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REVIEW 4 major objections 3 minor 43 references

Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent ?

T0 review · 4 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves that the PINN neural tangent kernel converges to a deterministic limit only when the output scaling $1/N^s$ has $s$ large enough—$s\ge1$ for non-homogeneous operators and $s\ge s_1$ for homogeneous ones—so the standard…

desk verdict A plausible and experimentally supported scaling rule for PINN initialization, but the main theorem rests on an unproven law-of-large-numbers step; worth refereeing with a clear demand for a rigorous proof. read the letter →

arxiv 2412.06158 v1 pith:L5GLRHTI submitted 2024-12-09 stat.ML cs.LGmath-phmath.MPnlin.PSphysics.comp-ph

Zijian Zhou , Zhenya Yan This is my paper · ORCID
classification stat.MLcs.LGmath-phmath.MPnlin.PSphysics.comp-ph MSC 68T0760F0535Q53
keywords neuraltangentkernelphysics-informednetworkspartialdifferentialequationsconvergencehomogeneityoutputscalingsine-GordonequationKdV
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

The paper asks whether the neural tangent kernel (NTK) of a physics-informed neural network solving a general partial differential equation always converges to a deterministic limit as the network width grows. It answers no: with the standard output scaling $1/\sqrt{N}$, the initialized NTK can diverge, and it proves sharp thresholds for the scaling exponent $s$ in $1/N^s$. For operators whose leading term is nonzero or that contain non-homogeneous monomials, as in the sine-Gordon equation with $\sin(q)$, convergence requires $s \ge 1$; for homogeneous operators such as KdV with $q q_x$, it requires $s \ge s_1$, where $s_1$ is determined by the maximum homogeneity degree of the operator. The paper also shows that during training the NTK stays frozen for $s > 1/4$, extending the range in which kernel-regime analysis applies. Numerical experiments on the sine-Gordon and KdV equations are presented as confirmation.

What carries the argument

The central object is the PINN kernel of (15), assembled from gradients of the operator $F[q]$ and the boundary trace $q_x^m$ with respect to all network parameters, with the network output scaled by $1/N^s$. The argument decomposes $\partial F/\partial\theta$ as $\sum_i F_i[q,q_x,\dots]\,\partial q_{ix}/\partial\theta$ and writes each kernel entry as a sum over the $N$ neurons; the homogeneity definition (33) assigns each monomial a degree $\sum_l t_{i_l}$, and the proof balances powers of $N$—the factor $N^{1-2s}$ coming from the sum against the degree-dependent growth of the $F_i$—so that the law of large numbers yields a deterministic limit. The threshold $s_1$ in (39) is exactly the exponent at which that balance holds.

What would settle it

Recompute the KdV threshold from formula (39): the monomial $q q_x$ has homogeneity degree $2$, so the formula gives $s_1 = (2\cdot2+1)/(2\cdot2+2) = 5/6$, whereas the paper states $3/4$. Simulating the initialized NTK of the KdV equation at a sequence of widths with scaling exponents $s=0.75$ and $s=0.83$ and checking whether $\|K_{N+k}-K_N\|_2$ decays in probability will show which threshold the dynamics actually follow.

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Extended reading notes

Core claim

Theorem 2.2 states that for the two-layer network (13) with output scaling $1/N^s$, the PINN kernel (15) converges in probability to a deterministic limiting kernel only above certain scaling thresholds. In case A—when $F_0[q,q_x,\dots,q_{nx}]F_0[\hat q,\hat q_x,\dots,\hat q_{nx}]\ne 0$, or when some monomial in the expansion $\partial F/\partial\theta = \sum_i F_i \,\partial q_{ix}/\partial\theta$ is non-homogeneous—the threshold is $s\ge 1$. In case B—when that product vanishes and every monomial is homogeneous—the threshold is $s\ge s_1$, where $s_1 = \frac{2\sum_l t_{i'_l}+1}{2\sum_l t_{i'_l}+2}$ and $\sum_l t_{i'_l}$ is the maximum homogeneity degree of the operator, with a secondary exponent $s_2$ for the cross-block $K_{fb}$. The standard choice $s=1/2$ therefore does not universally deliver convergence; the paper identifies the sine-Gordon equation, with the non-homogeneous term $\sin(q)$, as a case-A example and the KdV equation, with the quadratic term $q q_x$, as a case-B example, reporting thresholds $s_1=3/4$, $s_2=2/3$ for KdV. Theorem 2.3 supplements this by proving that, under uniform boundedness of parameters, operator derivatives, and integrated residuals, the NTK stays asymptotically constant during training whenever $s>1/4$.

