Unveiling the Mechanism of Continuous Representation Full-Waveform Inversion: A Wave Based Neural Tangent Kernel Framework
Authors: Ruihua Chen
Yisi Luo
Bangyu Wu
Deyu Meng
Venue: ICLR 2026 Poster (2026)
Keywords: Full-waveform inversion; Continuous representation; Implicit neural representation; Neural tangent kernel
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: This paper develops a theoretical framework to explain and optimize continuous representation FWI methods, and based on this, proposes some novel hybrid representations that strike a better balance between robustness and high-frequency convergence.

Abstract: Full-waveform inversion (FWI) estimates physical parameters in the wave equation from limited measurements and has been widely applied in geophysical exploration, medical imaging, and non-destructive testing. Conventional FWI methods are limited by their notorious sensitivity to the accuracy of the initial models. Recent progress in continuous representation FWI (CR-FWI) demonstrates that representing parameter models with a coordinate-based neural network, such as implicit neural representation (INR), can mitigate the dependence on initial models. However, its underlying mechanism remains unclear, and INR-based FWI shows slower high-frequency convergence. In this work, we investigate the general CR-FWI framework and develop a unified theoretical understanding by extending the neural tangent kernel (NTK) for FWI to establish a wave-based NTK framework. Unlike standard NTK, our analysis reveals that wave-based NTK is not constant, both at initialization and during training, due to the inherent nonlinearity of FWI. We further show that the eigenvalue decay behavior of the wave-based NTK can explain why CR-FWI alleviates the dependency on initial models and shows slower high-frequency convergence. Building on these insights, we propose several CR-FWI methods with tailored eigenvalue decay properties for FWI, including a novel hybrid representation combining INR and multi-resolution grid (termed IG-FWI) that achieves a more balanced trade-off between robustness and high-frequency convergence rate. Applications in geophysical exploration on Marmousi, 2D SEG/EAGE Salt and Overthrust, 2004 BP model, and the more realistic 2014 Chevron models show the superior performance of our proposed methods compared to conventional FWI and existing INR-based FWI methods.

Published as a conference paper at ICLR 2026
UNVEILING THE MECHANISM OF CONTINUOUS REP- RESENTATION FULL- WAVEFORM INVERSION: A WAVE BASED NEURAL TANGENT KERNEL FRAMEWORK
Ruihua Chen', Yisi Luo Bangyu Wu Deyu Meng!
'School of Mathematics and Statistics. Xi an Jiaotong University. an. an. Shaanxi, China MMacao Institute of Systems Engineering, Macau University of Science and Technology, Taipa, Macao Ruihua. Chen@stu. x.ruuxxjtu jjtu. edu.edu.c.. en, yisiluo1221 foxmail. com, {bangyuwu, dymeng @mmall xx ejt..edu.cn.
ABSTRACT
Full-waveform inversion (FWI) estimates physical parameters in the wave equa- tion from limited measurements and has been widely applied in geophysical ex- ploration, medical imaging, and non-destructive testing. Conventional FWI meth- ods are limited by their notorious sensitivity to the accuracy of the initial models. Recent progress in continuous representation FWI (CR-FWI) demonstrates that representing parameter models with a coordinate- -bsed neural network, such as implicit neural representation (INR), can mitigate the dependence on initial mod- els. However, underlying mechanism remains unclear, and INR-based FWI shows slower high-frequency convergence. In this work, we investigate the gen- eral CR-FWI framework and develop a unified theoretical understanding by ex- tending the neural tangent kernel (NTK) for FWI to establish wave-based NTK framework. Unlike standard NTK, our analysis reveals that wave-based NTK is not constant, both at initialization and during training, due to the inherent nonlin- earity of FWI. We further show that the eigenvalue decay behavior of the wave- based NTK can explain why CR-FWI alleviates the dependency on initial models and shows slower high-frequency convergence. Building on these insights, we propose several CR-FWI methods with tailored eigenvalue decay properties for FWI, including a novel hybrid representation combining INR and mustitresooution grid (termed IG-FWI) that achieves a more balanced trade-off between robustness and high-frequency convergence rate. Applications in geophysical exploration on Marmousi, 2D SEG/EAGE Salt and Overthrust, 2004 BP model, and the more realistic 2014 Chevron models show the superior performance of our proposed methods compared to conventional FWI and existing -Nsdd FWI methods.
(a) Seismic data acquisition
(b) Fale-waveformm inversion surfsce
source reeeivers
souecc rustivass
affoct
offect
sobsurficc
ma(x)
Acquistton
Update
roal wave propozzition
simuulated mavi propagation
Unkaows velicity medel
Lime Oibserved
Synthetic scismic data
Velueity model represented INR
Figure 1: (a) The acquisition process of observed seismic data obtained from the receivers placed on the surface; (b) FWI nonlincar inverse problem that inverts the subsurface models from observed seismic data. The CR-FWI framework synthesizes seismic data by solving the wave equation T where the model m(x) is represented by coordinate-based neural network mo (x), e.g., INR.
INTRODUCTION
Full-waveform inversion (FWI) an ill-posed and partial differential equation contttccos (i.e., wave equation) nonlinear inversion problem in seismic imaging, which estimates subsurface properties (e.g., velocity, impedance, or density models) by iteratively minimizing the discrepancy between observed seismic data (i.e., seismograms) and synthetic seismic data (i.e., the solution of the wave equation) Virieux & Operto. 2009). Since having a theoretically highest resolution,
Corresponding authors.

---

Published as a conference paper at ICLR 2026
Inittal veladity model
Conveniional FWT
INR Iasea FWT
IG-FWI (Ours)
Bigavalue spatrum compariee
33
SSIM: 0:161
SSIM: 0.517
SSIM: 0.752
33*
33
SSIM: 0.299
SSIM: 0.253
SSIM:
Rigetscdiin vndro
(A)
(b)
Figure 2: (a) Inversion results using conventional FWI (i.e., Multiscale ADFWI (Lu et al. 2025a existing INR-based FWI (i.e., IFWI (Sun et al., 2023a)), and our proposed IG-FWI methods with smooth (after 500 epochs) and constant initial models (after 1500 epochs) on the 2004 BP model; (b) Eigenvalue spectrum comparison of wave-based NTK using different FWI methods.
FWI has been widely applied across multiple domains, including geophysical subsurface explo- ration (Tromp) 2020), medical imaging (Guasch al.) (020), planetary imaging (Hanasoge) 2)114) and non-destructive testing (Nguyen & Modrak] (018). However, the nonlinearity of the inversion problem, complex geological structures (e.g., salt bodies and faults), and practical limitations in seismic data acquisition (e.g., sparse sampling, missing low-frequency components, and noise in- terference) may pose significant challenges to achieving stable convergence towards high precision inversion results in practical applications (Virieux al. 2017). An accurate and smooth initial veloc- ity model, including abundant cww-freuuency information, can be used to mitigate the ill-posedness and cycle-skipping (i.e., half-cycle waveform misfit causing inversion failure) (Chen et al. 2022). Still, obtaining such an accurate initial model remains significant challenge due to the complex near-surface conditions and heterogeneous subsurface layers (Yang et al. 2025).
To alleviate the aforementioned challenges, previous studies have explored various approaches, in- cluding initial model generation (Chen et al. 2016 Sun et al. 2017). misfit function modification Yang & Ma 2023). regularization design (Yan & Wang 2018). advanced optimization algorithms (Sun Alkhalifah] 2020), and multiscale inversion strategies (Fichtner et al., 2013). However, most of these conventional FWI methods are still sensitive to initial model accuracy and seismic data qual- ity. Recently, Sun et al. (2023a) and Yang & Ma (2025) utilized a specific coordinate-based neural network, i.e., implicit neural representation (INR) (Mildenhall et al. 2021 Sitzmann et al. (020)) to reparameterize the velocity model as a continuous function (see Fig. I]. This novel continuous rep- resentation FWI (CR-FWI) framework can enhance the robustness w.r.t. the initial model accuracy and seismic data quality. Previous studies (Sun et al., 2023a Yang & Ma 2025) have identified two main distinctions in inversion performance between conventional FWI and CR-FWI methods:
Robustness:
CR-FWI can recover satisfactory inversion results even with constant initial model and poor seismic data quality, whereas conventional FWI often fails under such conditions.
Convergence: CR-FWI exhibits slower convergence rates, particularly in the components, and requires larger number of iterations to achieve nigh-precision inversion results. contrast, conventional FWI converges more rapidly when smooth initial models are available.
We demonstrate this discrepancy in the 2004 BP model, as shown in Fig. (a). In light of the above distinctions, we propose two natural and intriguing questions: (i) Can unified theoretical frame- work be established to explain the differences in robustness and convergence between conventional FWI and CR-FWI? (ïi) Is there continuous represeniation that achieves bulanced trade-off be- tween robustness and convergence? ? Positive answers to these questions are provided in this study.
1.1 OUR WORK AND CONTRIBUTIONS
In this work, we dive into the general continuous representation FWI framework and establish a unified theoretical foundation for both conventional FWI and CR-FWI. This is accomplished by introducing and analyzing novel neural tangent kernel (NTK) tailored for FWI, i.e., the wave ker- nel for conventional FWI Proposition 2.1 and the wave-based NTK for CR-FWI in Proposition The eigenvalue of these NTKs provides a theoretical basis for understanding the convergence behavior and robustness conventional FWI and CR-FWI. Our analysis leads to two key insights:
3.1
Insight I: Distinct from standard NTK, both the wave kernel and wave-based NTK exhibit dy- namic behavior during training (see Theorem due to the nonlinearity of the inversion problem. Insight II: The eigenvalue decay of wave-based NTK is not slower than the wave kernel (see Theorem 4.2 due to the smooth kernel induced by the continuous representation (see Fig. (b)).
2

---

Published as a conference paper at ICLR 2026
The first insight establishes dynamic analytical foundation for the wave-based NTK, providing way for future theoretical research into NTK for PDE-constrained nonlinear inverse problems using stochastic analysis (Oksendal 2013). Especially, under the quasi-static assumption, the wave-based NTK can be locally linearized, thereby enabling analysis of local convergence and optimization be- havior based its eigenvalue decay rate. The second insight reveals that the training dynamics of FWI can be decomposed along the directions defined by the eigenvectors of the wave-based NTK, each associated with a distinct convergence rate determined by the corresponding eigenvalues (Jacol et al. 2018). The rapid eigenvalue decay of rave-based NTK reveals that CR-FWI can inherently achieve effective multiscale inversion from low frequencies to high frequencies adaptively, i.e., fre- quency principle (Xu et al.] 2025) spectral bias (Cao et al. 2019). This property explains why CR-FWI reduces reliance on initial models and exhibits slower higu-fryy convergence.
Motivated by the connection between the eigenvalue decay and its optimization behavior, we intro- duce several novel CR-FWI methods by incorporating low-rank tensor function representation (Luo) et al. 2023) (termed LR-FWI) and multi-grid parametric encoding (Müller et al.) 2022) (termed MPI-FWI). By encoding the inherent smoothness and low-rank properties of parameter models, LR-FWI improves inversion robustness and accelerates the convergence of high-frequency compo- nents due to an appropriate eigenvalue decay, as empirically verified in our numerical experiments (see Fig. (c)). Moreover, the multigrid parametric encoding has been proven yield a better eigen- value spectrum distribution in its NTK (Audia et al.] (025). Hence, compared to INR-based FWT (Sun al. 2023a Yang& & Ma 2025), the proposed MPE-FWI method exhibits a slower eigenvalue decay (see Fig. 2 (b) and Theorem 5.1] which facilitates faster convergence of high-frequency com- ponents and leads higher inversion accuracy (see Tab. 5) using smooth initial models. However, this gain in convergence speed and precision comes at the cost of reduced robustness. To inherit the robustness of ANR-bbsed FWI and the convergence benefits of MPE-FWI, we further propose a novel hybrid representation integrating INR with multi-resolution grids for FWI (termed IG-FWI, see Fig. 4.. This method induces a suitable eigenvalue decay tailored for FWI (see Theorem 5.2 and Fig. 2 (b)), thereby achieving trade-off between robustness and convergence (see Fig. (a)).
To our knowledge, this is the first work that analyzes the FWI problem from the NTK perspective and utilizes multigrid parametric encoding for FWI. In summary, our main contributions lie in threefold:
We develop a unified wave-based NTK framework for conventional FWI and CR-FWI. The eigenvalue analysis explains why CR-FWI reduces reliance initial models and exhibits slower high-frequency convergence. with numerical tests confirming these insights (see Fig. 5
Theory:
Method: Inspired by the eigenvalue decay analysis, we propose novel integrated representation combining INR and multi-grid for FWI. This method achieves a tailored eigenvalue decay that is trade-off (see Figs. and
more suitable for FWI, leading to better
Application: We extensively evaluate our methods under challenging scenarios, including inac- curate initial models, sparse sampling, seismic data with missing low frequencies and noise inter- ference, various benchmark datasets (i.e., Marmousi, SEG/EAGE Salt and Overthrust, and 2004 BP models), and more realistic 2014 Chevron blind data compared to conventional and existing CR-FWI methods. Inversion results show consistently superior performance (see Fig. (
comprehensive review and discussion of related work (e.g., including conventional FWI, neural representation-basel FWI, and NTK theory) is provided in Appendix A
FULL WAVEFORM INVERSION AND WAVE KERNEL
Let U and T represent the spatial and temporal domains, respectively. For E U and T, FWT aims to estimate the velocity model m(x) from the observed seismic data {uf (x, )) where up A O o Uj. Here, O denotes a linear observation operator, and (x t) corresponds to the full wavefield data with source wavelet function sy sx, t). Mathematically, the velocity model and the full wavefield data are governed by the wave equation (Engquist & Yang. 2022) expressed as follows:
m(x) (ay(()))
Px,z(x (x,,,,), E LET,
My (x. 0) = 0, Vu,
Buy
(x (xx0)=0) =0
X U, x U, L E,T,
(1)
where n is the unit outward normal to the boundary. This acoustic wave equation can be computed numerically via finite difference (Sun al. 2020) (see Appendix B.3). Define the forward operator
3

---

Published as a conference paper at ICLR 2026
9 : + uD juP uu...... ...:, UN_ and denote D as the sampling space domain. FWI aims to recover the true subsurface model m² by minimizing the misfit function between measurements oho - G[ml] and PDE-simulated seismic data Usyn A G[mll (Viiiexx & Operto 2009,. i.e.
min
[mm)
webe L2(DxT)
s.L. uoya (m) = G[ml]
where A is admissible set of velocity models and J is misfit functional. In the continuous-time limit (i.e., as the learning rate tends to zero), this gradient flow of velocity model is described by
dm
6J [m] ÔT 5m
59 [m] (u u
5m
where 7 is a pseudo-time. 6J /8m denotes the Fréchet derivative of the misfit functional J w.r.t. the parameter model m, and 89/ /8m is the sensitivity kernel operator. Applying the chain rule, we prove that the evolution of the wavefield during training is characterized by the following proposition.
Proposition 2.1 (Evolution of wavefield in FWI). Consider the loss function in equation 2 and the the synthetic data ugyn evolves according to the sollowing equation:
gradient flow in equation
Ou (x.
S1 Omme ((x, (x :x); (Mgn
( dt' dx';
(4)
where mave ((x, t), (x', t') = Su 10(x,) [ma] 50(x',e) (')
[midy is defined as the wave kernel.
smlly)
Remark 2.1. The proof is provided in Appendix E.1 The fitting of the synthetic data is driven by the
E..
data missa (u - w across the entire domain (data-driven), weighted by the wave kernel Q (physical guide). The wave kernel quantifies the coupling and correlation relationship between points (x, t) and (x', ,t' through point-wise perturbations in the model space. This often leads to uncoordinated updates and severe crosstalk, i.e., fitting one data point adversely affects others.
WAVE-BASED NEURAL TANGENT KERNEL
Let the space of neural network parameters be denoted by θ C R CR-FWI utilizes a coordinate- based neural network (-(-) U - R with learnable parameters 0 E θ, which maps a spatial coordinate x E U the corresponding velocity perturbation (x)x). Hence, the velocity model can be reparameterized as m²(x) (x) Fe (x) + mo(x) (where mo denotes the initial model) (Chen et al., 2025). Consider the following shallow fully-connected neural network with one hidden layer:
(x = V
w' <W"x x + 69) + b¹. 0 = (W', w . .'),
(5)
where X E U represents the coordinates input, a : IR + IR denotes element-wise nonlinear acti- vation function, 0 [W2, w 100 3 the complete set trainable parameters. In the usual initialization of NTK (Jacot et al.) 2018 Bonfanti et al., 2024a), all the weights and biases are ini- tialized to be independent and identically distributed as the standard normal distribution A 1). Here, we employ the NTK rescaling factor of 1/v as introduced in the original work (Jacot et al, 2018). This specific scaling is crucial for ensuring the convergence of the initialized neural tangent Kernel (Zhou & Yan 2024). Hence, the optimization variables of CR-FWI change from the discrete velocity model m in equation to neural parameters O. The objective function can be expressed as:
min 00e
J(O)
1 (0)
D
(DXT)))) S.L. un (0) 9 [m]]
When minimizing this PDE-constrained residual via gradient descent using the CR-FWI method, the evolution of the wavefield during this flow is characterized by the following proposition:
Proposition 3.1 (Evolution of wavefield in CR-FWI). Consider the loss function in equation and the gradient flow of the neural network in equation S with learning rate tends to zero, the synthetic data unn evolves according to the following equation:
Ju (x. = S.
1), (x), (u -.P--wPp -
w dd' dx',
where nrrk wave is the wave-based neural tangent kernel, defined as:
-Jc J.J. 89(x,t) t) [m]]
89(x', br((z)
Q mhe ((x, 1),
=
8mily)
[m]] K (y. 2: Oydydz.
(8)
where Kr (y, 2: Ei=
dmaly).
dmalz) is the standard NTK of networks with p parameters. d8, dB.
4

