (red dash lines) and reference [PITH_FULL_IMAGE:figures/full_fig_p024_23.png]
Figure 24
Figure 24. Figure 24: Lorentz system. (a), (b), and (c) are comparisons of the predicted (red dash lines) and reference solutions [PITH_FULL_IMAGE:figures/full_fig_p027_24.png]
Figure 25
Figure 25. Figure 25: Blue solid lines of (a) and (b) are dependences of relative [PITH_FULL_IMAGE:figures/full_fig_p027_25.png]
Figure 26
Figure 26. Figure 26: Harmonic Oscillator. (a) the left panel is the reference solution, (a) the middle panel is a prediction of [PITH_FULL_IMAGE:figures/full_fig_p030_26.png]
Figure 27
Figure 27. Figure 27: Harmonic Oscillator. (a) and (b) are dependences of relative [PITH_FULL_IMAGE:figures/full_fig_p030_27.png]
Figure 28
Figure 28. Figure 28: Harmonic Oscillator. Dependence of measure of generalization properties [PITH_FULL_IMAGE:figures/full_fig_p031_28.png]
Figure 29
Figure 29. Figure 29: Lorentz system. (a) the left panel is the reference solution, (a) the middle panel is a prediction of a trained [PITH_FULL_IMAGE:figures/full_fig_p032_29.png]
Figure 30
Figure 30. Figure 30: Lorentz system. (a) and (b) are dependences of relative [PITH_FULL_IMAGE:figures/full_fig_p033_30.png]
Figure 31
Figure 31. Figure 31: Lorentz system. Dependences of measure of generalization properties [PITH_FULL_IMAGE:figures/full_fig_p033_31.png]
Figure 32
Figure 32. Figure 32: Harmonic oscillator. (a) is a comparison of the predicted (red dash lines) and reference solutions (blue solid [PITH_FULL_IMAGE:figures/full_fig_p033_32.png]
Figure 33
Figure 33. Figure 33: Lorentz system. (a) is a comparison of the predicted (red dash lines) and reference solutions (blue solid [PITH_FULL_IMAGE:figures/full_fig_p034_33.png]
Figure 34
Figure 34. Figure 34: Schematic diagram of the neural network for the SPINN approach at 2D problem. [PITH_FULL_IMAGE:figures/full_fig_p036_34.png]
Figure 35
Figure 35. Figure 35: Allen–Cahn equation. (a) is reference solution, (b) is a prediction of a trained physics-informed neural [PITH_FULL_IMAGE:figures/full_fig_p037_35.png]
Figure 36
Figure 36. Figure 36: Advection equation. (a) is reference solution, (b) is a prediction of PIDD initialized neural network, (c) is [PITH_FULL_IMAGE:figures/full_fig_p038_36.png]

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

    math.NA 2024-12 conditional novelty 2.0 of 10

    Rectified sigmoid (hard sigmoid) activation is reported to cut PINN solution errors by about an order of magnitude on two ODE benchmarks, but the result may be an interpolation artifact because the paper never disclos...

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