Load-bearing premise

The load-bearing premise is that the sums over neurons making up the kernel entries obey a law of large numbers toward deterministic limits, even though the summed terms contain the operator coefficients $F_i$, which are nonlinear functions of the whole random network output; no concentration bound is given for those dependent, nonlinear terms.

Editorial extensions

If this is right

  • Under the paper's result, NTK-based convergence analyses of PINNs that silently assume $s=1/2$ do not transfer to general nonlinear PDEs; a practitioner must either raise the output scaling to $s\ge 1$ or first verify that the operator is homogeneous and compute its maximum degree.
  • For homogeneous operators such as KdV, there is a window $s_1 \le s < 1$ in which the initialized kernel converges even though the classical scaling does not, so kernel convergence at initialization and kernel frozenness during training are governed by different exponents ($s_1$ versus $1/4$).
  • The homogeneity degree of the differential operator becomes a computable design parameter: operators with higher-degree nonlinear terms demand larger scaling exponents, so the same architecture may need different $s$ for different PDEs.
  • The theorem gives a sufficient condition for the kernel gradient-flow description (5) to be well posed, which is a prerequisite for using NTK-based diagnostics to explain or fix PINN training failures.

Reading between the lines

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

  • A rigorous proof of the sharp thresholds would require concentration bounds: the law-of-large-numbers step in the proof averages terms that are nonlinear, dependent functions of all $N$ random parameters, so the thresholds $s_1,s_2$ are best treated as scaling predictions until such bounds are supplied.
  • Formula (39) and the paper's stated KdV threshold $3/4$ are in tension if the quadratic monomial $q q_x$ is assigned homogeneity degree $2$, since the formula then gives $s_1=5/6$; a direct check of which value the KdV kernel follows would settle the discrepancy.
  • The same balancing argument suggests a practical design rule beyond the paper's examples: compute the maximum homogeneity degree of the PDE's nonlinear terms and set $s$ accordingly, which would also apply to other homogeneous equations such as Burgers' equation.
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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 / 3 minor

Summary. The paper studies the neural tangent kernel (NTK) of physics-informed neural networks (PINNs) for general nonlinear PDEs. It introduces an output-layer scaling 1/N^s and claims that the initialized NTK converges in probability to a deterministic kernel only above thresholds that depend on the homogeneity of the differential operator (Theorem 2.2): s >= 1 for non-homogeneous operators (case A) and s >= s1 for homogeneous operators (case B), with s1 defined in Eq. (39). It further claims that the NTK stays constant during training for any s > 1/4 (Theorem 2.3). The claims are tested on the sine-Gordon and KdV equations by plotting ||K_{N+k} - K_N|| for widths up to 50000 at s = 0.5 and s = 1. The paper reports divergence at s = 0.5 and convergence at s = 1, and interprets this as confirmation of the theory.

Significance. If the quantitative thresholds were correct, the paper would provide useful design guidance for PINN initialization and identify a genuine limitation of standard NTK scaling for nonlinear PDEs. The basic qualitative phenomenon is plausible and the numerical experiments, while limited to two equations and a proxy quantity, show a clear difference between s = 0.5 and s = 1. The paper also has a useful negative message: the standard NTK convergence argument does not automatically extend to nonlinear PDEs. However, the theoretical derivation of the thresholds is the load-bearing part of the manuscript, and it contains substantial gaps: the law-of-large-numbers step is not justified for dependent nonlinear terms, the scaling analysis appears to ignore the N^{1/2} fluctuations of the hidden-layer sum, and the norm estimates used in the proof of Theorem 2.3 are not valid for the claimed range of s. I therefore cannot regard the main quantitative claims as established.