---

Published as a conference paper at ICLR 2026
Remark 3.1. The proof can be found in Appendix E.2 When set m²(x) m(x), the proposition 3.1 degrades into 2.1 (see Proposition E.3) The wave kernel can be expressed as follows:
(0x''x1" [m]] Ksly, z)dydz, J..o 6G(x, 1) (m] 8m(z)
Oware ((x,t), (x', (x,+'); f)):
6m(l)
where Koly,2 z) A 8(y - z) is Dirac kernel. Consequently, the wave-based NTK framework provides a unified framework for the simultaneous analysis of conventional FWI and CR-FWI.
Remark 3.2. Unlike the Dirac kernel, the wave-based NTK encodes smooth and architecture- dependent kernel that incorporates a joint guidance mechanism derived from the product of sensi- rivity kernels across different velocity model points, rather than relying on point-wise computations in the wave kernel, which enables coordinated global updates and may help mitigate cycle-skipping.
4 MAIN THEORETICAL RESULTS
Distinet from standard NTK, the wave-based NTK defined in Proposition 3.1 cannot converge to deterministic kernel at initialization and during the training process (see the following Theorem 4.1) when the width of networks tends to infinity, due the nonlinearity of FWI.
Theorem 4.1. Consider the network defined in equation [ which satisfies Assumption D.1 Let m be the convergent velocity model, and suppose that for every k E N. there exists a time Tk such that ( - m= S &K with Ej + 0 as k + c0. Let 0 be the neural parameters under Gaussian random initialization, i.e., {(0) (O 1), and denote by Q 1" the corresponding wave-based wave
NTK with width n at time t. Then, as the network width n. + 0o, the following hold:
(I) The wave-based NTK initialization converges in distribution
non-deterministic kernel:
@utk D Outk wave l
wave!
as 72 + 00.
(II) The wave-based NTK during training satisfies:
lim n n-+00 TE(0,00)
sup
|@wakel nM - ntk
almost surely.
0
Remark 4.1. The proof is provided in Appendix E.3 The proof core lies in the fact that the wave kernel changes in response to variations in the velocity model. Here, the global minimum conver- gence assumption can be guaranteed for the output of nonlinear functions (Luu et al.) [2020] (222), while having the potential to be extended to nonlinear operators future work. By Theorem 4. we conclude that the wave kernel is not a deterministic kernel during the training process. The conclu- sion that NTK is not deterministic kernel has also been presented in other literature and research areas, like nonlinear PINN (Bonfanti et al. 2024b) and classification problems (Yu et al. 2025).
Remark 4.2. Although the training dynamics of the wave-based NTK cannot be uniformly charac- terized by a fixed NTK matrix due to the stochasticity throughout the training trajectory, the time- varying NTK can still capture the local convergence behavior within sufficiently small intervals around time T using tools such as stochastic disferential equations and probability bounds (Evans) 2012), which is promising direction for future research. Specifically, within sufficiently small training window, the velocity model changes small, keeping the wave-based NTK nearly constant. This local stability allows the kernel's 's eigenvalue spectrum to quantitatively estimate local conver- gence rates. The wave-based NTK is a self-adjoint, compact, and non-negative definite operator (see Proposition F.2 which has spectral decomposition K(@
wave) =Et dR0k & @& (where K. is wave-bused NTK operator, {>x)e denotes eigenvalue, { is the corresponding eigen- function). The subsequent substitution of this decomposition into equation 8 leead to the following data missi evolution equation along different spectral directions (see Proposition E.4);
QQ( (u = =e -A (M 1.
(10)
where A = =dia((A1, X2....) is eigenvalue sequence and Qf (K. denotes the spectral projection operator. Hence, larger NTK eigenvalues yield faster error reduction along their corre- sponding directions, i.e., the eigenvalue decay rate determines the convergence behavior.
To explain the performance difference between conventional and INR-based FWI, the following theorem demonstrates that the latter exhibits faster eigenvalue decay rate, i.c., INR-based FWI can capture large-scale low- wow-frequency components to enhance the robustness and reduce the dependency on initial model accuracy, while its resolution of high-frequency details is consequently limited.
5

---

Published as a conference paper at ICLR 2026
Theorem 4.2. Consider the wave kernel θ wave in equation 9 and the wave-based NTK O equation with eigenvalues {> and {uz 1.0 Assume that the operator norm of K satisfies ||~]| S 1. Then, for all E N, we have μ1j S X3-
wave in 1==1 (arranged in non-incredne -on-incregsing order). respectively.
Remark 4.3. The proof is provided in Appendix [.4.1 is reasonable to assume ||K,|| S because the NTK is often normalized scaled such that its largest eigenvalue is bounded by Jacot et al.
2018 which is a common practice in theoretical analysis.
Theorem 4.2 indicates that the smooth kernel incorporated the continuous representation truncates the rate along its high-frequency convergence direction. As a result, the rapid decay of eigenvalues in the wave- based NTK implies that frequeeyn components (corresponding to larger eigenvalues) optimized rapidly, thereby alleviating the risk of cycle-skipping and decreasing reliance accurate initial model. In contrast, high-frequency components, associated with the sharply decaying slowly due their small eigenvalues, which lead to
tail of the eigenvalue spectrum, converge a slower reduction in error. By connecting the eigenvalue decay behavior of the wave-based NTK optimization performance, such robustness and convergence, it becomes essential to develop novel CR-FWI methods with deliberately tailored eigenvalue decay properties.
NOVEL CONTINUOUS REPRESENTATION FWI METHODS
We propose novel low-rank tensor function (LR-FWI) and multi-grid parametric encoding (MPE-FWT)-based FWI methods using contin- representations, shown in Fig. LR-FWI and MPE-FWI improve the high-frequency convergence rate by reducing the attenuation rate of eigenvalues (Theorem 5.I] To fur- ther achieve trade-off between robustness and convergence, we propose an integrated INR and multigrid representation (IG-FWI), whose eigenvalue decay rate is between INR-base and MPE-based FWI methods (Theorem 5.2
Continuous Representation FWI
INR-FWI
(b) LR-FWI
MPE-FWI
Cuurdinatos
Separate cosrdinate vectors
Multigrid
X
Z
Qunkk paini
P
ENR
ENE
INR-based FWI Traditional INR methods Sitzmann et al., 2020; Sun et al. 2023ai Yang & Ma 2025) use coordinate-bass
Velocity model
PDB Jereant medeling
Full Waveform Inversion
neural net-
work (e.g.. MLP) with special activation func- tions to parameterize the velocity model (Fig. (a)). The INR can be expressed as
Obijective function
Synthetie data
Observed data
Fo (x)
(11)
Figure 3: Pipeline of CR-FWI, CR-FWI employs (a) implicit neural representation, (b) low rank tensor function, (c) multi-grid parametric coding to represent the velocity parameter model and integrate the wave equation in a loop.
where = (H.) are weights of the MLP
and (-)) is scalar activation function. The
choice activation function significantly im-
pacts the high-resolution representation capa- bility. In seismic FWI, [Sun et al.] (2023)) de- veloped IFWI using SINRE's activation function (Sitzmann et al. 2020), while Yang & Ma 2025 proposed WinFWI incorporating Gabor wavelet activation functions (Saragadam et al. 2023
LR-FWI. The reparameterized subsurface geophysical parameters typically exhibit inherent struc- tural constraints, such as low-rank and non-local similarity (Li et al., 20244).. To embed these prop- erties, I.R methods decompose the velocity model using tensor factorization methods (e.g., Tucker and CP decomposition) and represent the low-dimensional tensor separately using INRs (Luo et al. 2023 as shown Fig. (b). The general formulation of 2D LR-FWI can be expressed as:
(x)x) = [C; Feg (x) = Fe, (a1) x C x Fo (22)
(12)
where x = (x1, x22 denotes spatial coordinates, Fo. R R lxn Fez : Xxa are one mensional coordinate networks (c.g., INRs), and C E R xr2 is the matrix. Hence, the neural parameters are 0 = {0,00, 02, C}. Moreover, the LR- FWI method enhances the convergence rate for high-frequency components due to appropriate eigenvalue decay rate, which have empirically

---

Published as a conference paper at ICLR 2026
verified through numerical experiments (see Fig. 5 (c)). However, providing rigorous mathemati- cal proof of the eigenvalue decay rate remains challenging due to the complex structure of the tensor product, which will be key objective of our future research based on tensor decomposition theory (Kolda & Bader 2009) and matrix analysis (Horn & Johnson 2012).
MPE-based FWI. Multi-grid parametric encoding employs trainable auxiliary data structures embedding spaces. As illustrated in Fig. 3 (c), MPE-based FWI leverages multigrid hash encoding (Müller al. (022) represent the velocity model. The hash function ((-) : U + 2 Xxy maps a coordinate point x E U to a feature vector via hxx) " where ny denotes the number of multigrid levels and ny the number of features per grid. Then, these interpolated features are passed through a lightweight INR to produce the physical velocity value. The overall representation can be expressed as:
(e.g.. grid-based representations) to construct higher-dimension
Fx(x) MLP (h(x); where h(x) ①
T mod T,
(13)
where Ti are large prime numbers, T is the hash table size, and @ denotes the bit-wise exclusive-or operation. For each query point the feature is obtained by interpolating the embeddings from adjacent grid vertices across multiple resolution levels (Audia et al.) 2025 Luo et al. 2025).
The MPE approach integrates the representational flexibility of INR with the efficient spatial lookup offered by multigrid hash encoding. leveraging a grid structure to accelerate convergence and alle- viate spectral bias (Audia et al., 2025). The effectiveness of MPE-FWI is explained in Theorem 5.1 which provides a lower bound on the eigenvalues of its wave-based NTK.
Theorem 5.1. Let TMR and Ouu be the wave-based NTK INR-based FWI and MPE-based FWI with eigenvalues ('i(O ) and {><(OMMP)) MPF)) == in non-increasing order, respectively. Then, for a dataset of size n, the i-th eigenvalues satisfy >e MPE)) 2 >x INR).
Remark 5.1. The proof can be found in Appendix E.4.2 The proof of this theorem is bused on the fact that the eigenvalues of NTK of MPE are not smaller than those in INR, i.e., the inequality (Audia 2 >e where IIN and KMPE are the NTKs for INR and the same INR composed with MPE, respectively, and K is a positive semi-definte matrix arising from the learnable grid parameters in MPE. This theorem guarantees strict elevation of the entire eigenvalue spectrum, accelerating the learning of high-frequency components, especially starting from smooth initial velocity model. However, the eigenvalue decay rate is inadequate to address the ill- ill-posedness and nonlinearity inherent in FWI (see Tab. 7 Therefore, a novel continuous representation tailored for FWI needs be developed. one that incorporates a suitable eigenvalue decay rate to achieve a more balanced trade-off between convergence and robustness.
et al. (2025) X₂ (KMPE)
5.1 INTEGRATED INR-MULTIGRID HYBRID REPRESENTATION
Based eigenvalue analysis for wave ker- nel and wave-based NTK, the high-frequency convergence of INR-based FWI methods in- herently limited due to faster eigenvalue de- cay (Theorem 4.2). whereas the robustness of classical FWI and MPE-based FWI methods remains constrained due to slower eigenvalue decay (Theorem 5.T). To achieve balanced robustness aobustnes-coonvergenee convergence trade-off, we propose novel continuous representation by integrating the INR and MPE for FWI, termed IG-FWI, as shown in Fig. 4 IG-FWI employs a tiny INR to implicitly encode smooth features into the la- tent space, which are then combined with multi- grid encoding features. Next, these features are combined using a tiny MLP The reparameterized velocity model can be expressed as follows:
INR-Grid Representation FWI
MLP
Multigrid
POR Surward medeling
Quarted
Synthetic data
INR
Velucity model
Observed data
Figure 4: Pipeline of the proposed INR-Grid hy- brid representation for FWI This method com- bines encoding features from INR and MPE, which are then fused using an MLP network.
(x) = MLP(v(x)), where v(x) = Va h(x) @ (1 - a). T(x),
(14)
where lt(-) is a hash encoding defined in equation 13 I(-) is tiny INR defined in equation 11 and hybrid features E +n.) is formed by splicing h(.)) and 7(-) with scaling fa and v 0,
7

---

Published as a conference paper at ICLR 2026
respectively, where n, denotes the output dimension of I()). The following theorem indicates that the eigenvalue decay of IG-FWI is, as desired, between that of INR-based FWI and MPE-FWI.
Theorem 5.2. Let OiG be the NTK of IG-FWI with eigenvalues (Nr(ON))) When INR and MPE features are normulized such that their gradient norms are comparable, then all i E N,
(,(O)N)) INR) S (,(6))) S >.(O
Hence, the eigenvalue decay rate of (O lies between those of (; and X₂ (g
Remark 5.2. The proof is provided in Appendix [.4.3 By balancing the eigenvalue decay in this manner, IG-FWI successfully inherits the robustness of INR-based methods and the superior con- vergence of MPEEbasel methods, thereby achieving more accurate and reliable inversion results.
EXPERIMENTAL ANALYSIS AND APPLICATIONS
Baselines and Implementation Details. We carefully select widely used conventional FWI meth- ods, including automatic differentiation-based FWI (i.e., ADFWI) (Liu et al.) 2025a). ADFWI with TV regularization (i.e., ADFWI-TV) (Esser et al. 2018), ADFWT with weighted envelope correlation- based loss function (i.c., WECI-FWI) (Song al. 2023), and frequency-basee multi- scale inversion FWI (i.c., MS-FWI) (Fichtner et al. 2013), and existing state-of-the-an CR-FWI methods, including INR-based FWI with SINRE's activation function (i.e. IFWT) (Sun et al., 2023a) and Gabor wavelet activation function (i.e., WinFWT) Yang & Ma (2025) to compare with our proposed novel CR-FWI methods, including multigrid parametric encoding FWI (i.e., MPE- FWT), low-rank tensor decomposition-based FWI (i.e., LR-FWII, and INR-Grid based FWI (i.e.. IG-FWI). We provide more implementation details for each baseline in the Appendix B. More- over, we also provide comparisons with InversionNet (Wu & Lin 2019b) on OpenFWI datasets.
Datasets and Scenarios. We evaluate the baselines and our proposed methods on Marmousi [Brougois et al. 1990). 2D SEG/EAGE Salt and Overthrust (Aminzadch) 1996), and 2004 BP (Bil- lette& & Saaandsber--Dhii 2005) models using different initial velocity models (i.e., smooth, linear, and constant velocity models). Moreover, we test the robustness w.r.t. the seismic data quality (i.e., noise interference, missing low frequencies, and sparse sampling) for the Marmousi model. The detailed forward modeling and acquisition system settings are provided in Appendix B.2
6.1 EMPIRICAL VALIDATION OF OUR WAVE BASED NTK THEORETICAL RESULTS
PRADDV ribooas mom
100
63
300
200
Buoch
(c)
Figure 5: (a) Mean and standard deviation of the Frobenius norm of the wave-based NTK at initial- ization. (b) Relative Frobenius norm during training. (c) Eigenvalues decay using different methods.
To verify that the wave-based NTK does not converge to a fixed kernel either at initialization or during training, we conduct NTK experiments on one-dimensional FWI problem. The detailed experimental sctup is described in Appendix B.4] with the results illustrated in Fig. (a) and (b). Furthermore, we compare the eigenvalue decay of the wave-based NTK using ADFWI (Liu et al.,
2025a), based FWI (Sun et al., 2023a), and our proposed MPE-FWI, FR- and IG-FWI methods, shown in Fig. (c). From these experiments, we draw two key observations:
Obs 1: The wave-based NTK is non-stationary, not only during the inversion process, but also at initialization, even when the width tends to infinity, which supports Theorem 4.]
Obs 2: The eigenvalue decay is slowest in conventional FWI, faster in baE-based FWI, and fastest in INR- FWI. The proposed IG-FWI exhibits a decay rate between those of MPE- based and INR-based FWI, consistent with Theorems 4.2 5.1 552] The spectrum of LR-FWI lies a moderate region as well, and its theoretical analysis warrants discussion the future.
6.2 ROBUSTNESS COMPARISONS WITH INITIAL MODELS AND DATA QUALITIES
To evaluate the performance the FWI methods, we conduct a series of tests on both the baseline and our proposed CR-FWI methods. The experiments are carried out using four challenging datasets
8