major comments (4)
  1. [Appendix B, Eqs. (31)-(40)] The proof of Theorem 2.2 invokes the law of large numbers for sums whose summands are not independent. In the expression for K_ff, the coefficients F_i[q,q_x,...] are nonlinear functions of q(x,theta), which is itself a sum over all N hidden units, so the summands indexed by k are dependent. No variance or concentration bound is supplied for these terms. This is not a cosmetic issue: for s = 1/2 and F[q] = sin(q), q converges to a nondegenerate Gaussian process, and cos(q(x))cos(q(hat x)) has positive variance, so a deterministic limiting kernel cannot exist. For s > 1/2 a Slutsky-type argument might be possible, and for the homogeneous case a delta-method argument controlling the normalized fluctuations of F_i would be needed. Since neither is provided, the thresholds s1 and s2 in Theorem 2.2 are not proved as stated.
  2. [Appendix B, Eqs. (34), (39)-(40)] The scaling analysis appears to neglect the N^{1/2} fluctuation of the sum defining q. For mean-zero iid summands, q(x,theta) = N^{-s} sum_k W^{(1)}_k sigma(...) is of order O_P(N^{1/2-s}), not O_P(N^{-s}). For a homogeneous coefficient F_i of degree d = sum_l t_{il}, this gives F_i = O_P(N^{d(1/2-s)}), and the corresponding K_ff contribution has size N^{1+d-2s(d+1)}. Balancing this yields s = 1/2, not the formula s1 = (2d+1)/(2d+2) in Eq. (39). Consequently, the claimed KdV thresholds s1 = 3/4 and s2 = 2/3 do not follow from the displayed scaling; for the homogeneous terms in KdV the contribution would already vanish for every s > 1/2. The experiments only test s = 0.5 and s = 1.0, which cannot distinguish the paper's thresholds from the simpler condition s > 1/2.
  3. [Appendix C, proof of Theorem 2.3] The proof of Theorem 2.3 uses Frobenius-norm estimates that are not sufficient for the claimed range of s. Lemma C.1 gives a uniform bound O(N^{-s}) on each entry of the Jacobian, but the Jacobian has 3N+1 columns, so ||J(t)||_F = O(N^{1/2-s}), which diverges for s < 1/2. The step following Eq. (72), which treats ||J(t)||_F as bounded for all s > 1/4, is therefore unjustified. Similarly, the bound ||J(t)-J(0)||_F = O(N^{1/2-2s}) leads to a product in Eq. (72) that vanishes under these estimates only for s > 1/3, and a spectral-norm or concentration argument would be needed to reach s > 1/4. As written, Theorem 2.3 is not established for s in (1/4, 1/2].
  4. [Sec. 3.2 and Fig. 5] The experimental validation of Theorem 2.3 is internally inconsistent: the text states that the convergence coefficient s is set to 0.3, while the caption of Fig. 5 says s = 0.2. Since 0.2 is below the claimed threshold 1/4, the figure as captioned would not validate the theorem. The authors should correct this mismatch and, more importantly, provide experiments at several values of s in the claimed critical ranges (for example s = 0.55, 0.6, 0.7 for the KdV equation) to distinguish between their thresholds and the simpler s > 1/2 condition.
minor comments (3)
  1. [Theorem 2.2 statement] The statement of Theorem 2.2 says the kernel converges in probability to a deterministic limiting kernel K*, but K* is never defined in the theorem; it appears only implicitly in the proof. The statement should be made precise.
  2. [Sec. 1, Fig. 1 and Sec. 3] The paper uses ||K_{N+k} - K_N|| as a proxy for ||K_N - K*||. Convergence of the former is necessary but not sufficient for convergence to a fixed limit, and this caveat should be stated more carefully in the interpretation of the experiments.
  3. [Throughout] There are several typos and minor inconsistencies, including 'Theorem 3' in Sec. 3.2 instead of Theorem 2.3, and the LaTeX corruption in Eq. (59). The notation in Remark 1 is also difficult to parse because the case distinction between 'F0 ... != 0' and 'every monomial is homogeneous' is not presented cleanly.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the s-thresholds derive from operator homogeneity (Eqs. 33-39), the experiments validate rather than fit the theory, and self-citations are contextual; the unproven law-of-large-numbers step is a proof gap, not a circular reduction.