---

Published as a conference paper at ICLR 2026
Table 1: Performance comparison (MSE) of conventional FWT (ADFWI, ADFWI-TV, WECI-FWI, MS-FWII and CR-FWI (IFWI, WinFWI, LR-FWI, MPE-FWI, IG-FWI) methods. Best results are highlighted in bold, and second-bor are underlined. Complete results see Appendix B.6.1
Marmousi Model
Overthrust Model
Salt Model
2004 BP Model
Compating Resource
Method
|Smooth Constant G-Noise F-Cutoff S-Shots S-Shot:|Smoot |Smooth
Constant
Smooth Constant
Smooth Constami
Time Params Memory
ADFWI
0.2132
1.1522
0.5422
0.4975
0.4730
0.0592
1.3364
0.6462
0.6337
0.1003
0.4281
1.89 94,00
17.09
ADFWI-TV
0.2449
0,9386
0.3875
0.3764
0.3547
0.1154
1.3122
0.5041
0.5318
0.0995
0,4299
1.92 94,00
17.09
WECI-FWI
0,2977
1,2537
0.6366
0,3220
0.3357
0.1182
1,5526
0,3032
1.0293
0.0951
0.3847
3.12 94,00
17.56
MS-FWI
1.1313
0.3547
0.3325
0.0612
1.0802
0.5358
0.9234
0.0672
0.4667
17.09
JFWI
0.1907
0,9474
0.3374
0,3358
0.3483
0.0683
0.6432
0.1201
0.7412
0.0719
0.1412
1.88
50.05
17.48
WinFWI
0.2013
0,4689
0,3514
0.3460
0.3239
0.0682
0.5738
0.1223
0,3549
0,0812
0.1083
1:90
60.63
18.20
LR-FWI
0,1638
0.2893
0.2553
0.2276
0.2641
0.0548
0.1592
0.2785
0,2368
0.0481
0.1248
1.92
77.91
17.10
MPE-FWI
0.1427
2.2266
0.2990
0.3322
0.3358
0.0613
1 1,2887
0.8534
1.1008
0.0586
1.602
14.26
17.25
IG-FWI
0.1423
0.2961
0.2151
0.1846
0.1654
0.0587
0.5724
0.1181
0.1883
0,0521
0.0843
191
39.34
17.41
G-Noise: 8:g Gaussian noise added to data; F-Cutoff: Data with missing frequencies below 6 Hv; S-Shots: Only 5 shot gathers used;
Time: average running time for one epoch (s): Params: Parameter count (K): Memory: GPU memory overhead (GB).
Velocity Model Datasets
Conventional FWI Methods
Continuous Representation FWI Methods
Ground truth
Intia model
ADFWI
MS-FWI
IFWI
LR-FWI
IG+FI
Marmaell modil
SENO IEA meli
MER OZIN
MST-A1997
1SF. CI1I
MSE ENT
MAD 8390
mahi
kkiinski
VSE BM
MSE 03221
MBE: 92214
MAP-
Conora Bod
SEE BA31
NSE EX24
VSG 67412
MSE 3:338
NORE LING
Figure 6: Conventional FWI and CR-FWI performance comparison using different initial velocity
models on different velocity model datasets. More results are in the Appendices from B.6.2 to B.6.6
with different initial models, as well three data-degraded scenarios based on the Marmousi model. Partial results are shown in Fig. 6 and Tab. (computing resource metrics are evaluated using a single shot of the BP model per training run). From these results, we draw three main observations:
Obs 3: Conventional FWI (e.g.. ADFWI and MS-FWI) and MPE-based FWI methods can re- cover high-precision velocity models with rapid high-frequency convergence when an accurate and smooth initial velocity model is available. However, the performance of these methods is highly sensitive to the quality of the initial model and the seismic data. This limitation can be attributed to the slow decay rate of eigenvalues, established in Theorems 4.2 and Obs 4: baRed FWI methods (e.g., IFWI and WinFWII can recover satisfactory inversion results even when starting from constant initial model. However, this enhanced initial model robustness comes at the expense of a slower convergence rate and reduced resolution, which can be attributed to the relatively fast eigenvalue decay rate of the wave-based NTK (Theorem 4.2
Obs 5: Our proposed FWI methods (i.e., I.RFFW and IG-FWI) achieve a trade-off between high- precision results and robustness w.r.t. the initial model and data quality, which can be attributed to the suitably controlled eigenvalue decay rate of the wave-based NTK for FWI (Theorem 5 5.2).
Considering the inversion crime and computational challenges, we conduct comparison on a more realistic 2014 4 Chevron model (see Appendix C.I) and the 3D Overthrust model (see Appendix C.2)
which show that our proposed CRFWI can be used in more realistic large-scal, scale inversion problems.
6.3 CONVERGENCE BEHAVIOR ANALYSIS FOR CRFWI
To validate the convergence behavior using different FWI methods, we provide convergence curves of velocity model error (see Fig. 7 and generated velocity visualizations corresponding different
9

---

Published as a conference paper at ICLR 2026
(2) Marmousi model
(b) Overtarust model
[C) Salt body model
(d) 2004 UP model
Figure 7: Convergence curves of velocity model MSE using different FWI methods and different
datasets with constant initial models. (a) Marmousi. (b) Overthrust. (c) Salt body. (d) 2004 BP
50 spach
130 epach
200 spoch
400 epoch
700 spach
1000 epocis
IG-WI MPL-INE IFWI
Figure 8: Generated velocity visualizations corresponding to different iterations using IFWI, MPE- FWI, and IG-FWI methods on 2004 BP model. More detailed results are in Appendix (B.677
iterations (see Fig. 8). detailed in Appendix B.6.7 These results indicate that conventional FWT and MPE-FWI methods achieve rapid convergence of high-frequenee details at the cost of increased sensitivity, while IFWI and WinFWI enhance robustness at the expense of high-frequency details of inversion results. Our proposed IG-FWI and LR-FWI methods achieve a more balanced trade-off, maintaining robustness without sacrificing the convergence rate for high-frequency information.
6.4 ABLATION EXPERIMENTS FOR IG-FWI
We conduct a comprehensive ablation study to investi- gate the impact of key parameters (c.g., different INR fre- quency bases, different grid scales, and weighting factor), shown in Fig. The results show that both exces- sively high and low grid resolutions in MPE, as well as frequency settings in INR, may reduce the quality of in- version results, which is consistent with our theoretical results. Moreover, our proposed IG-FWI is robust to the weighting factor, which is typically set to 0.5.
12
Fregancy
XBE
Rosokzion
RZ2
6.5 COMPARISONS WITH DATA-DRIVEN METHOD Evaluations on the OpenFWI dataset show that our pro- posed methods outperform InversionNet (Wu & Lin 2019b) in reconstructing velocity structures from noisy data (see Tab. 2 as detailed in Appendix C.3
64
05
60
Weightin
(n) Marmousi model
(3) Salt model
Figure 9: Ablation study for frequency, grid resolution, and weighting factor.
7 CONCLUSION
Table 2: MSE of data-driven FWI and CRFWI. Best results are highlighted in bold, and second- are underlined.
In this study, establish unified theoretical analy- sis framework based on the wave kernel and wave-based NTK to investigate the convergence and robustness of both conventional and continuous representation FWI. Our analysis reveals that the wave kernel and wave-based NTK are non-stationary and exhibit distinct eigenvalue decay patterns, i.e., the wave kernel shows slow decay, whereas the wave-based NTK decays rapidly. From the theory of spectral analysis, this difference explains why conventional FWI achieves fast convergence but lacks robustness, while INR-based CR-FWI offers improved robustness at the cost slower convergence. Motivated by these findings, we generalize the INR-based CR-FWI to a broader continuous representation FWI framework, which includes the proposed FWFWI and MPE-FWI methods. Furthermore, we introduce a hybrid representation that integrates INR with multigrid strategy for FWI, aiming balance robustness and convergence by better controlling the eigenvalue decay. Tests on challenging datasets and diverse seismic data scenarios show the superior performance of the proposed CR-FWI methods.
Methods
|InwNer ADFWI FR-FWI IG-FWI
FaliVel
36.78
21.30
11.72
6.65
6.94
Curve
94.16
57.07
16.87
9.89
7.93
FaltPault
113.18
48.23
46.59
19.59
28.84
CurveFault
10009
99.14
Style
53.76
27.66
48.25
14.56
17.08
10

---

Published as a conference paper at ICLR 2026
ACKNOWLEDGEMENTS
This work was supported by the Key Program of the National Natural Science Foundation of China (grant 42530802), the National Natural Science Foundation of China (No. 124B2029), the Key Program of the National Natural Science Foundation of China (grant 62476214), Funda- mental and Interdisciplinary Disciplines Breakthrough Plan of Ministry of Education China (YYB205XDXM101) and Tianyuan Fund for Mathematics of the National Natural Science Foun- dation of China (Grant No. 12426105).
REFERENCES
Ayrat Abdullin and Umair Bin Waheed. Generalization capability of data-driven deep learning models for seismic full-waveform inversion: An example using the openfwi dataset. In Third International Meeting for Applied Geoscience & Energy, pp. 1083-1087. Society of Exploration Geophysicists and American Association of Petroleum ... 2023.
Sina Alemohammad, Zichao Wang, Randall Balestriero, and Richard Baraniuk. The recurrent neural tangent kernel. Xiviv preprint arXiv:2006. 2020.
Fred Aminzadeh. 3-d salt and overthrust seismic models. 1996.
John Anderson, Lijian Tan, and Don Wang. Time-reversal checkpointing methods for rtm and fwi. Geophysics, 77(4):S93-SD S103, 2012.
Alcksandr Aravkin, Tristan Van Leeuwen, and Felix Herrmann. Robust full-waveform inversion using the student's t-distribution. In SEG Technical Program Expanded Abstracts 20/ pp. 2669- 2673. Society Exploration Geophysicists, 2011.
Sanjeev Arora, Simon Du, Wei Hu, Zhiyuan Li, and Ruosong Wang. Fine-grained analysis of op- timization and generalization for overparameterized two-layer neural networks. In International conference on machine learning, pp. 322-332. PMLR, 2019.
Samuel Audia, Soheil Feizi, Matthias Zwicker, and Dinesh Manocha. How learnable grids recover fine detail in low dimensions: A neural tangent kernel analysis of multigrid parametric encodings. arXiv preprint 2rXiv:2504.13414 2025.
Alberto Bietti and Julien Mairal. On the inductive bias of neural tangent kernels, Advances Neural Information Processing Systems, 32, 2019.
FJ Billette and Sverre Brandsberg- Dahl. The 2004 bp velocity benchmark. 67th EAGE Confer- ence & Exhibition, pp. cp-1. European Association of Geoscientists & Engineers, 2005,
Patrick Billingsley. Convergence of probability measures. John Wiley Sons, 2013.
Andrea Bonfanti, Giuseppe Bruno, and Cristina Cipriani. The challenges of the nonlinear regime for pwysies-inoommneeeeee informed neural networks. Advances in Neural Information Processing Systems, 37: 41852-41881, 2024a.
Andrea Bonfanti, Giuseppe Bruno, and Cristina Cipriani. The challenges of the nonlinear regime for physics-informed neural networks. Advances in Neural Information Processing Systems, 37: 41852-41881, 2024b.
A Brougois, Marielle Bourget, Patriek Lailly, Michel Poulet, Patrice Ricarte, and Roelof Versteeg, Marmousi, model and data. In EAEG workshop aspects of seismic data inversion, pp- cp-108. European Association of Geoscientists & Engineers, 1990.
Carey Bunks, Fatimetou M Saleck, Stephane Zaleski, and Guy Chavent. Multiscale seismic wave- form inversion. Geophysics, 6055)::457-1433, 1995.
Yuan Cao, Zhiying Fang, Yue Wu, Ding- Xuan Zhou, and Quanquan Gu. Towards understanding the spectral bias of deep learning. arXiv preprint 1912,09 2019.

---

Published as a conference paper at ICLR 2026
Fuqiang Chen, Daniel Peter, and Matteo Ravasi. Cycle-skipping mitigation using misfit measure- ments based on differentiable dynamic time warping. GEOPHYSICS, 87(4) R335, 2022. doi: 10. 10.11999022221-0598
Ruihua Chen, Bangyu Wu, Meng Li, and Kai Yang. Initial model incorporation for deep learning fwi: Pretraining or denormalization? arXiv preprint arXiv:2506.0544 2025.
Yangkang Chen, Hanming Chen, Kui Xiang, and Xiaohong Chen. Geological structure guided well log interpolation for high-fidelity full waveform inversion. Geophysical Supplements to the Monthly Notices of the Royal Astronomical Society, 207())1113-1331, 2016.
Zixiang Chen, Yuan Cao, Quanquan Gu, and Tong Zhang. A generalized neural tangent kernel analysis for two-layer neural networks. Advances in Neural Information Processing Systems, 33: 13363-13373, 2020.
S Cui. Introduction modern theory of partial differential equations, 2015.
Chengyuan Deng, Shihang Feng, Hanchen Wang, Xitong Zhang, Peng Jin, Yinan Feng, Qili Zeng, Yinpeng Chen, and Youzuo Lin. Openfwi: Large-scale multi-structural benchmark datasets for full waveform inversion. Advances in Neural Information Processing Systems, 35:6007-6020, 2022.
Bo Du, Jian Sun, Anqi Jia, Ning Wang, and Huaishan Liu. Physics-informed robust and implicit full waveform inversion without prior and low-frequency information. IEEE Transactions on Geoscience and Remote Sensing, 2024.
Björ Engquist and Yunan Yang. Optimal transport based seismic inversion: Beyond cycle skipping. Communications on Pure and Applied Mathematics, 75(10):2201-2244, 2022.
Ernie Esser, Lluis Guasch, Tristan van Leeuwen, Aleksandr Y Aravkin, and Felix Herrmann Total variation regularization strategies in full-waveform inversion. SIAM Journal on Imaging Sciences, 11(1):376-4000 2018.
Lawrence C Evans. Partial differential equations: Second edition. Wadsworth & Brooks/cole Math- ematics, 19(1):211-223 2010.
Lawrence Evans. An introduction to stochastic differential equations, volume 82. American Mathematical Soc., 2012.
Gabriel Fublen-Ouellet, Erwan Gloaguen, and Bernard Giroux. A stochastic I-bfgs approach for full-waveform inversion. SEG Technical Program Expanded Abstracts 2017, pp. 1622-1627. Society of Exploration Geophysicists, 2017.
Andreas Fichtner, Jeannot Trampert, Paul Cupillard, Erdine Saygin, Tuncay Taymaz, Yann Capdev- ille, and Antonio Villasenor. Multiscale full waveform inversion. Geophysical Journal Interna- tional, (4<(1)55345556 2013.
Weiye Gan, Yicheng Li, Qian Lin, and Zuoqiang Shi. Neural tangent kernel of neural networks with loss informed by differential operators. arXiv preprint arXiv:2503.1 1029, 2025.
Lluis Guasch, Oscar Calderón Agudo, Meng-Xing Tang, Parashkev Nachev, and Michael Warner Full-waveform inversion imaging of the human brain. NPJ digital medicine, 3(1):28, 2020.
Naveen Gupta, Medha Sawhney, Arka Daw, Youzuo Lin, and Anuj Karpatne. unified framework for forward and inverse problems in subsurface imaging using latent space translations. arXiv preprint arXiv:2410.1122 11247, 2024.
Shravan M Hanasoge. Full waveform inversion of solar interior flows. The Astrophysical Journal, 797(1):23, 2014.
Leon Hermann, Tim Bürchner, Felix Dietrich, and Stefan Kollmannsberger. On the use of neural networks for full waveform inversion. Computer Methods in Applied Mechanics and Engineering, 415:116278,, 2023. ISSN 0045-7825. doi: https://bo.org//0.011.jio 2023.116278 URL
www. htt.:///ceeettteteeec../.eeeaaaa
SD645782252X04410
12

---

Published as a conference paper at ICLR 2026
Roger A Horn and Charles R Johnson. Matrix analysis. Cambridge university press, 2012.
Arthur Jacot, Franck Gabriel, and Clément Hongler. Neural tangent kernel: Convergence and gen- eralization in neural networks. Advances in neural information processing systems, 31, 2018.
Navnit Jha and Ekansh Mallik. Gpinn with neural tangent kernel technique for nonlinear point boundary value problems. Neural Processing Letters, 56(3):192, 2024.
Peng Jin, Xitong Zhang, Yinpeng Chen, Sharon X Huang. Zicheng Liu, and Youzuo Lin. Unsuper- vised learning of full-waveform inversion: Connecting CNN and partial differential equation in loop. In International Conference Learning Representations, 2022.
Tamara G Kolda and Brett W Bader. Tensor decompositions and applications. SIAM review, 51)3): 455-500, 2009.
Jiahang Li, Hitoshi Mikada, and Junichi Takekawa. Inexact augmented lagrangian method-based full-waveform inversion with randomized singular value decomposition. Journal of Geophysics and Engineering, 21(2):572-597, 2024a.
Shiqian Li, Zhi Li, Zhancun Mu, Shiji Xin, Zhixiang Dai, Kuangdai Leng, Ruihua Zhang, Xiaodong Song, and Yixin Zhu. Globaltomo: A global dataset for physics-ml seismic wavefield modeling and fwi. arXiv preprint arXiv:2406. /8202, 2024b.
Chaoyue Liu, Libin Zhu, and Misha Belkin. On the linearity of large non-linear models: when and why the tangent kernel constant. Advances Neural Information Processing Systems, 33: 15954-15964, 2020.
Chaoyue Liu, Libin Zhu, and Mikhail Belkin. Loss landscapes and optimization in over- parameterized non-linear systems and neural networks. Applied and Computational Harmonic Analysis, 59:85-116 16, 2022.
Feng Liu, Haipeng Li, Guangyuan Zou, and Junlun Li. Automatic differentiation-based full wave- form inversion with flexible workflows. Journal of Geophysical Research: Machine Learning and Computation, 20)xc2024J0000542, 2025a.
Fengliang Liu, Hui Zhou, Hanming Chen, Lingqian Wang, and Yuxin Fu. A full- inversion method based on structural tensor constraints. IEEE Transactions on Geoscience and Remote Sensing, 2025b.
Youshan Liu, Jiwen Teng, Tao Xu, José Badal, Qinya Liu, and Bing Zhou. Effects of conjugate gradient methods and step-length formulas the multiscale full waveform inversion in time domain: Numerical experiments. Pure and Applied Geophysics, 174:1983-2006, 2017.
Yisi Luo, Xile Zhao, Zhemin Li, Michael K Ng, and Deyu Meng. Low-rank tensor function repre- sentation for diletiidmmensionll data recovery. IEEE transactions on pattern analysis and machine intelligence, (6655)33311-336) 2023.
Yisi Luo, Xile Zhao, and Deyu Meng. Continuous representation methods, theories, and applica- tions: An overview and perspectives. arXiv preprint arXiv:2505.15222. 2025.
Levi D McClenny and Ulisses M Braga-Neto. Self-adaptive physics-informed neural networks. Journal of Computational Physics, 474:111722, 2023.
Ludovic Métivier, Romain Brossier, Stéphane Operto, and Jean Virieux. Full waveform inversion and the truncated newton method. SIAM review, 59(1):153-195 2017.
Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng. Nerf: Representing scenes neural radiance fields for view synthesis. Communications of the ACM, 65(1):99-106, 2021.
Thomas Müller, Alex Evans, Christoph Schied, and Alexander Keller. Instant neural graphics prim- itives with a multiresolution hash encoding. ACM transactions on graphics (TOG), 41(4):1-15, 2022.
13

---

Published as a conference paper at ICLR 2026
Luan T Nguyen and Ryan T Modrak. Ultrasonic wavefield inversion and migration in complex heterogeneous structures: numerical imaging and nondestructive testing experiments. Ulra- sonics, 82:357-370, 2018.
Mike Nguyen and Nicole Mücke. Optimal convergence rates neural operators. arXiv preprint 1rXiv:2412.17518, 2024.
Bernt Oksendal. Stochastic differential equations: an introduction with applications. Springer Science & Business Media, 2013.
Damir Pasalic and Ray McGarry. Convolutional perfectly matched layer for isotropic and anisotropic acoustic wave equations. In SEG International Exposition and Annual Meeting, pp- SEG-2010. SEG, 2010.
Shaoxiang Qin, Fuyuan Lyu, Wenhui Peng, Dingyang Geng, Wang, Xing Tang, Sylvie Leroyer, Naiping Gao, Xue Liu, and Liangzhu Leon Wang. Toward better understanding of fourier neural operators from a spectral perspective. arXiv preprint arXiv:2404.07206 2024.
Majid
Christian Huber, Khemraj Shukla, and George Em Karniadakis. Physics- informed neural networks (pinns) for wave propagation and full waveform inversions. Journal of 2022.
Geophysical Research: Solid Earth, (27(5):2001JB022111
Vishwanath Saragadam, Daniel LeJeune, Jasper Tan, Guha Balakrishnan, Ashok Veeraraghavan, and Richard G Baraniuk. Wire: Wavelet implicit neural representations. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 18507-18516, 2023.
Gerard T Schuster, Yuqing Chen, and Shihang Feng. Review of physics-informed machine learning inversion of geophysical data. Geophysics, 89(6): 1-1,, 2024.
Matthieu Simeoni, Sepand Kashani, Paul Hurley, and Martin Vetterli. Deepwave: recurrent neural- network for real-time acoustic imaging. Advances In Neural Information Processing Systems, 32, 2019.
Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wetzstein. Im- plicit neural representations with periodic activation functions. Advances neural information processing systems, 33:7462-744 -473,,7773, 2020.
Chao Song and Tariq A Alkhalifah. Wavefield reconstruction inversion via chhysics-inffrmee neural networks. IEEE Transactions on Geoscience and Remote Sensing, 60:1-12, 2021.
Chao Song, Yanghua Wang. Alan Richardson, and Cai Liu. Weighted envelope correlation-based waveform inversion using automatic differentiation. IEEE Transactions on Geoscience and Re- mote Sensing, 61:1-11, 2023.
Bingbing Sun and Tariq Alkhalifah. MI-descent: optimization algorithm for full-waveform inversion using machine learning. Geophysics, (566):477-R492 2020.
Jian Sun, Zhan Niu, Kristopher A Innanen, Junxiao Li, and Daniel O Trad. A theory-guided deep- learning formulation and optimization of seismic waveform inversion. Geophysics, 85(2):887- R99, 2020.
Jian Sun, Kristopher Innanen, Tianze Zhang, and Daniel Trad. Implicit seismic full waveform inversion with deep neural representation. Journal of Geophysical Research: Solid Earth, 128(3): e2022J8025964, 2023a.
Mengyao Sun, Jie Zhang, and Wei Zhang. Alternating first-arrival traveltime tomography and wave- form inversion for near-surface imaging. Geophysics, 82(4) R257, 2017.
Pengpeng Sun, Fangshu Yang. Hongxian Liang, and Jianwei Ma. Full-waveform inversion using
learned regularization. IEEE Transactions on Geoscience and Remote Sensing, 61:1-15, 2023b.
Jeroen Tromp. Seismic wavefield imaging of earth's 's interior across scales. Nature Reviews Earth & Enviroument, 1(1):40-53, 2020.
14

---

Published as a conference paper at ICLR 2026
Dirk Philip van Herwaarden, Christian Boehm, Michael Afanasiev, Solvi Thrastarson, Lion Krischer, Jeannot Trampert, and Andreas Fichtner. Accelerated fall-wwveform inversion using dynamic
Geophysical Journal International, 221(2):1427-1438, 2020.
Roelof Versteeg. The marmousi experience: Velocity model determination on synthetic complex data set. The Leading Edge, 13(9):927-95) 1994.
Jean Virieux and Stéphane Operto. An overview of full-waveform inversion in exploration geo- physics. Geophysics, 7466:WCCIWCC26, 2009.
Jean Virieux, Amir Asnaashari, Romain Brossier, Ludovic Métivier, Alessandra Ribodetti, and Wei Zhou. An introduction to full waveform inversion. In Encyclopedia of exploration geophysics, pp. RI-1. Society of Exploration Geophysicists, 2017.
Sifan Wang, Xinling Yu, and Paris Perdikaris. When and why pinns fail to train: A neural tangent kernel perspective. Journal of Computational Physics, 449:1 10768, 2022.
Yukai Wo, Jingjing Zong, Hua- Zhou, Kai Li, Yubo Yue, and Xuri Huang. Near-surface struc- tural regularization for full waveform inversion using directional total variation. IEEE Transac- fions on Geoscience and Remote Sensing, 2025.
Ru-Shan Wu, Jingrui Luo, and Bangyu Wu. Seismic envelope inversion and modulation signal model. Geophysics, 79()):AAI-WA224 2014.
Yue Wu and Youzuo Lin. Inversionnet: An efficient and accurate data-driven full waveform inver- sion. IEEE Transactions on Compulational Imaging, 6:419-433, 2019a.
Yue Wu and Youzuo Lin. Inversionnet: An efficient and accurate data-driven full waveform inver- sion. IEEE Transactions on Computational Imaging, 1:419-433, 2019b.
Xianliang Xu, Ye Li, and Zhongyi Huang. Convergence analysis of wide shallow neural operators within the framework of neural tangent kernel. arXiv preprint arXiv:4412.05545, 2024.
Zhi-Qin John Xu, Yaoyu Zhang, and Tao Luo. Overview frequency principle/speccraal bias in deep learning. Communications Applied Mathematics and Computation, 7(3):827-864, 2025.
Zichao Yan and Yanfei Wang. Full waveform inversion with sparse structure constrained regulariza- tion. Journal of Inverse and Ill-posed Problems, 26(2):243-257, 2018.
Dinghui Yang. Xingpeng Dong, Jiandong Huang, Zhilong Fang, Xueyuan Huang, Shaolin Liu, Mengxue Liu, and Weijuan Meng. High-resolution full waveform seismic imaging: Progresses, challenges, and prospects. Science China Earth Sciences, pp. 1-28, 2025.
Fangshu Yang and Jianwei Ma. Wasserstein distance-based full-waveform inversion with regu- larizer powered by learned gradient. IEEE Transactions on Geoscience and Remote Sensing, 61: 1-13, 2023.
Fangshu Yang and Jianwei Ma. Gabor-wavelet-aciivtion implicit neural learning for full waveform inversion. Geophysics, 90(3):1-78, 2025.
Zixiong Yu, Songtao Tian, and Guhan Chen. Divergence of empirical neural tangent kernel in classification problems. arXiv preprint arXiv:2504.11130 2025.
Zhongping Zhang, Yue Wu, Zheng Zhou, and Youzuo Lin. Velocitygan: Subsurface velocity image estimation using conditional adversarial networks. In 2019 IEEE Winter Conference on Applica- tions of Computer Vision (WACV), pp. 705-714. IEEE, 2019.
Hongkai Zheng, Wenda Chu, Bingliang Zhang, Zihui Wu, Austin Wang, Berthy T Feng, Caifeng Zou, Yu Sun, Nikola Kovachki, Zachary Ross, et al. Inversebench: Benchmarking and--and- play diffusion priors for inverse problems in physical sciences. arXiv preprint arXiv:2503. 1043, 2025.
Zijian Zhou and Zhenya Yan. Is the neural tangent kernel of pinns deep learning general partial differential equations always convergent? Physica D: Nonlinear Phenomena, 457:133987, 2024.
15

---

Published as a conference paper at ICLR 2026
Min Zhu, Shihang Feng, Youzuo Lin, and Lu Lu. Fourier-deeponet Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness. Computer Methods in Applied Mechanics and Engineering, 416:116300, 2023.
16

---

Published as a conference paper at ICLR 2026
SUPPLEMENTAL MATERIAL
This supplemental material is divided into the following five appendices.
Appendix A: Related works about existing FWI, NTK, and discussion.
Appendix B: Experimental details about baselines, datasets, forward modeling settings, the solu- tion of the wave equation, wave-based NTK experiments, and main inversion results.
Appendix C: additional experiments including 2014 Chevron blind test, 3D SEG/EAGE Over- thrust model, and comparisons with supervised data-driven FWI method.
Appendix D: Preliminary about standard NTK and hyperbolic PDE theory. Appendix E: Proof of Propositions 2.1 3.1 and Theorems 4.1
and 5.2
Comprehensive details on hyperparameters and model architectures are provided ensure all ex- periments can be exactly replicated. For the theoretical components, all proofs and mathematical derivations are included in the appendix with detailed step-by-step explanations.
A RELATED WORKS
In this section, we review relevant work covering conventional FWI methods, neural representation- based FWI methods, and the neural tangent kernel theory.
A.1 CONVENTIONAL FWI METHODS
To alleviate the ill- ill-posedness of FWI, many techniques like misfit functions, prior regularization,
multiscale inversion strategies, and optimization algorithms have been developed. For instance, de- signing more convex misfit functions using correlation information (e.g.. envelope (Wu et al. 2014).
global correlation (Song et al. )023)) and probability distribution (c.g., Student-T distribution (Ar-
avkin et al. 2011.. optimal transport (Yang & Ma 2023 to reduce the reliance on initial models. Additionally, incorporating prior knowledge using explicit regularization enhance robustness, in-
cluding local smoothness (e.g., total variation (Esser et al. (2018), Tikhonov (Yan & Wang 2018)
and geological constraints (e.g., structural dips (Wo et al. (025),, structure tensor (Liu et al. (025b) ) Furthermore, progressively inverting velocity models from low-frequency to high-frequency (Bunks cet al. 1995), or from coarse-gri grid to fine-grid (Fichtner et al., 2013), can mitigate the local minima problem; Using high-order optimization algorithms like the truncated Newton method (Métivier ctt al., 2017). the quasi-Newton method (Fabien-Ouellet et al. 2017), and the conjugate gradient method (Liu et al. 2017) also stabilize the inversion process.
Discussion. Although existing conventional FWI methods can invert velocity models from inac- curate linear initial models, they still struggle to produce satisfactory results using constant ini- tial models. Moreover, their performance remains highly sensitive to the seismic data quality. A promising future research direction involves understanding more conventional FWI methods using our proposed wave-based NTK framework. Such analysis would help build a deeper understand- ing of the roles played by the misfit function, regularization, and optimization algorithms in the inversion process, ultimately facilitating the development of more advanced FWI methods.
A.2 NEURAL REPRESENTATION-IE BBSED FWI METHODS
Deep neural reparameterized FWI generates the velocity model using neural networks, which can be divided into data-driven supervised and physics-informed unsupervised FWI methods.
A.2.1 DATA-DRIVEN SUPERVISED REPRESENTATION
Data-driven supervised FWI methods directly learn the inverse operator of the wave equation by training a network from pairs of seismic data and corresponding label velocity models, like Inver- sionNet (CNN) (Wu& & Lin 2019a), VelocityGAN (GAN) (Zhang et al. 2019). Fourier-Deep (Neural Operator) (Zhu al.) 2023), PnPDP-based FWI (Diffusion model) (Zheng et al.) 2025). Due to the limited availability of geophysical labeled data, numerous synthetic datasets have been developed and widely adopted in existing studies (Abdullin & Bin Waheed 2023 Gupta ct al. 2024
Jin et al. 2022). such acoustic OpenFWI (Deng al.) 2022), elastic EE" (Müller et al. 2022
and GlobalTomo (Li et al. 2024b) datasets.
Discussion.
However, the generalization performance of such supervised frameworks is constrained by the distribution shift between synthetic training data (c.g., OpenFWI) and real-world velocity models (Schuster et al., 2024), particularly in regions characterized by complex fault systems and
17

---

Published as a conference paper at ICLR 2026
salt bodies. Although data-driven methods that generate initial models represent a promising alter- native, this study focuses on achieving stable and such as the Marmousi and 2004 BP models. Our experiments with data-driven and proposed CR-FWI methods on the OpenFWI datasets (Deng et al.] 2022) demonstrated that our proposed unsupervised methods can outperform supervised approaches (see Appendix
inversion for challenging datasets
A.2.2 PHYSICS-INFORMER UNSUPERVISED REPRESENTATION
Distinet from pure data-driven FWI methods, physics-ifformed informed unsupervised representation utilizes a neural network, e.g.. coordinate- based neural network, to represent the velocity model and in- tegrate the wave equation in loop. For instance, Sun et al.] (2023a) proposed implicit FWI, which represents the velocity model using an implicit neural representation with SINRE's activation func- tion Sitzmaan et al. 2020). Subsequent advances include using INR with GELU activation (Du) ct al. 2024) and Gabor wavelet activation functions (Yang & Ma 2025) to represent the velocity model. This INR-based CR-FWI method can reduce the dependence
initial models at the cost of a slow convergence rate. To our knowledge, only INR is used to represent the velocity models in FWI continuously. Moreover, [Song& & Alkhalifah (2021); [errmaan al. (2023); Ressst et al. (022) introduced a physics-informed neural network (PINN)-based FWI methods, which use two INRs represent the velocity model and the seismic wavefield, respectively. Physical con- straints are incorporated via a penalty term derived from the wave equation within the loss function. In a different vein, Jin et al. (2022) proposed an unsupervised learning framework for FWI (termed UPFWT), which employs a CNN to predict velocity models directly from observed seismic data.
Discussion. Our research focuses on elucidating the mechanisms underlying the performance differ- ences between conventional FWI methods and existing CR-FWI methods, as discussed in (Sun et al.) 2023a Yang& & Ma 2025). Building on these insights, we propose several novel CR-FWI methods specifically tailored for enhanced inversion performance. Although PINN-based FWI and UPFWI fall outside the immediate scope of this study, our proposed wave-based NTK framework can be naturally extended to these approaches, which can be promising direction for future research.
A.3 NEURAL TANGENT KERNEL
Standard NTK provides a mathematical framework for analyzing the training dynamics of infinitely wide neural networks and argues that the gradient descent dynamics converge to a linear kernel regression problem governed by a deterministic kernel as network width tends to infinity (Jacot et al.)
2018 Arora al.) 2019). This fundamental result holds for various neural network architectures, including multilayer perceptrons (Chen et al. 2020). CNNs (Bietti & Mairal 2019). recurrent neural networks (Alemohammad et al. 2020), and linear PINNS (Wang et al. 2022; Gan al. (025).
However, some studies indicate that the NTK is not a deterministic kernel under some conditions and scenarios, like the last layer of the network is non-linear (Liu et al. 2020), nonlinear physics-
informed neural networks (Bonfanti et 2024b). and classification problems using fully connected neural networks and residual neural networks (Yu et al. 20025). NTK analysis has been applied to many fields, e.g.. physics- innfore- neural networks (Jha & Mallik 2024|
& Braga-Neto
2023) and neural operator (Qin et al., 2024: Xu et al. 2024 Nguyen & Mücke 2024).
Discussion. The proposed wave-based NTK also does not remain constant at initialization or dur- ing training, due to the inherent nonlinearity of FWI. It offers a novel theoretical perspective on continuous representation FWI as EDE-constrained nonlinear inverse problem. In contrast to the NTK theory developed for nonlinear PINNs in the context of forward PDE problems (Bonfanti) al. 2024b), our work focuses specifically the inverse problem and incorporates the numerical solution of the nonlinear wave equation through neural network guidance.
B EXPERIMENTAL DETAILS AND MAIN RESULTS
B.1 BASELINES AND HYPERPARAMETER SETTINGS
We carefully select widely used conventional FWI methods and existing state-of-the-art CR-FWI methods to compare with our proposed novel CR-FWI methods, shown in Tab. 3 Moreover, give more explanations and implementation details for each baseline and proposed method.
(H (Hyperparameter settings of conventional FWI methods) For ADFWI with TV regularization, the isotropic constraint weights are set to o₂ = 2e -9 and a = 2e-9, 9 respectively, with a maxi- mum step size of 20 and scaling factor gamma of 0.9. Moreover, for WECI-FWI methods,
18