full rationale

The derivation chain is self-contained. The central thresholds of Theorem 2.2 are computed analytically from the differential operator's homogeneity: Eq. (33) defines homogeneity of the coefficients F_i, and the critical exponents s1 (Eq. (39)) and s2 (Eq. (47)) are algebraic functions of the maximal homogeneity degree only. For the KdV operator (22), the degree-1 coefficients F0 = 6q_x and F1 = 6q give s1 = 3/4, matching the paper's value; the value is consistent with Eq. (39) as written, which uses the degree of the coefficient F_i rather than of the monomial q q_x. Nothing in the sine-Gordon or KdV experiments feeds back into this computation, so the predictions (divergence at s = 1/2, convergence at s >= 1 in case A and s >= 3/4 in case B) are not fitted inputs renamed as predictions. The 1/N^s scaling is transparently introduced at Eq. (13) as a modification of the standard initialization, not smuggled via citation, and no uniqueness theorem is imported from prior work. Self-citations (refs [28] and [30]) are application examples in the introduction and do not carry the proof chain. The genuinely fragile loci are proof gaps rather than circular reductions: Appendix B (Eqs. (31)-(40)) invokes "the law of large numbers" for sums whose summands include F_i[q] dq/dtheta_k dq_hat/dtheta_k, where F_i depends on all N hidden units through q, and no variance or concentration bound for these dependent, nonlinear terms is supplied; and Theorem 2.3's assumption (iii), Eq. (20), bounds the time-integral of residuals as an explicit premise rather than a derived fact. The Figure 5 caption (s = 0.2) also disagrees with the Section 3.2 text (s = 0.3) for the KdV training experiment, which weakens that validation but is not circular. Under the analytical rules, an omitted or unjustified proof step is a correctness risk, not circularity, because the theorem's claim does not reduce by construction to its own inputs. No circular step is therefore identified; the score of 2 merely records minor non-load-bearing self-citations.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. The free parameters are the scaling exponent s and the loss weights alpha, beta; these are design choices, not fitted constants. The axioms are the concentration/LLN step, the boundedness assumptions in Theorem 2.3, and the unstated determinism of the output bias. The most problematic is the LLN step, which is asserted without proof and is central to the convergence thresholds.

free parameters (2)
  • s (convergence coefficient) = Not fitted; chosen by the user (0.5, 1, 0.25, 0.3, etc.)
    The paper's central claim is a condition on s, but s itself is a design parameter introduced in (13) and varied in experiments. It is not fitted, but it is a free parameter of the theoretical framework.
  • alpha, beta (loss weights) = unspecified
    Introduced in the loss (14) to balance equation and boundary terms. The numerical experiments use a 'fixed weighted loss' but do not state values of alpha and beta, which affect the kernel block scaling.
assumptions (5)
  • ad hoc to paper The law of large numbers applies to the kernel sums even when the operator coefficients F_i are nonlinear functions of the network output q.
    This is the core step in the proof of Theorem 2.2 (Appendix B, after Eq. (34)). No concentration bound is provided for sums involving products of N random variables through F_i(q, q_x, ...). The claim is load-bearing and unproven.
  • domain assumption Network parameters remain uniformly bounded during training (Assumption (i) of Theorem 2.3).
    Theorem 2.3 assumes sup_t ||theta(t)||_infty <= C, independent of N. This is not derived from the gradient flow (4), and for the divergent NTK cases it may not hold. It is a strong assumption needed for the training-time kernel constancy result.
  • domain assumption All derivatives of the PDE operator up to order n are uniformly bounded along the training trajectory (Assumption (ii) of Theorem 2.3).
    This assumption excludes operators with unbounded terms (e.g., logarithmic nonlinearities, as noted in Remark 3). It is used in Lemmas C.1 and C.4 to bound the Jacobian differences.
  • domain assumption The activation function is smooth and all its derivatives are bounded (Assumption (iv) of Theorem 2.3).
    This is standard for NTK analyses; it is used in the mean-value theorem arguments in Lemmas C.2-C.4.
  • ad hoc to paper The output-layer bias b^(1) is deterministic (implicitly zero) so that the limiting kernel is deterministic.
    In the proof of Theorem 2.2, q converges to E[q] + b^(1) as N grows. If b^(1) were random, the limit F0(b^(1)) would be random, contradicting the claimed convergence to a deterministic kernel. The experiments initialize biases to zero, but the theorem does not state this assumption.