---

Published as a conference paper at ICLR 2026
Table 3: Conventional FWT, existing CR-FWI, and proposed CR-FWI methods.
Category
Abbreviation
Method Name
ADFWI (Luu et al. 2025a)
Automatic differentiation based FWI
Conventional
ADFWI-TV Esser et al.
2018
ADFWI with total variant regularization
FWI Methods
WECI-FWI (Song et al.
2025
Weighted envelope correlation-based FWT
MS-FWI (Fichtner ct al.
2013
Frequency-basee multiscale inversion FWI
Existing
IFWI (Sun et al. (2223)
INR-hased FWI with SINRE's activation
CR-FWI Methods
WinFWI (Yag & Ma (2025)
INR-hased FWT with Gabor wavelet activation
MPE-FWI (Ours)
secoding-based based FWI Low-rank tensor decomposition based FWI
Multigrid parametric encoding-
Proposed
LR-FWI (Ours)
CR-FWI Methods
IG-FWI (Ours)
Hybrid INR-Grit based FWI
weighted envelope misfit and global correlation misfit (Song et al. 2023), where the weights change according to the logistic function as the number of iterations iccrassss following (Song et al. 2023). Furthermore, for the MS-FWW method, we divide the observed seismic data into frequency bands, ranging from 4 Hz to 14 Hz, with an interval of Hz.
(Hyperparameter settings of existing CR-FWI methods) For IFWI, we use the sine activation function with frequency hyperparameter Wo = 30 (where we controls the frequency of the sine function) for MLP, and the depth of the MLP set 4. The number of neurons of the MLP is set to 128, following the recommended settings in the paper (Sun et al. 2023a). Moreover, for WinFWI, we use the Gabor wavelet activation function with hyperparameters wo and 8g = (where wh controls the frequency of the wavelet and 50 controls the frequency of the wavelet). The depth of the MLP is set to 4, and the number neurons of the MLP is set to 200, following the recommended settings in the paper (Yang & Ma 2025).
(Hyperparameter settings of our proposed CR-FWI methods) For the proposed LR-FWI method, we employ the same sine activation function as IFWI. The rank of the low-rank ma- trix factorization is set to half of the model dimension. The depth of the MIPs is set to 3, and the number of neurons in the MLPs is set 128. Moreover, the MPE-FWI method utilizes multi-resolution hash grid (Müller et al. 2022) encoding with 16 levels, a base resolution of 50, and per-level scale factor of 1.05. This feature grid is combined with a compact MLP with two hidden layers of 64 neurons each to model the inverse problem. Furthermore, the IG-FWI model employs hybrid architecture, combining a multi- mesti-resooltion hash grid encoding (base resolution 50 with 16 levels) with a sinusoidal feature network (2 layers of 128 neurons with wo 30). These extracted features are subsequently fused and processed by a compact MIP (2 layers of 64 neurons) generate the final physical property output.
The Adam optimizer is used to optimize the learnable parameters in all methods. The learning rate is set to 5 for conventional FWI methods and 0.0001 for CR-FWI methods. All numerical experiments are conducted using an RTX 3090 GPU, leveraging the PyTorch framework.
B.2 DATASETS AND FORWARD MODELING SETTINGS
We evaluate baselines and our proposed methods on Marmousi (Brougois et al. 1990, Versteeg 1994), 2D SEG/EAGE Salt and Overthrust Amiinzaee (199)), and 2004 BP models (Billette & Brandsberg-Daie 2005) (see Fig. [0) using different initial velocity models (i.e., smooth, linear, and constant velocity models). The detailed forward modeling parameters are as follows:
Marmousi model: A 94 x 288 modified Marmousi model is used (15 m grid spacing) with 13 shots deployed at 300 m intervals (8 Hz Ricker wavelet). Receivers are spaced 15 m apart along the surface. Wave propagation employs a 1.9 ms time step, recording 1,000 samples, following the recommended setting in the paper (Sun et al. 2023a).
2D SEG/EAGE Salt model: A 75 x 250 slice of the SEG/EAGE Overthrust model (10 m grid spacing) is used, with 24 shots at 100 m intervals (10 Hz Ricker wavelet). Receivers are placed at 10 intervals on the surface. Wave propagation uses 1.5 time steps and 2,500 samples.
2D SEG/EAGE Overthrust model: A 94 x 401 slice of the SEG/EAGE Salt model (15 m grid spacing) is used, with 20 shots 270 m intervals (10 Hz Ricker wavelet). Receivers are placed at 15 m intervals on the surface. Wave propagation uses 4.0 ms time steps and 1, 1,200 samples.
2004 BP model: 188 x 500 slice of the 2004 BP model (10 m grid spacing) is used, with 20 shots at 200 m intervals (10 Hz Ricker wavelet). Receivers are placed at 10 m intervals on the surface. Wave propagation uses 4.0 ms time steps and 2,000 samples.
19

---

Published as a conference paper at ICLR 2026
Narmoull Mocul
Smooth
Unver
Constant:
Dverthount Model
Salt Mocel
2004 SP Model
Con
Figure 10: Datasets and Initial velocity model, including smooth, linear, and constant initial models.
Marmousi, SEG/EAGE Salt and Overthrust models configuration features a free-surface top bound- ary and perfectly matched layer (PML) absorbing boundaries the remaining three sides. More- over, we set PML absorbing boundaries on all sides for the 2004 BP model to reduce the impact of wavefield erosstalk. We employ the eighth- finite difference method to solve the using deepwave (Simeoni et al. 2019). Representative observed seismic data are shown in Fig.
equation
(6) Marmousi model
(b) Overthrust model
(c) Salt model
(d) 2004 BP model
Figure 11:
Observed seismic data using different velocity models, including (a) Marmousi model, (b) SEG/EAGE Overthrust model, (c) SEG/EAGE Salt model, and (d) 2004 BP model.
Furthermore, the robustness tests can be divided into three scenarios using the Marmousi model, including noisy seismic data, seismic data with missing low frequencies, and sparse acquisition.
(Seismic data with noise) To evaluate the robustness of our proposed CR-FWI method against noise, conduct noise perturbation experiments using constant initial models. Gaussian noise with standard deviations a = 40 and o 8ōn is added to the observed seismic data generated from the Marmousi velocity model, where ao denotes the standard deviation of the original seismic data. Noise contamination common issue in field data acquisition due to environmental factors, which often obscure meaningful signals and degrade the inversion quality.
(Seismic data with missing low-frequencies We further assess the robustness of CR-FWI under the challenge of absent low-frequency components, a frequent limitation in field surveys caused by bandwidth constraints of sources or receivers, and attenuation effects. The observed seismic data are high-pass filtered at cutoff frequencies of 4 Hz and Hz to simulate realistic scenarios where low-frequency information is unavailable. The lack of low frequencies often leads cycle- skipping and convergence to local minima conventional FWI.
(Seismic data with sparse acquisition) To mimic realistic acquisition constraints such as limited survey access, high costs, or logistical barriers, which often result in spatially ander-sampled data, decimate the source-receiver configurations in the seismic dataset derived from the Marmousi model. Sparse acquisition leads to inadequate spatial sampling, posing significant challenges for high-resolution velocity model building. This experiment tests the ability of CR-FWI recover accurate subsurface structures under such spatially limited conditions.
B.3 NUMERICAL SOLUTION ACOUSTIC WAVE EQUATION
We define the discrete wavefield solution, velocity model, and source function as tensors 2 R NNN,N,N,, M E I N.NN, and S N.N, respectively, where N, and N, denote the number of discrete points in the and directions, and N; represents the number of time steps. The deriva- tives equation] T are approximated using the second-order central finite difference method:
84
u++ 24 4e-1
A' u K,
(15)
012
At
20

---

Published as a conference paper at ICLR 2026
where 4 denotes the wavefield tensor at time step E, At is the time step size, and K is the discrete convolution kernel for the Laplacian operator. Under the assumption that Ax = Az Ah, the kernel K is given by:
0
1
07
1
K
1
-4
1
(16)
A
0
1
0
The iterative update scheme for the wavefield without considering boundary conditions is:
u++ 22 -4-- A
M (u' K +-)
(17)
where O denotes the Hadamard (eeement-wise) product, and S is the source term time step f. To prevent unwanted reflections from the boundaries, Perfectly Matched Layer (PML) is incorporated. Following the approach in the provided content, we introduce auxiliary variables p and 2 to handle the PML (Pasalic & McGarry 2010). The modified update scheme in the PML region becomes:
u = 211 21-111111 + A [MMM (V + Vp 2 55)]
p² 00 V
(18)
=00 (v'
where a and b spatially varying coefficients within the PML, and V denotes the spatial deriva- tive in the z-direction. The time-stepping scheme can be interpreted as a recurrent process, anal- ogous recurrent neural network (RNN), where each time step corresponds to layer in the network (Sun et al. 2020 Simeoni et al. 2019).
B.4 WAVE-BASED NTK FOR 1D FWI EXPERIMENTAL SETTING
Our experiments are based on the following one-dimensional acoustic wave equation:
m(x)0fu(x, 1) 0, s(x, (),
(19)
where x E IR denotes the spatial coordinate, t E R represents time, m is the velocity model, w is the wavefield, and is the source function. The target velocity model is constructed from trace data extracted from the Marmousi model with a grid spacing of 15 m (see Fig. 12 (a)). The seismic
observations are generated using a finite-difference method (see Fig. 12 (b)). An 8 Hz Ricker wavelet is employed as the source function, with time step of 1.9 ms and 1000 samples (see Fig.
Eerathed Hennd
Ricles sonneer
01
(b)
(c)
Figure 12: (a) ID velocity model. (b) observed seismic data. (c) Ricker source function.
To compare the eigenvalue decay across different FWI methods, including ADFWI, INRebased FWI, MPE-FWI, LR-FWI, and IG-FWI. We compute the wave-based NTK for CR-FWI using network width of 100,000 and a wave kernel for ADFWI. Furthermore, we test the wave-based NTK relative nuclear norm variation initialization and during the training process, i.e.,
NTK wave
(T)
@NTK((O) (0)
(T
(20)
ONT (O)]l
over the network's width, for 10 independent experiments. The results are presented in Fig.
B.5 USAGE OF LLMS
We employed large language models solely for the purpose of polishing and refining the writing of this paper. The models assisted with improving grammatical accuracy, enhancing sentence flu- ency, and ensuring clarity of expression, while strictly preserving the original technical content and scientific meaning. All ideas, methodologies, results, and conclusions remain entirely
21

---

Published as a conference paper at ICLR 2026
B.6 MAIN EXPERIMENTAL RESULTS
B.6.1 EVALUATION METRICS
Table 4: Comparisons between different conventional FWI (i.e., ADFWI, ADFWI-TV, WECI-FWI, and MS-FWI) and CR-FWI (i.e., IFWI, WInFWW, LR-FWI, MPE-FWI, and IG-FWI) methods across different evaluation metries for the 2D Marmousi model. Here, bold represents the best evaluation result, while underline represents the second- best evaluation
Metric
Method
Initial velucity model
Gaussian noise
Frequency cutoff
Shols number
Smooth
Linear
Constant
4eto
8erg
4Hz
6 Hz
9 shots
shots
ADFWI
0.6260
0.2689
0.0549
0.2167
0.1401
0.2594
0.2287
0.2499
0.1969
ADFWI-TV
0,5466
0.2735
0.1503
0.2628
0.2456
0.2607
0.2462
0.2718
0.2289
WECL-FWI
0.4239
0.3638
0.0220
0.1193
0.0948
0.3581
0.3475
0.3240
0.2926
MS-FWI
0.7093
0.6559
0.0952
0.5274
0.4218
0.5298
0.4193
0.5362
0.4798
SSIM
IFWI
0.6510
0.6173
0,5252
0.5410
0.4607
0.5587
0,5124
0.5802
0.4234
WinFWI
0.6498
0.5982
0.5617
0.5220
0.3960
0,5442
0.4514
0.5938
0,5002
LR-FWI
0.6853
0.6662
0,5911
0.5714
0,5079
0.6414
0.6166
0,6250
0.5665
MPE-FWI
0.7177
0.3681
0.1888
0.3620
0.3008
0.2828
0.2723
0.2837
0.2745
IG-FWI
0,7183
0.7130
0.6773
0.6182
0.6048
0.6774
0.6650
0.7030
0.6808
ADFWI
0.2132
0.4921
1.1522
0.4965
0,5422
0.4965
0.4975
0,4831
0.4730
ADFWITVV
0,2449
0.3780
0.9386
0,3875
0,3807
0.3764
0.3741
0.3547
WECI-FWI
0.2977
0.3347
1.2537
0.5414
0.6366
0,3320
0.3220
0.3338
0.3337
MS MS-FWI
0,1683
0.2651
1,1313
0.2824
0,3547
0,5536
0.4198
0,2867
0.286770000
0,3325
M
IFWI
0.1907
0.2375
0,9474
0.3072
0,3374
0,4040
0,3358
0,2857
0,3483
WinFWI
0.2013
0.2917
0,4689
0.3042
0.3514
0,2029
0.3460
0.2401
0.3239
LR-FWI
0.1638
0.1733
0.2893
0.2218
0,2551
1998
0.2276
0,2064
0.2641
MPE-FWI
0,1427
0.3755
2.2266
0.327T
0,2990
0.3322
0,3320
0.3338
0.3393
IG-I'I
0.1423
0.1534
0.2961
0.2124
0.2151
0.1653
0.1846
0.1602
0.1654
ADFWI
0.2682
0.4571
0.8866
0.4678
0,5084
0.4626
0.4757
0.4708
0.4555
ADFWI-TV
0.2800
0.3979
0.8074
0.4028
0.4084
0.4005
0.4034
0.4077
0.3874
WECI-FWI
0.3389
0.3628
0.9170
0.5719
0.6251
0,3475
0. 0.3605
0.3726
0.38
MS-FWI
0.2201
0.2719
0,8590
0.3314
0,3917
0.2941
0.3996
0.3146
0.3435
MAE J
IFWI
0.2468
0.2786
0.5568
0.3224
0.3504
0.3165
0.3415
0.3053
0,3737
WinFWI
0.2480
0.3097
0.3979
0.3227
0,3633
0,3182
0.3536
0,2744
0,3131
LR-FWI
0.2278
0.2472
0,3149
0,2841
0,3090
0.2495
0.2679
0,2561
0.2921
MPE-FWI
0.2114
0.3977
1.1312
0.3649
0.3514
0.3791
0.3833
0.3811
0.3863
IG-FWI
0,2109
0.2196
0.2989
0.2644
0,2706
0,2312
0,2368
0,2243
0,2307
Table 5: Comparisons between conventional FWI (i.e., ADFWI, ADFWI-TV, ADFWLLL WECI-FWI, and MS- FWT) and CR-FWI (i.e., IFWI, WinFWI, LR-FWI, MPE-FWI, and IG-FWI) methods for the other challenging models. Here, bold represents the best evaluation result, while underline represents the second-best evaluation.
Metric
Method
2004 BP model
Salt model
Overthrust model
Smooth
Linear
Constant
Smooth
Linear
Constant
Smooth
Linear
Constant
ADFWI
0.5104
0.2391
0.0920
0.1717
0.3770
0.2334
0.7334
0.4699
0.0324
ADFWI-TV
0.5042
0.2573
0.0963
0.2778
0.5272
0.3315
0.4497
0.3098
0.1055
WECL-FWI
0.4392
0.2111
0.0973
0.3290
0.2525
0.0877
0.4958
0.1413
0.0931
MS-FWI
0.6872
0.4157
0.0493
0.2590
0.4515
0.3020
0.7033
0.6802
0.0456
SSIM
IFW1
0.6168
0.5807
0.5533
0.5882
0,5177
0.4189
0.7124
0.6849
0.6398
WinFWI
0.5926
0.3579
0.6322
0.5507
0.5205
0.5123
0.6946
0.6737
0.6148
LR-FWI
0,7003
0.4992
0.4868
0.4222
0.4944
0.4636
0.7305
0.7005
0.6985
MPE-EWI
D.6351
0.3221
0.0968
0.1554
0.3053
0.2100
0.6571
0.4506
0.0557
IG-FWI
0.7316
0.6459
0.7274
0.5910
0.5265
0.5138
0.7272
0.6985
0.6501
ADFW]
0.1003
0.2568
0.4281
0.6462
0.3292
0.6337
0.0592
0,1902
1.3364
ADFWLTTV
0.2593
0.4259
0.2143
0.5318
0.1154
0.1985
1.3142
WECI-PWI
0.0951
0.2748
0.3847
0.3032
0.9294
1.0293
0.1182
0,2987
1.5526
MS-
0.0672
0.1631
0.4667
0.5358
0.3271
0.9234
00752
007797
1,0802
IFW1
0.1143
0.1412
0.1201
0.2214
0.7412
0,0683
00800
0.6482
WinFWI
0.0812
0.1827
0.1083
0.1223
0.2241
0.3549
0.0682
0.5738
LR-FWI
0.0481
0.1
0.1248
0.2785
0,2143
0.2368
0.0548
0.0724
0.1592
MPE-FWI
0.0586
0.3387
L.6024
0.2942
1,100K
0.0613
0.2115
1.2887
IG-FWI
0.0521
0.1057
0.0843
0.1181
0.1997
0.1883
0.0587
0.0762
0.5724
ADEWI
0.1957
0.3697
0.5690
0.6168
0.4024
0.5412
0.1710
0.3182
0.9107
ADFWI-TV
0.2006
0.3672
0.5685
0.5051
0.3071
0.4852
0.2653
0.3479
0.9184
WECL-FWI
0.2024
0.4064
0.5045
0.3791
0.6999
0.7845
0.2536
0.4314
1.0064
MS-PWI
0.1595
0.2717
0.5612
0.5007
0.3794
0.5705
0.1802
0.2118
0.8087
MAE
IFWI
0.1637
0.2230
0.2030
0.2129
0.3200
0.5210
0.2239
0.2028
0.4672
WinFWI
0.1883
0.3018 8
0.2061
0.2152
0.3284
0.3325
0.2392
0.1962
0.4564
LR-FWI
0.1322
0,2128
0.2295
0.3272
0,3011
0.2962
0.1671
0.1907
0.2793
MPE-EWI
0.2099
0.3918
0.6462
0.6667
0.3734
0.6513
0.1974
0.3392
0.8742
IG-FWI
0.1386
0,2085
0.1839
0.2118
0.3096
0.2700
0.1702
0.1870
0.4088
22