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Cite this review

Pith. "Pith review of Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent ?." pith.science (2026). https://pith.science/paper/L5GLRHTI

@misc{pith2026241206158,
  author       = {Pith},
  title        = {Pith review of: Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent ?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L5GLRHTI}},
  note         = {Machine review of arXiv:2412.06158}
}
read the original abstract

In this paper, we study the neural tangent kernel (NTK) for general partial differential equations (PDEs) based on physics-informed neural networks (PINNs). As we all know, the training of an artificial neural network can be converted to the evolution of NTK. We analyze the initialization of NTK and the convergence conditions of NTK during training for general PDEs. The theoretical results show that the homogeneity of differential operators plays a crucial role for the convergence of NTK. Moreover, based on the PINNs, we validate the convergence conditions of NTK using the initial value problems of the sine-Gordon equation and the initial-boundary value problem of the KdV equation.

Figures

Figures reproduced from arXiv: 2412.06158 by the authors.

Figure 1
Figure 1. The trends of kKN+k − KNk2 of the initial value problem of the sine-Gordon equation . The blue area is the variation range of the results of 50 independent experiments, and the dark line is the mean of these experiments. where Kf f =  ∇θF[q](x i f , θ) T ∇θF[q](x j f , θ)  i,j=1,...,Nf , Kf b = K T b f =  ∇θF[q](x i f , θ) T ∇θq(x j b , θ)  i=1,...,Nf ; j=1,...,Nb , Kbb =  ∇θq(x i b , θ) T ∇θq(x j b , θ) … view at source ↗
Figure 2
Figure 2. The trends of kKN+k − KNk2 of the initial value problem of the sine-Gordon equation (10) when (a) s = 0.5, (b) s = 1. The blue area is the variation range of the results of 50 independent experiments, and the black line denotes the mean value of these experiments. (iii) There exists a constant C > 0, Z T 0 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. The trends of sup t∈[0,n] kKN(t) − KN(0)k2 of the initial value problem of the sine-Gordon equation (10) when s = 1/4, where n represents the maximal training step and N the width of the neural networks. 3.1 NTK convergence of sine-Gordon equation In the first experiment, we continue to examine the initial value problem given by Eq. (10). As discussed in Re￾mark 2 of Theorem 2.2, the convergence of the NTK is influe… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The trends of kKN+k − KNk2 of the initial value problem of the KdV equation (22) when (a) s = 0.5, (b) s = 1. The blue area is the variation range of the results of 50 independent experiments, and the black line is the mean value of these experiments. Periodic boundary…
Figure 5
Figure 5. Figure 5: The trends of sup t∈[0,n] kKN(t) − KN(0)k2 of the initial value problem of the KdV equation when s = 0.2, where n represents the maximal train step and N the width of the neural networks. Appendix A. Proof of Lemma 2.1 Proof. In case A, the loss function is L(θ) = Lf (…

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Works this paper leans on

43 extracted references · 37 canonical work pages

  1. [1]

    LeCun, Y

    Y . LeCun, Y . Bengio, G. Hinton, Deep learning, Nature 521 (2015) 436

    work page 2015

  2. [2]

    Goodfellow , Y

    I. Goodfellow , Y . Bengio, A. Courville, Deep Learning (M IT Press, 2016)

    work page 2016

  3. [3]

    Dissanayake, N

    M. Dissanayake, N. Phan-Thien, Neural-network-based a pproximations for solving partial differential equations , Commun. Numer. Methods Eng. 10 (1994) 195-201

    work page 1994

  4. [4]

    I. E. Lagaris, A. Likas, D. I. Fotiadis, Artificial neural networks for solving ordinary and partial differential equ ations, IEEE Trans. Neural Netw . 9 (1998) 987-1000. 16

    work page 1998

  5. [5]

    Cybenko, Approximation by superpositions of a sigmoi dal function, Math

    G. Cybenko, Approximation by superpositions of a sigmoi dal function, Math. Control Signal Systems, 2 (1989) 3

    work page 1989

  6. [6]

    Hornik, Approximation Capabilities of Muitilayer Fe edforward Networks, Neural Networks, 4 (1991) 251-257

    K. Hornik, Approximation Capabilities of Muitilayer Fe edforward Networks, Neural Networks, 4 (1991) 251-257

    work page 1991

  7. [7]

    Sirignano, K

    J. Sirignano, K. Spiliopoulos, DGM: a deep learning algo rithm for solving partial differential equations, J. Compu t. Phys. 375 (2018) 1339-1364

    work page 2018

  8. [8]

    Raissi, P

    M. Raissi, P . Perdikaris, G. E. Karniadakisa, Physics-i nformed neural networks: A deep learning framework for solv ing forward and inverse problems involving nonlinear partial d ifferential equations, J. Comput. Phys. 378 (2019) 686-707

    work page 2019

Show all 43 references
  1. [9]

    W. E, B. Yu, The Deep Ritz Method: A Deep Learning-Based Nu merical Algorithm for Solving Variational Problems, Com- mun. Math. Stat. 6 (2018) 1-12

    2018

  2. [10]

    Z. Long, Y . Lu, X. Ma, B. Dong, PDE-Net: Learning PDEs fro m Data, PMLR 80 (2018) 3208-3216

    2018

  3. [11]

    Z. Long, Y . Lu, B. Dong, PDE-Net 2.0: Learning PDEs from d ata with a numeric-symbolic hybrid deep network, J. Comput. Phys. 399 (2019) 108925

    2019

  4. [12]

    Bar-Sinaia, S

    Y . Bar-Sinaia, S. Hoyerb, J. Hickeyb, and M. P . Brennera , Learning data-driven discretizations for partial differ ential equa- tions, PNAS, 116 (2019) 15344-15349

    2019

  5. [13]

    Y . Wang, Z. Shen, Z. Long and B. Dong, Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning, (2019) arXiv:1905.11079

    2019 arXiv

  6. [14]

    L. Lu, P . Jin, and G. E. Karniadakis, DeepONet: Learning nonlinear operators for identifying differential equatio ns based on the universal approximation theorem of operators, (2019) a rXiv:1910.03193

    2019 arXiv

  7. [15]

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhatt acharya, A. Stuart, and A. Anandkumar, Fourier Neural Opera tor for parametric partial differential equations, (2021) arXiv: 2010.08895v2

    2021 arXiv

  8. [16]

    K. Wu, D. Xiu, Data-driven deep learning of partial diff erential equations in modal space, J. Comput. Phys. 408 (2020) 109307

    2020

  9. [17]

    Y . Khoo, J. Lu and L. Ying, Solving parametric PDE proble ms with artificial neural networks, Euro. J. Appl. Math. 32 (2 021) 421-435

  10. [18]

    G. M. Rotskoff, E. Vanden-Eijnden, Learning with rare d ata: using active importance sampling to optimize objectiv es domi- nated by rare events. PMLR 145 (2022) 757-780

    2022

  11. [19]

    D. Pfau, J. S. Spencer, A. G. Matthews, W. M. C. Foulkes, A b initio solution of the many electron Schr ¨ odinger equation with deep neural networks. Phys. Rev . Res. 2 (2020) 033429

    2020

  12. [20]

    Goswamia, M

    S. Goswamia, M. Yinb, Y . Yu, G. E. Karniadakis, A physics -informed variational DeepONet for predicting crack path i n quasi-brittle materials, Comput. Methods Appl. Mech. Engr g. 391 (2022) 114587

    2022

  13. [21]

    S. Lin, Y . Chen, A two-stage physics-informed neural ne twork method based on conserved quantities and application s in localized wave solutions, J. Comput. Phys. 457 (2022) 11105 3

    2022

  14. [22]

    S. Lin, Y . Chen, Physics-informed neural network metho ds based on Miura transformations and discovery of new local ized wave solutions, Physica D 445 (2023) 133629

    2023

  15. [23]

    J. Li, Y . Chen, A deep learning method for solving third- order nonlinear evolution equations, Commun. Theor. Phys. 72 (2020) 115003

    2020

  16. [24]

    J. Li, J. Chen, B. Li, Gradient-optimized physics-info rmed neural networks (GOPINNs): a deep learning method for s olving the complex modified KdV equation, Nonlinear Dyn. 107 (2022) 781-792

    2022

  17. [25]

    Z. Miao, Y . Chen, VC-PINN: Variable Coefficient Physica l Information Neural Network For Forward And Inverse PDE Problems with Variable Coefficient, arXiv preprint arXiv:2 305.07479

  18. [26]

    J. Pu, Y . Chen, Complex dynamics on the one-dimensional quantum droplets via time piecewise PINNs, Physica D 454 (2023) 133851

    2023

  19. [27]