---

Published as a conference paper at ICLR 2026
B.6.2
INVERSION RESULTS MARMOUSI MODEL WITH DIFFERENT INITIAL MODELS
Smooth initial model
Linear initial model
Constant initial model
MSE: 0.289
MSE: 0.316
MSE: 0.947
ADFWI
ADFWI-TV
MSE: 0.245
MSE: 0.378
MSE: 0.939
WECI-FWI
MSE:
MSE:
MSE: 1.254
MS MS-FWI FW-
MSE: 0.168
MSE: 0.265
MSE: 1.131
IFWI
MSE: 0.191
MSE: 0.238
MSE: 0.947
Wir.WW
LR-FWI
MSE: 0.164
MSE: 0.173
MSE: 0.289
MPE-EWI
MSE:
MSE:
IG-FWI
MSE: 0.142
MSE: 0.153
MSE: 0.296
3.0
3.5
Velocity (km/s)
Figure 13: Conventional FWI, based FWI, and our proposed FWI methods (bold) performance comparison using different initial velocity models (i.c., smooth, linear, and constant) the 2D Marmousi model.
23

---

Published as a conference paper at ICLR 2026
B.6.3
INVERSION RESULTS MARMOUSI MODEL WITH DIFFERENT SEISMIC DATA QUALITY
8 Go Caussian noise
Hz frequency cutoff
5 shot gathers
MSE: 0.542
MSE: 0.198
MSE: 0.473
ADPWI-TV
MSE: 0.388
MSE: 0.376
MSE: 0.355
WECL-PWI
MS-FWI
MSE: 0.355
MSE: 0.420
MSE: 0.333
IPW1
MSE:
MSE:
MSE: 0.348
WhFWI
MSE: 0.351
MSE: 0.346
MSE: 0.324
LR-FWI
MSE: 0.255
MSE: 0.228
MSE: 0.264
MIPE EWI
IG-FWI
MSE: 0.215
MSE: 0.185
MSE: 0.165
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Velocity (km/s)
Figure 14: Conventional FWI, INR-based FWI, and our proposed FWI methods (bold) performance comparison using linear initial model with different seismic data quality on the 2D Marmousi model.
24

---

Published as a conference paper at ICLR 2026
B.6.4
INVERSION RESULTS OF OVERTHRUST MODEL WITH DIFFERENT INITIAL MODELS
Smooth initial model
Linear initial model
Constant initial model
MSE: 0.178
MSE: 0.429
MSE: 2.074
ALEWI
MSF: 0.059
MSE: 0.190
MSE: 1,336
WEDC FINT ADPWETVV
MSE: 0.115
MSE: 0.199
MSE: 1.314
MSE: 0.118
MSE: 0.299
MSE: 1,553
MS-FWI
MSE: 0.068
MSE: 0.080
MSE: 0.643
WInWW
MSE 0.068
MSE: 0.093
MSEI 0.574
LR-FWI
MSE 0.055
MSE: 0.072
MSE: 0.139
MIPE- NIPE-EWT
IG-FWI
MSE:
MSE: 0.076
MSE: 0.572
2.5
3.0
3.5
4.0
4.5
5.0
5.5
Velocity (kmns)
Figure 15: Conventional FWI, INR-based FWI, and our proposed FWI methods (bold) performance comparison using different initial velocity models (i.c., smooth, linear, and constant) on the 2D SEG/EAGE Overthrust model.
25

---

Published as a conference paper at ICLR 2026
B.6.5
INVERSION RESULTS SALT MODEL WITH DIFFERENT INITIAL MODELS
Smooth initial model
Linear initial model
Constant initial model
MSE: 0.507
MSEs
MSE: 0.850
ADFWT
MSE: 0.646
MISE: 0.329
MSE:: 0.634
ADPWETV
MSE: 0.504
MSF: 0.214
MKE: 0.532
WEEE SEVE
MIS
MSE: 0.536
MSE: 0.327
MSE: 0.923
JFW
Win PWI
MSE: 0.122
MSE: 0,221
MSE: 0.355
MSE:: 0.279
MSE:: 0.214
MSE: 0.237
MPE-EWI
MSE: 0.110
MIE 0.200
MSE: 0.180
1.5
2.0
2.5
3.0
3.5
4.0
4.5
(km/s)
Figure 16: Conventional FWI, based FWI, and our proposed FWI methods (bold) performance comparison using different initial velocity models (i.c., smooth, linear, and constant) the 2D SEG/EAGE Salt model.
26

---

Published as a conference paper at ICLR 2026
B.6.6 INVERSION RESULTS 2004 BP MODEL WITH DIFFERENT INITIAL MODELS
Linear initial model
Constant initial model
Smooth initial model
ADENI
MIF: 0.100
MSE: 0.257
MSF: 0.428
WECIEWI
MSE: 0.095
MSE: 0.275
MSE: 0.385
MS- MSS-PWY
JFWY
MSE: 0.072
MSE:: 0.114
MSE: 0.141
WinFWI
LR-FWI
MSE: 0.058
MISS: 0.119
MSE: 0.125
MPE-EWI
IG-FWI
1
MIS: 0.052
XISE: 0.106
MSE: 0.081
1.5
2.0
2.5
3.0
3.5
Velocity (kaus)
Figure 17: Conventional FWI, based FWI, and our proposed FWI methods (bold) performance comparison using different initial velocity models (i.e., smooth, linear, constant) on the 2004 BP model.
27

---

Published as a conference paper at ICLR 2026
B.6.7
GENERATED VELOCITY CORRESPONDING TO DIFFERENT ITERATIONS
spoct:
100 spoch
200 spoch
400 spoch
700 epoch
1000 spoch
4DPW!
LR-PMI IG-FW NIP--WW IPW
Figure 18:
Generated velocity model corresponding to different iterations with conventional FWI, IFWI, MPE-FWI, IG-FWI, and LR-FWI using Marmousi model
50 spoch
100 apoch
200 epoch
400 spoch
700 epoch
1000 spoch
IPW1 ADPNI
IC-PWZ MPP- FWNE
ERCIOI
Figure 19: Generated velocity model corresponding to diffcrent iterations with conventional FWI, IFWI, MPE-FWI, IG-FWI, and LR-FWI using Overthrust model
50 spoch
100 epech
200 apoch
400 spech
700 spech
1000 spoch
LR-FWI IG-FWI MPE-FW BPNV 1997
Figure 20: Generated velocity model corresponding to different iterations with conventional FWI,
IFWI, MPE-FWI, IG-FWI, and LR-FWI using Salt model
28

---

Published as a conference paper at ICLR 2026
100 200 epoch
400 opoch
700 epoch
1000 epoch
A3FM
1830
MPR-EWF
LR-IWI IG-FW1
Figure 21: Generated velocity model corresponding to different iterations with conventional FWI, IFWI, MPE-FWI, IG-FWI, and LR-FWI using 2004 BP model
C ADDITIONAL EXPERIMENTS
In this Appendix, we provide additional numerical results, including a more realistic 2014 Chevron blind test, 3D SEG/EAGE Overthrust model test with computational challenges, and comparisons with data-driven method on the OpenFWI dataset. Finally, we present two failure cases using our proposed methods.
C.1 2014 CHEVRON BLIND TEST AND RESULTS
To further validate our proposed IG-FWI method on more realistic field data and consider the in- version crime problem, applied multiscale ADFWI (Liu et al. 2025a Fichtner al. 2013) INR-based FWI (Sun et al. 2023a), and IG-FWI to the realistic Chevron 2014 benchmark dataset.
This 2D marine towed-streamer dataset with maximum offset of km, provided by Chevron Oil Company for FWI, 2D isotropic, elastic wave equation with a free surface at the top and absorbing boundaries on the sides and the bottom. It contains noise, with the signal predominating above Hz. We used 146 shots with spacing of 150 m, where each was recorded by 321 receivers spaced apart, all at a depth of 15 The veloc- ity model was discretized at m in both di- rections. Data were recorded step over seconds. The source wavelet and its spectrum are shown in Fig. 22 A 1D initial velocity model and a well log were provided, but the model was withheld.
(a) Ricker Wavelet
modeled using
Amplitude
-20
50
100
150
200
250
(ms))
(b) Amplitude Spectrum
4-ms time
Specteal Power
Power
For the multiscale FWI method, we adopt five frequency bands ranging from Hz 11 Hz with a 2 Hz interval and 50 iterations. The rel- evant hyperparameters for both the based FWI and our proposed IG-FWI methods are set as described epochs. Here, we produce the synthetic seismic data using the acoustic wave equation rather than the isotropic and elastic wave equation, which avoids the inversion crime problem. Inversion results on the Chevron blind test data (see Fig. 23 show that all methods accurately recover the shallow struc- tures. However, challenges emerge at greater depths. For instance, velocity profiles (see Fig. 23 indicate that the obtained model aligns well with well-log data from km to 2.5 km, though some discrepancies appear below 2.0 km. Despite these difficulties, IG-FWI delivers superior performance compared to both multiscale FWI and INR-based FWI. Because the model unknown,
Frequency
(Hi)
Figure 22: (a) Initial source wavelet and (b) its corresponding normalized spectrum, as provided by Chevron Oil Company for FWI.
Appendix B and run 500
29

---

Published as a conference paper at ICLR 2026
extensively assessed the inversion performance by comparing common-souree shot gathers, shown in Fig. 23 (b). This improvement can be attributed to the suitable eigenvalue decay of IG-FWI, thereby enhancing the stability and accuracy of the inversion.
(a)
lard model
Hil based
3.0
-3.5
(b)
Depth (cm)
Depth
2.0
20
25
30
35
40
45
Distance
Distance
Figure 23: Inversion Results of 2014 Chevron blind test model. Left: (a) 2014 Chevron initial velocity model. (b) Inversion results using multiscale ADFWI (Fichtner et al. 2013 Liu et al. 2025a). (c) Inversion results using INR-based FWI (Sun et al. 2023b). (d) Inversion results using our proposed IG-FWI; Right: (a) Velocity profiles of the well log, initial model, and inverted models using different FWI methods. (b) Comparison of common- source shot gather between the true and modeled records. Here denotes simulated data and O denotes the observed data.
C.2 3D SEG/EAGE OVERTHRUST MODEL TEST
Considering the computational time and memory challenges of 3D domains, we test conventional FWI, INR-based FWI, and proposed CRFWI methods the 60 x 160 160 modified 3D SEG/EAGE Overthrust velocity model (50 m grid spacing) with 36 shots deployed at 1.2 km in- tervals (12 Hz Ricker wavelet). Receivers are placed at 50 m intervals on the surface. and wave propagation uses 2.5 ms time steps and 1000 samples. Here, we maintain the same hyperparameters and neural structure as FWI settings. Moreover, we utilize the finite difference method solve the 3D acoustic wave equation, and the representative observed seismic data are shown in Fig. 24
200
3.008
(a) Acquisition systems
(b) Observed seismic data
Figure 24: (a) 3D acquisition systems. (b) Representative seismic data.
To alleviate the computational burden of the domain, we introduce some advanced techniques like checkpointing (Anderson et al.] (0022). gradient accumulation (Simeoni et al.) 2219). and ran- dom mini-batch strategy (van Herwaarden et al. 2020j. Hence, our proposed CRFWI methods can be implemented in one RTX 3090 GPU to conduct the 3D FWI experiments. The inversion results are shown in Fig. 25 using a smooth initial velocity model with 500 iterations. We can see that our proposed CR-FWI methods achieve
robust and nigh-precision inversion results. Moreover, provide a detailed comparison of computational resources, as shown in Tab. 6 We can see that proposed LR-FWI and IG-FWI reduce the memory usage of the continuous representation
30

---

Published as a conference paper at ICLR 2026
compared the existing INR-based CRFWI method, which is attributed the decoupling low- rank representation in LR-FWI and the compact hash representation in IG-FWI. These results show that our proposed CR-FWI methods do not impose additional computational burden compared with conventional FWI methods.
Table 6: Performance comparison of different FWI methods 3D Overthrust model. Here, bold represents the best evaluation result, while underline represents the second-best evaluation.
Method Normalized MSE + SSIM 1
Memory (GB)
Time (s/it)
ADFWI
0. 0.1469
0.5576
15.89
13.82
IFWI
0.1302
0,4634
30.61
15.08
IG-FWI
0.1151
0.5729
22.84
14.61
LR-FWI
0.0968
0.5914
16.01
14.09
03
13
26
24
17
46
mmm
80 08
(a) Ground truth
(b) Smooth initial model
(c) ADFWI
57
13
76
28
77
17
2
is
20
(d) IFWI
(e) IG-FWI (Ours)
(f) LR-FWI (Ours)
Figure 25: 3D SEG/EAGE Overthrust model test using smooth initial model. (a) Ground truth.
(b) Smooth initial model. (c) ADFWI. (d) INR-based FWI. (e) IG-FWI. (f) LR-FWI.
C.3 COMPARISONS WITH DATA-DRIVEN FWI
We evaluate our proposed CRFWI approaches with the data-driven FWI method InversionNet Wu & Lin 2019b) on the synthetic Fpln dataset (Deng et al. (022). a large-scale benchmark designed for seismic imaging tasks. Specifically, we predict velocity models. including FlatVel, FlatFault, CurveVel, CurveFault, and Style, from noisy seismic data (see Fig. 26) using pre-trained models provided by the original InversionNet. We compare the performance of conventional FWI, INR- based FWI, and our proposed LR-FWI and IG-FWI methods inverting velocity models from the same noisy seismic data. The inversion results indicate that both LR-FWI and IG-FWI can achieve superior performance compared to some supervised CNN-based approaches (see Figs. 27 3I).
While supervised frameworks suffer from generalization issues caused by the distribution shift be- tween synthetic training data (e.g., from OpenFWI) and complex field models (e.g.. Marmousi and Overthrust models), their computational speed remains highly appealing. A promising pathway is to embed such methods within our proposed CR- FWI framework to rapidly generate initial models, which we plan explore in future work.
31

---

Published as a conference paper at ICLR 2026
FlaNVe!
CurreVal T
Style T
FlatFault IT
CurveFault TT
Figure 26: OpenEWI datasets and noisy seismic data input.
Tree Model
Inversion
ADPVI
IPWI
LN-PII
Ki-FWI
9SE:
MSE: 30 A4S
M:E: 4.052
M:E 2.331
MSE: 1,495
M.C:1.775
PSE:
HSE:
NSC:
MIC: 7.055
NSE: 6.505
MSC: 4,009
9S6: 1000
MSs:: 7.700
NSC 337
H:C 4.500
HSS: 3.258
MS:: L4.357
PS8:
RIE US0
MA2: 53.506
MSE:
MSS 6.42
Figure 27: Inversion results on Flat dataset using different FWI methods.
32

---

Published as a conference paper at ICLR 2026
True Model
Inversionfiet
ADVWI
IPWI
LR-FWI
IC-FWI
137.133
MSST 101
MSE 28.547
SOD: 0.002
NSE 60 274
MSS: 39 401
PSE: 43.707
MSS 31.733
PSE. 27A35
VSE: 0.002
MIS:: 31,590
MSR 73
MSC: 33.250
NSE 42.344
9252 51610
POR: 0.003
PORT 147 283
Poe 2::55
MSRI 12.913
PSe 80412
188,200
MSZ AOG
MSE: 37-354
20.011
VS2: 30.422
Figure 28; Inversion results FlatFault dataset using different FWI methods.
True Model
Inversionflet
ADIVI
IFWI
LR-FWI
IG-FWI
3130
2.370
SIDE: 0,000
ASE 47.423
Moc: 435
H30: 2762
ASC 32-222
KSC 5.35
SISE: 0.000
PISE: 163 224
MSC 5.453
MSSC: 7.267
MSE: 4.089
MSE 4,300
Figure 29: Inversion results on Curve dataset using different FWI methods.
33

---

Published as a conference paper at ICLR 2026
True Model
Inversionfiet
ADVWI
IPWI
LR-FWI
IG-FW
MSC:
72.535
195994
MSE: 541051
MODI: 0000
SOSE 378.219
MSC: 124.59
PS2: 77-340
NSC 55-154
NS2: 71.459
SISE: 0,002
NIE. 90.22
MSS: 79 500
MSC 51.300
NSC 38.764
MSE: 20.153
MOR: 0.003
ASEE 41.50
VSS: 20.443
PSF 19:189
MSE 16.145
HSSI 6.120
PSS: 140,502
MSZ 074
74,354
43
22.372
Figure 30: Inversion results CurveFault dataset using different FWI methods.
True Model
Inversionfliet
ADFVI
IFWI
LB-PWI
IG-FWI
37.302
SOD: 0.000
RSS:: 30 220
MSS: 23
RSt: 35.349
MSR: 12.508
SS:: 22.005
SSS:: 0.002
ASE $4.303
MSZ: 22 A47
MSC: 58.333
MSE: 22.531
PS: 23193
Figure 31: Inversion results on Style dataset using different FWI methods.
34

---

Published as a conference paper at ICLR 2026
C.4 Two FAILURE FWI CASES FOR CR-FWI
Here, we present two failure cases using our proposed methods using the Canadian Foothills model with irregular topography (See Fig. 32) and the Overthrust model with near-surface low-speed layer (See Fig. 33 The performance degradation in these challenging scenarios can be attributed to complex wavefield propagation patterns (Yang et al. (025) that our current velocity parameteri- zation struggles to accurately capture, which guide subsequent algorithm design and integration with existing seismic techniques (c.g., illumination compensation techniques) future research.
(a) Ground truth
(b) Initial model
ADFWI
Difonce Oerd
Dakace Fer
(d)
(c) LR-FWI
(c) IG-FWI
Figure 32: Inversion results on the Canadian Foothills model with irregular topography
(a) Ground truth
(b) Initial model
(c) result
Figure 33: Inversion results on the Overthrust model with near-surface low-speed layer
C.5 CONTRIBUTION TO SEISMIC COMMUNITY
Our theoretical analysis not only explain the underlying mechanism of CRFWI, but also provide a methodology continuous representation structure design. For instance, we can develop more novel hybrid CR-FWI method by replacing the tiny INR in IG-FWI with other representation, c.g., a low-rank representation (termed LRG-FWI). Here, we provide experiments using the Marmousi, Salt and Overthrust models with constant initial model using LRG-FWI (see Fig, 34 and Tab. 77. We see that this novel CR-FWI achieves high-precision inversion results and achieves performance comparable that of IG-FWI.
05
24
10
15
Figure 34: Inversion results using LRG-FWI method.
Table 7: Comparisons with LRG-FWI and other CR-FWI methods.
Method
MPE-FWI LR-FWI LRG-FWI IG-FWI
Marmousi (MSE J)
2,2266
0.2893
0.1995
0.2961
Overthrust (MSE ))
1.2887
0.1592
0.1354
0.5724
Salt body (MSE !)
1.1008
0.2368
0.2179
0.1883
35

---

Published as a conference paper at ICLR 2026
THEORETICAL PROOF
Next, we present the proofs of the main theorems introduced in this paper. The overall structure begin with preliminary foundations, including the con-
of the proofs is illustrated Fig. 35
Framework of Theoretical Proof
Proof of Theoeens:..1 (Apperdic Ci)
Eigeavvblue Comparison (Apperalis H)
Preliminary (Appendia
Lipechitz Continulty of the Gieam's Fanation (Lemma
Infniic with Convergence of Neural Netwreis for FWI (I (Lamin O. Jaor, 2018)
Spectral Compatioor Theorem (Theorero H.l1
Convergance Propertye Standand NTK
Regularity of Hoperhalic
Inciaal-Beurrdary Proisamm (Thecee D.,, Doons 2010
Thoomum 20181
Lipschitz Contimiity the Sensitivity Kemel (Lecono
Senstivvity Kemel Coneergnnee Dirriomos (Prposition
A o Alf
Ev Evalantion Equation (Apperilis E)
Continuors Mapping Thoorom (Billingsle, 2013)
Wave Kemel For Conventional FW]
Wave based NTX
Cortimous FWI
Wase-baasd Consegence Daring Traising
Wave-haaed Convengnnce
(Prepesitinn 2.1)
(Peepoiition 3.1)
ADNED i y A(9 [Tzcoeem 5:2)
Figure 35: Framework of our theoretical proof.
vergence properties of the standard Neural Tangent Kernel, regularity conditions for hyperbolic initial-boundary problems, and the formulation of the evaluation equation. Subsequently, the proof of Theorem [.1 is structured around key lemmas such as the Lipschitz continuity of the Green's function and the sensitivity kernel, supported by the continuous mapping theorem. This section also establishes the convergence the wave-based NTK during training and at initialization, alongside distributional convergence of the sensitivity kernel. The final part focuses eigenvalue compar- isons, proposing and applying a spectral comparison theorem to demonstrate several inequalities regarding eigenvalues under different parameter settings, stated in Theorems 4.2
5.1 and 5.2
D PRELIMINARY
D.1 NOTATIONS
We denote scalars, vectors, and matrices by x, and X, respectively. The spatial domain is repre- sented as an open set C Rd, with boundary 0U, and the space-time domain as UT U x [0, T)) Scalar-valued and vector-valued functions are denoted by U + I and U + IRm respec- tively. The first and second derivatives a function (..) are denoted by d'(.) and o"(.) The gradient of function f(w,x x) with respect to is written as Vwf(w, x) Rd For function spaces, CUU refers to the space of k-times continuously differentiable functions, and (*(U)) de- notes the Lebesgue space of p-integrable functions, where 1
S +00. Regarding norms, for vectors, || and || |/ represent the Euclidean norm and the infinity norm, respectively; for matri- ces, -|- and || || denotes the spectral norm and the Frobenius norm. For functions, the L and the essential supremum norm are denoted by || (lze(() and (C)) respectively.
D.2 STANDARD NTK THEORY
Assumption D.1. The neural network Fe defined equation satisfies the following properties:
1. Parameter boundedness: There exists constant C' > O, independent of the hidden layer width n, such that the network parameters remain uniformly bounded:
sup [0(T))l SC
2. Smoothness of activation function: The activation function belongs to (*++((R) for some such that for any scalar input E R.
2 2, and there exists a constant C
sup (0) C.
max OS1Sk+1 DER
Remark D.I. This is common technical condition in related literature WWng et al. 2022
36

---

Published as a conference paper at ICLR 2026
Next, summarize prior theoretical results concerning the standard NTK. Consider neural net- work or, generally, machine learning model fr (x, 0) with learnable parameters 0 E R", like equation The empirical NTK is defined as the inner product of the gradients of the network's output w.r.t. all of its parameters. Denoted by
Rd x Rd - IR, it is given by:
(x, 0) (f-(''0)
K= x'; 0) (Vo[-xx00) Vol= Σ
(21)
00;
00,
Under the infinite- width limit, the empirical NTK converges to a deterministic kernel (x,x') that remains constant throughout training, a phenomenon referred to as the "lazy training" regime.
Theorem D.1 (Convergence Property of Standard NTK). Consider a neural network f+(x, 0) of width n like equation under Assumption D.I. Suppose the parameters O(0) are initialized as i..dd draws from N( 1). Then, as the width a + oo, the empirical NTK (x x'; 0) converges in probability to deterministic kernel ((x, x'), both at initialization and during training, iee,
xx - (xxx'')
Moreover, this convergence uniform over all imput pairs (x, x') and all time points T 2 O. Specif- ically, denote 0g as initial parameters, under certain regularity conditions, with high probability,
sup sup (x, x,x' 720
K Kx 0o)) _ O,
where F. denotes convergence in probability.
Remark D.2. The proof can be found in (Jacot et al. 2018). This theorem elucidates the fun- damental concept of the lazy training regime. The convergence of the empirical NTK to a static, deterministic limit K implies that the function f=(x, 0) evolves in a manner asymptotically equiv- alent to a linear model in the feature space defined by the gradient initialization. Consequently, the training dynamics of an infinitely wide neural network under gradient descent are governed by the deterministic kernel K, simplifying the analysis of its convergence and generalization.
D.3 HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION THEORY
Let Ω C I" be a bounded domain with C2 boundary an. Consider the second-order linear partial differential operator depending on time
72
82
n
Ô Dx,
L(t)
Σ
any(t, x) dz,On,
Σ bt(,,))
c(t,x),
(22)
And the hyperbolic initial-boundary value problem:
Ofu(t, + Lt)uu(t, x) = xt,, u(t,x)= = u((,,)) = p(x), 0,u(0, x) (x(x,,
x E 22 t 0, in, 0,
(23)
x f.
Next, we consider the regularity for hyperbolic initial-boundary value problems in equation 23 Theorem D.2 (Regularity for Hyperbolic Initial-Bounde
Value Problems). Assume the co- efficients satisfy Gij, daaij, Sasi (Srr) (i.j = 1.2...... n. and bi, O,b, 0.c E L ([rr) (i = 1,2,..., There exists > such that for all (1, x) E [o, T] x and & R""
Σ Canst, xxE 2 0,
and that the source and boundary functions satisfy:
E H ((0, oc), (()),, E (())), ve ())).. Let u be a weak solution of (4.4.20). Then:
u E I (10, 00), H²( n ())),
0,u E L (10, 00), H (0)), 0R E 1 (10, xo), (2)).
(24)
37

---

Published as a conference paper at ICLR 2026
and the the following estimate holds for all T > 0:
(10))19 + [0.0 + 1 CI) (0.7 + (1ll + | (1))
where C(T) is constant depending T and the coefficients L(t).
Proof. This proof can be found in (Cu 2015) and
Evans 2010).
Assumption D.2. Let U C. gd bounded domain with C2 boundary, and let T O. Consider the acoustic wave equation equation
7 The velocity model m belongs the admissible set:
A = (m E C' (RR)) 0 < my S mmx) S m2 < co for a.e. X EED ER"),
where 1 and m2 are positive constants. The source term 8j is assumed to be in H (O, The domain D (where the receivers are located) is disjoint from the source domain U, i.e., there exists a constant > 0 such that dist( Soa(D,ue D,U) 2
Remark D.3. This assumption can be easily satisfied using an implicit neural representation (Sitz mann et al. (2020) using an activation functiion with C continuity, such as IFWI (Sun et al. 2023a
and WinFWi (Yang & Ma 2025
Proposition D.1 (Regularity for the Acoustic Wave Equation). Under Assumption D.2 consider the acoustic wave equation equation | with zero initial conditions:
u(0, x) - O, 0,u(0,x) =0 f
Then, for any T > 0, there exists unique weak solution uj satisfying the regularity properties:
U E (0,T; (z)())
0.u ( 8lu E L (0, L(U),,
and the following energy estimate holds:
|| 1 (O.T::H(()) + +10yy + oo
((0T)):+(())- S CCT.UMM, m2) (m)))))l((l(())
Proof. We rewrite the acoustic wave equation equation in the form:
of Lu s,(1,x),
(25)
where the spatial operator is Lu Ax) This fits the framework of Theorem D.2 with
coefficients ai (x) = m² mx)o; (UT), and b; = 0. The uniform ellipticity condition is satisfied since m(x) 2 m2 > implies:
Σ an (x)(x) mx))) 2 ml for all & Rd 1
(26)
The zero initial conditions satisfy yp Applying (.2 yields the desired regularity and the estimate, where the constant C' depends on T, the domain U, and the bounds m1, m2.
and the source term 8j has the required regularity.
Remark D.4. This proposition implies that the wavefield u, and its second time derivative oR,, are bounded the corresponding norms, which is important for subsequent estimates.
E THEORETICAL PROOF
E. PROOF OF PROPOSITION 2.1
Proof. For a fixed data point (x, t), the derivative of the synthetic data uSy with respect to the flow parameter is given by the chain rule:
D Ou syn (x, t) (g]m(()](xx1)
f(
8g(x,1) Omlly)
Sm(y) [m].
Ør dy.
(27)
Ør
Ør
38

---

Published as a conference paper at ICLR 2026
Substituting the gradient flow dynamics from equation 3 into equation 27 yields:
@u (x,t) t)
lo 6g(x, 1)
8J
[m]
im]dy
6m(y)
6y(y)
So (x
[m]] [ f 69 .t')
[m]] (u - w dx') |dy
6ylly)
5mm(y)
J.
ög(xx, [m]] 8G(x', 1') [m]dd) (uSyn " dx', 8m(()
(28)
8mm(y)
Here, the last step of the equation is due to Fubini's theorem, where we assume the integrand is absolutely integrable over U x D. We now define the wave kernel θ as the inner product of the Fréchet derivatives:
J. 80(x,1) [m] 5mlly)
8g(x', [m]dy. dm(y)
θ ((x, t), (x, m):
(29)
This kernel is symmetric and characterizes the interaction between wavefield perturbations differ- ent data points (x, t) and (x',t) t') due to model changes.
E.2 PROOF PROPOSITION 3.1
Proof. For a fixed data point (x, t), the derivative of the synthetic data u (xx t) with respect to the flow parameter is given by the chain rule:
Dugy (x,
0g @g[mm](xx 8)
00:
(30)
00,
Ör
Substituting the gradient flow dynamics
80 07
0J 00,
[0] yields:
OuLy (x,
P aglmol(x.t) 0J
00, (0)
(31)
00,
The derivative of the cost functional I given by:
0J 0
J,
0g [m|mo] (x',1)
(u (x)(1) - u o (x')) di' dx'xx'
(32)
00,-
00;
Substituting this back, obtain:
adD
syn (x,1)
00 (m)) t)
Un 0g[mol(x',
(u wibbs) di' dx'
00;
00;
=-
Σ
ag 00[mo](x, t)
0g [me] (x',
(u (Ddnn) dt' dx'.
(33)
00:
00;
The interchange of the sum and integrals is justified provided the sum converges absolutely. We now define the wave-based NTK as the inner product of the parameter gradients:
ag[mo (x,,) t). og @g/mol f')
θ Wave (xx, t), (x', 0)
(34)
a0,
00;
This kernel be further expanded by applying the chain rule. The derivative with respect network parameter is related to the Fréchet derivative with respect to the model m:
0g(x, t)
89(x,t) t)
Jm(l) [ma] dm
00, dy.
(35)
00,
Substituting this form into the definition of Omkk gives:
(1.
tg(
@moly) dy
8g @mo(z) dz
O mmk wave
Σ
(me)
[me]
8mm(y)
00;
dm(z))
00,
==
8g(x, t)
[mo) 89 5g('',t' 1')
Omoly) @m)
δm((z) [mo]
dydz.
(36)
8m(ly)
00; 00:
39

---

Published as a conference paper at ICLR 2026
The term in parentheses is recognized the standard NTK for the network output me:
dma(y) @mp(z)
K (y,, z;
(37)
00;
00,
Therefore, the wave-based NTK is expressed as double integral of the physical wave kernels cou- pled with the neural NTK:
Jo 60 t) dy) [me] 8m((z)
8G [me] Ky z; dyddz
O Wave ((x, t), (x', (); (x,+);
(38)
Substituting this back into the evolution equation concludes the proof.
E.3 PROOF OF THEOREM
To establish the stochastic properties of the wave-based NTK both at initialization and during train- ing, we first prove the convergence of the continuous representations under the infinite-width limit. We then demonstrate that the wave- based NTK is Lipschitz continuous, and employ the Continu- ous Mapping Theorem (Oksendal 2013) to prove the convergence distribution of the wave-based NTK. Accordingly, we begin by analyzing the initial distribution of the velocity model using a con- tinuous representation under the infinite-width assumption.
Lemma E.1 (Infinite-width Convergence of Neural Networks). Consider a fully-conmected deep neural network (x as defined in equation 5 under Assumption D.I, Suppose the parameters O are initialized according to proper scaling. Then, as the widths of all hidden layers sequentially or jointly, for any finite collection of inputs {x1, X2,...,xa. .x) C U, the oulput function Fo(x) converges distribution to zero-mean Gaussian process:
(Fo(x1).... ...., Fo(xx)) D, (O.
where the covariance matrix E has entries Σij I(x;, given by a deterministic kernel function E U x U TR. that can be computed recursively through the network's layers. Consequently, the reparameterized velocity model mo(x) = mo(x) + Fo(x) using a continuous representation converges in distribution to a Gaussian process GP (mo (xx,,(x)xx)) x').
Proof. The proof relies on the application of the Central Limit Theorem (CLT) and induction over the network's layers. For detailed proof, see (Bonfanti et al.) (024b) or (Jacot et al.) 2208)). Since mo(x) is a deterministic function, adding it to the zero-mean GP Fo results in a GP with mean function mo and the same covariance kernel Σ.
Next, we prove the continuity of the Green function and the sensitive kernel.
Lemma E.2 (Lipschitz Continuity of the Green Function). Let Gm denote the fundamental so-
lution (Green function) associated with the wave operator L., V mm())0f Under Assumption D.2 in particular given the lower bound mi 0 and the separation ó >> 0 between the source re- gion U and the receiver region D, the mapping m + Gml DxT is Lipschitz continuous from L (U) L T), ie., there exists a constant Lg (m,, m2, 8, D) 0 such that:
||Gms (o, Gmg(s Xo, 10)l $ I.gllm m2 (=)))=
for any mi, m2 E A.
Proof. Let m, m2 E AA be two velocity models and define their difference as Am = my m2g Consider the difference of Green functions w(x, Xn, to) Gmy (x, xo, 1o) Gmg (x, t; Xo, to). which satisfies the following wave equation:
Lm, w(x, f) = Am(x) (x,
4o)-
(39)
We apply energy estimates to the wave equation satisfied by w. Define the energy functional:
1 1 / [mi(x)]0,u(x 2) + [w [ dx. E(t) 2
(40)
40