    L. Wang, Z. Yan, Data-driven peakon and periodic peakon solutions and parameter discovery of some nonlinear disper sive equations via deep learning, Physica D 428 (2021) 133037

    2021

  20. [28]

    Z. Zhou, Z. Yan, Deep learning neural networks for the th ird-order nonlinear Schr?dinger equation: bright soliton s, breathers, and rogue waves, Commun. Theor. Phys. 73 (2021) 1 05006

    2021

  21. [29]

    J. Song, Z. Yan, Deep learning soliton dynamics and comp lex potentials recognition for 1D and 2D PT-symmetric satur able nonlinear Schr ¨ odinger equations, Physica D 448 (2023) 133729

    2023

  22. [30]

    Zhong, S

    M. Zhong, S. Gong, S.-F. Tian, Z. Yan, Data-driven rogue waves and parameters discovery in nearly integrable PT-sym metric Gross-Pitaevskii equations via PINNs deep learning, Physi ca D 439 (2022) 133430

    2022

  23. [31]

    Shuning Lin a, Yong Chen, A two-stage physics-informed neural network method based on conserved quantities and app li- cations in localized wave solutions, J. Comput. Phys. (2022 )

    2022

  24. [32]

    S., Li, Z., Salakhutdinov , R., Wang, R

    Arora, S., Du, S. S., Li, Z., Salakhutdinov , R., Wang, R. , and Yu, D. (2019c). Harnessing the power of infinitely wide d eep nets on small-data tasks. arXiv preprint arXiv:1910.01663

    2019 arXiv

  25. [33]

    Y . Shin, J. Darbon, G. E. Karniadakis, On the convergenc e of physics informed neural networks for linear second-ord er elliptic and parabolic type PDEs, Commun. Comput. Phys. 28 ( 2020) 2042-2074

    2020

  26. [34]

    T. Luo, H. Yang, Two-layer neural networks for partial d ifferential equations: optimization and generalization t heory . (2020) arXiv:2006.15733

    2020 arXiv

  27. [35]

    Jacot, F

    A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel : convergence and generalization in neural networks. Adv . Neural Inf. Process. Syst. 31 (2018) 8571-8580

    2018

  28. [36]

    Martens, A

    J. Martens, A. Ballard, G. Desjardins, G. Swirszcz, V . D alibard, J. SohlDickstein, and S. S. Schoenholz, Rapid trai ning of deep neural networks without skip connections or normali zation layers using deep kernel shaping. arXiv preprint arXiv:2110.01765, (2021). 17

    2021 arXiv

  29. [37]

    Dyer and G

    E. Dyer and G. Gur-Ari, Asymptotics of wide networks fro m feynman diagrams. In International Conference on Learnin g Representations. (2020)

    2020

  30. [38]

    Meanti, L

    G. Meanti, L. Carratino, L. Rosasco, and A. Rudi, Kernel methods through the roof: handling billions of points effici ently . arXiv preprint arXiv:2006.10350.(2020)

    2020 arXiv

  31. [39]

    Arora, S

    S. Arora, S. S. Du, W. Hu, Z. Li, R. R. Salakhutdinov , and R . Wang, On exact computation with an infinitely wide neural ne t. In Advances in Neural Information Processing Systems, (201 9b) pages 8141-8150

  32. [40]

    S.Wang, X. Yu, P . Perdikaris, When and why PINNs fail to t rain: a neural tangent kernel perspective. J. Comput. Phys. 449 (2022) 110768

    2022

  33. [41]

    Tancik, P

    M. Tancik, P . P . Srinivasan, B. Mildenhall, S. Fridovic h-Keil, N. Raghavan, U. Singhal, R. Ramamoorthi, J. T. Barro n, and R. Ng, Fourier features let networks learn high frequenc y functions in low dimensional domains. arXiv preprint arXiv:2006.10739.(2020)

    2020 arXiv

  34. [42]

    S. Wang, H. Wang, P . Perdikaris, On the eigenvector bias of Fourier feature networks: From regression to solving mul ti-scale PDEs with physics-informed neural networks, Comput. Metho ds Appl. Mech. Engrg. 384 (2021) 113938

    2021

  35. [43]

    M. J. Ablowitz, P . A. Clarkson, Solitons, Nonlinear Evo lution Equations and Inverse Scattering (Cambridge Univer sity Press, Cambridge, 1991). 18

    1991

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