---

Published as a conference paper at ICLR 2026
Differentiating with respect to time and using the wave equation:
dE
(x) Vw. Vd.w] dx
di
(0,o(m10p V2 w)) V (B. dx
Rd
(41)
1- C., w + V (W. dx Rd
{ Rd
[-dm (mmdfGma) + V (d:wVw)] dxc
Integrating space RR and applying the divergence theorem (the boundary term vanishes due to finite propagation speed), we have:
dE
| Bew (mm + V (OwwVw) dx
dt
Rd
J dew
Gm. )dx + of (Qw nds
(42)
Oew Gmn))
Using the boundedness Am and the Schwarz incquality:
dE
Ammlle J(c 18,u|l0fGma Gma fm
(F gg 18
1/2
(/
1/2
18. 2dx
(43)
$ [Amill=
/ E(t) Gmz (v, Xo, to) to) (2(()))
where the last inequality holds due to the energy definition and the lower bound m, > 0, i.e.,
2 E(())
10,0
(44)
m
This gives us the differential inequality:
d
1
E(I) <
(45)
di
2m1
Integrating from to to
1
I' (1
E(t)
Amile=
(46)
v 22m
From the theory of hyperbolic equations and under Assumption D.2 we have the regularity results for the Green function Gmz- For a fixed source point (Xo, to) and away from the singularity, Gun enjoys higher regularity. Specifically, for D where dist(D,
0, have:
||0Gm, (s, 111x;xx xo. (0)lL>(() Ci m1, m2, To).
(47)
The energy of wavefield decays sufficiently fast away from the source due to finite propagation speed and geometric optics approximations. This ensures that:
S (. (2) (0 S C2(m1, m2, To).
(48)
41

---

Published as a conference paper at ICLR 2026
Combining the previous estimates and using Poincaré type inequalities, obtain:
[ (x,
||w( 4; xo. (0: =
(2(D)
- (10,w(;
(49)
2
F((()dr - To) v ((((
1o V m1
Cs(D,T) 22m Lm m2, To)
Integrating over time 1 [0, To]:
[ - /2/2 sup
|w( (0)
EET
S T// (D, To) /2m1
1
C2(mi, m2 (o0
(50)
A T.glmm - -m2 -22l1=-U)=
T/ depends my, m2, 6, D, To- This completes the proof.
where Lg
V2mi
Lemma E.3 (Lipschitz Continuity of the Sensitivity Kernel), Consider the acoustic wave equa-
tion equation under Assumption D.2 The Fréchet derivative omily) [m] is Lipschitz continuous
69(x.))
with respect to the velocity model That is, there exists constant L L(mi, m2, D, > such that for any mi, 1122 E A::
(x(xtt) 8m
(x
[m1]
(mz)
Tl|ma mallz=(0)
8m((y)
(2(D)T)
Proof. The Fréchet derivative of the wavefield uy w.r.t. the model parameter m is given by:
6u,(x, 1) [m]] = =- G. t:y. (
(51)
6m(y)
where Gm is the Green function for the wave operator. For m. m2 E A, consider the difference:
8ug 8m
du, 8m
[m]]
[mz]
/ (x, (mm,))))
(x, ( -
1')] di
(52)
/ (Gm, - G (, J. Gms (mzu,(m1) - (kur((z)) dt'
A A, By.
Using the Sauchy-Schward inequality and the properties of the wave equation:
/' ||(Gm, (. dt'
(53)
T
< f
|[mm, -ww.--/llleelll@@@mmmmlllll
From Lemma E.2 we have:
||Gm, G (mall1 S Lyllm m2lla=(0))
(54)
From Proposition D.1 we have:
100u, (m1))l S S
(55)
42

---

Published as a conference paper at ICLR 2026
Therefore:
||)]|E*(DXT) $7L.GGlls)llrrooooeeoool - me)l
(56)
Similarly:
l [[G [[Gmmllaacp))0888ymm))) u (mz)
(57)
- 5/ (ml - (m2))
where C2 is bound on [|GmallZ((D) which exists due to the regularity of the Green function away from the source. To estimate (m0)))(m)) - 8Ru, (m2)))l===())))) we consider the difference V/ (m(m)) u,(m2) which satisfies:
Cm, by (m1 m2 m2)00f, (m2).
(58)
Using energy estimates for hyperbolic equations and the regularity from Proposition D.1
we obtain:
[10,u,lle~(1) & Callma ( (ma))l)m
(59)
Therefore:
[Bll=<DDXTT S TC2C3C4 (Ise:llsslll1:0_)))))pp (177))))) || m2
(60)
where C2 is a bound for ( L (U). (u))- Combining the estimates for A, and Bj, we obtain:
5ug
ouj
[m]]
m2
I Al[ma - m2llz=(U))
(61)
8m
8m
(P((TT))
where = C2CCCL (Q.2))
This completes the proof.
Then, focus the convergence of the initial sensitive kernel the width tends to infinity. Proposition E.1 (Sensitivity Kernel Convergence in Distribution). Consider a fully-connected neural network Fo as defined in equation 5 under Assumptions D.1 [.2] Suppose the param- eters are initialized with appropriate scaling. Then, as the width n co, the finite-dimensional distributions of the sensitivity kernel converge:
t1(x1.t1) [me).....
8g(x,,, tx)
(89(x1, t1)
8g((k: tx)
[ma]
[GP),
(gP)
8m(ly)
8m((yk)
5m(y
8m((K)
For any finite collection of points (x,, to y.) Gaussian process (P 2) from Lemma E.1
D I....,k, where GP denotes the
Proof. By Lemma E. as the width n + oc, the finee-admeesional distributions of ma converge to those of GP (mo; )). That is, for any finite set of points (x1,-. xx3 U,
mx(x))) D ((, E), where l (moo))),.... mxx)))) and Eij = E(x,, x,) Furthermore, by Lemma E,3 the Fréchet
(62)
derivative operator
① m 8g(x, 1)
[m]
(63)
8m((y
Is Lipschitz continuous from (L -1==)) to (L2(D x T), || (22). In particular, for any fixed
finite collection of points (x,, t. yi). the mapping
66((1, (1)
[m]] 89(xx, tx)
[m]
(64)
8m((y))
8m((ys)
continuous from (U), 11-11 to I Since me converges to GP(mg, Σ) in dimenssonal distributions, and the mapping & is continuous, by the continuous mapping theorem for finite- dimensional distributions (Billingsley 2013), we have:
88((1,,1) imo)...., 8G(x,, (x) 8m1(y1) This completes the proof.
2, (89(x1,41) [gP),.... dm(ly))
8g(xe, (GP).. dm(y))
(mol)
(65)
6m(y))
Next, we begin to prove the conclusions and (II) Theorem 4.1
43

---

Published as a conference paper at ICLR 2026
E.3.1 PROOF OF THEOREM 4.1
Proof. Recall the definition of the wave-based neural tangent kernel from Proposition E.2
wave ((x, 1), (x', ('): J/lo 6g( [m]
89(x', (') (m] K-(y, 2; O)dydz.
(66)
5m(y)
8m(z)
By Lemma 4.1 the standard NTK K2 (y 2; 0) converges in probability to a deterministic limiting kernel, i.e., (y, 2; 0) + K (yy, as n + oc. Furthermore, by Proposition E.1 the following sensitivity kernels converge in distribution to the following random process as 7 oo, i.c.
(g(, t) [me]= 2, 80(x,t) t) [CP(mmo Σ)] and 8g(x''t') [me] 2, 60 t') (g Σ)] (67) dm(y)
8m((y
òm((z
8m(ly
Applying Slutsky's theorem, we conclude that for each fixed 2), the following product:
(x, t)
5g(x',t')
(y) [ma]
(m) [ma] K- (r
(68)
converges in distribution to the following random product under the widhh condition:
8g 1) [GP(mm, Σ)]
(0(''(') (gP(mo: E) (y,
(69)
8m((y)
8m((z)
Since this convergence holds point-wise in (y,z). z), and the integral is continuous mapping, we apply the continuous mapping theorem to conclude:
wave ((x, 1), ())
2
2 f. 6g(x.t) (gP) 8m(y)
5g( (GP) K(y, z)dydz. 5m(z)
(70)
The limiting expression is random variable for each fixed input pair (x, x t'), which we denote by Oatk ((x, (x', t')). This kernel is noneetteriiiistic due the randomness in the Gaussian process realizations of the sensitivity kernels. Therefore, we have established the point-wise conver- gence in distribution, i.e.,
mtk wave 7=00
Amik wave
as 72 + 00;
(71)
where wave is random kernel.
E.3.2 PROOF OF THEOREM 4.1 (1I)
Proof. By Assumption D.I and the properties of gradient flow, assume that there exists quence тn -> oo such that mø(r.) - m² almost surely as n >> oc, where m is minimizer of the loss function. Define the sensitivity kernel operator for fixed (x, k)) as:
5g(x,t) f)
A(m)= = 8m(y) [m].
(72)
By Proposition E.1 surely, we have A(ma(=..)) )) - A(m") almost surely for each fixed (x,t,y). The wave-based NTK at time t and width is defined as:
is Lipschitz continuous w.r.t the L norm. Thus, since mo(F.) ) + m² almost
[ Ju Ammo()))xx K2 (y, A(mo(=))(2 dy dz,
θ
Ontk wave
(73)
where K2 is the standard NTK for the model output. By Theorem 4.1 at initialization, the standard NTK Ko converges uniformly to deterministic kernel K as the width tends to infinity. Moreover, in the infinite-widih limit, the standard NTK remains constant during training, Kr - K for all T 2 0. Hence, the infinite-width limit of the wave-based NTK at initialization is:
- -fc (x, K(y,z) z) A(mo(o)) (x'((,2)) dy dz.
Outk 0c wavelo
(74)
44

---

Published as a conference paper at ICLR 2026
Now, consider the difference in the sequence Tn, and apply the triangle inequality, we have:
@ wavelr,
-Otk |o0
wavelo
- 11 + I2.
(75)
where
1 Je A(mo(sa)) Kr., A(me(r.)) M(mo(r))) K A(me(r)) dy da
(76)
1 J1 A(mo(r.)) K A(mo(T.))) A(maro)) K A(mo(o)) dy dz
(77)
and we use shorthand A(mo(T.)) for A(mo(r.)) , t' z). For 11, since Kr, K (by Theorem
and ((ma(r.)) uniformly bounded (by Lemma E.3 and the boundedness of A), we have:
1 S [|A(me(r..))) ( Mls | - | |Up 0 almost surely as - oc.
(78)
For I2, define the function f A + IR for fixed (x,t,x' t by:
[(m) [ / (,,)( K(y,,) 2) A(m)(x' , 22) dy dz.
(79)
By the continuity of A and the boundedness of U, f is continuous on A. Then:
I2 (f(ma(r.)) (mero)))
(80)
Since mø(r.) - m almost surely, we have (marr..) + f(m") almost surely. At initialization, mø(o) is random with non-degeeerate Gaussian distribution (by the initialization assumption and the wide network theory), and ʃ is not constant on A (because the sensitivity kernel depends non- trivially on m). Therefore, the set {m E A f(m) = f(mm)) has measure zero Hence, almost surely, ()) * J'(m"), ) and thus for large n, (f(m((.)) (mero))) > 0. Then, combining the
estimates for I and I2, we conclude that almost surely:
lim inf Qutk =
Qutk pc wave 0
> 0.
(81)
wave
This implies that for large n, the supremum over satisfies:
sup Ontk In ntk wave wave lo
O,
(82)
since the value at Tn is positive. Therefore, almost surely:
lim inf sup @" 1n θ ntk
100
> 0.
(83)
-00 720
This completes the proof.
E.4 PROOF OF THE EIGENVALUE DECAY COMPARISON THEOREM
The spectral properties of the wave kernel and wave-hased NTK are fundamental to understanding the convergence behavior and robustness in FWI and CR-FWI. To analyze the eigenvalue of the wave kernel and the wave-based NTK, we define the Fréchet derivative integral operator Fm : LU) + LD(D x 7') and its adjoint Fü L²(U)) + L²(D x T') as follows:
Fm),)(xtt) Sf
(G(x,t) [m]h(y)dy, Fllg)(y) S J 89(x, 1) [m]g(x, t)dtdx. ōm(y)
(84)
8m(y)
Moreover, we define the kernel integral operator K IT? - (U) as
K[N((y) = =fo K(y, 2)h(
(85)
Hence, the wavefield evolution equation of CR-FWI can be expressed as follows:
dd (x,t) t)
= Fm K F [4 d T(K) - d-
(86)
where T(K) A Fm o K e F denotes the integral operator of wave-based NTK.
Next, compare the eigenvalue for two operators of wave-based NTK, i.e., T(Ki) and T(K2).
45

---

Published as a conference paper at ICLR 2026
Theorem E.1 (Spectral Comparison Theorem of CR-FWI). For two self-adjoint and positive semi-definite operators K1, K2 (2) + (U),, define the wave-based NTK operators as:
T, FmKiKK i 1,2.
(87)
Let Xy( denote the eigenvalues of arranged in non- non-increasing order. tf operators satisfy the dominance condition K1 = K2 (i.e., K1 K2 is a positive semi-definite operator), then the eigenvalues of the corresponding wave-based NTK operators are similarly ordered:
Xy(Ti) 2 (T(2 for all 1.2.....
(88)
Proof. The proof follows directly from the min-max principle and the properties of the operators. Since K1 = K2, we have that K is positive semi-definite. This means that for any g E (U)
(K19.9) (V) 2 (K29, g) LA(U).
(89)
Now, for any f LD x T), g= Filf E (U)). Then the quadratic forms of the wave-based NTK operators satisfy:
(TT.f) = (Fi (II((TT))
= (KFF
2 KKaFm.iFaa
(90)
(FiKFF (IDADTT)
Thus, we have established that:
(TIS, (T2f. for all E LD T).
(91)
By the Courant-Fisss Fischer (min- mmx) theorem, for each j = 1,2....:
>y(T1)= min VDLT FEV dim(V)=j [f]=1
max (TiJ,
2 min
max (T25,
(92)
VCL'(DXT) JEV dim(V)=j [/|=1
= [T)(
The inequality follows because for any fixed j-dimensional subspace V, we have:
max SEV [f]=1
(Tif.f)
JEV 1/|=1 (Te
max
(93)
and taking the minimum over all such V preserves the inequality.
Remark E.1. According to the Spectral Comparison Theorem (Theorem E.1y. if two self-adjoint positive semi remi-definite kernel operators satisfy K₁ > K2, then the eigenvalues of the corresponding wave-based neural tangent kernel operators T₁ and Tz satisfy >y(TT) 2 >y(T2). In particular, both the wave kernel θ wave defined in Proposition 2.7 and the wave neural tangent kernel Qrtk defined in Proposition 3.1 can be expressed in the form T(K)
in the theorem, where K corresponds to the identity operator Kg (see the following Proposition EE2) and the neural network's s NTK kernel Kr, respectively. Therefore, if the NTK kernels of two neural networks satisfy KT,1 Z KT then the eigenvalues of their wave neural tangent kernels will also satisfy the same ordering relation.
To apply Theorem E.1 we need to rewrite the formulation of the wave kernel and prove the self- adjoint and positive semi-definite of wave-based NTK.
Proposition F.2. The wave-based NTK Quik defined in Proposition 3.1 1 is symmetric and semi- positive definite, under the assumption that NTK K,, is a symmetric and semi-positive definite kernel.
Proof. We aim to prove that the wave-based neural NTK defined by
mmk (xx, t), ('); [.
89(x,1) dmly) [m] 8G(x', ,'),
6m(() [m] K- d)dydz
(94)
46

---

Published as a conference paper at ICLR 2026
Symmetry. To show symmetry, we observe that the expression for O Wave is symmetric in (x. and tx), Explicitly, interchanging (x, t) and (x', ,') yields:
O ((x,, ('); (x, 0);0) Solo 89 ,t) [m]] 0g t)
[m] Kr O)dydz
8mn(y)
8m(2)
So (x(x,1)
[m]] 6g(x', (')
[m]] Kr (z; 0ddzdy
dm(y)
dm((z)
O wave ((x, t), (x', ())
(95)
where (y, 0) (y, 0) by the symmetry of Kn.
Semi-Positive Definiteness. To show semi-positive definiteness, we must verify that for any square- integrable function f(x, t), the double integral
// f(x, t) wave ((x, t), (x: d')dxdddd' dt!
/] f(xx, [/o (g(x,t)
[ml 5g(x',t) δm((z) [m]]
dm(y)
Kr O)dydz) f(x, ddxdtdx'dd dt'
So K- 2; 0)
Jux f(x, // 8g(x,1) 1) Sm(y) [m]] dx dt
[/]
U [(x', 09(x',t')
[m]] dx' de di) dy de
Öm(())
SC [. Kyly, n; 0) g(y) g(2) dy du 2
(96)
where g(y) & SfeET fxx, t) snnty) [m]] dx dt and K, is semi-positive definite. Thus, 1 2 0, prov- ing that Omkk is semi-positive definite.
Proposition E.3. The wave-based NTK Qu wave defined in Proposition 3.1 degrades into the wave
kernel Wove defined in Proposition 2. without neural reprexentation.
Proof. The proof consists of showing that under the given conditions, the parameter-space inner product that defines the wave-based NTK reduces to the model-space inner product that represents the wave kernel. The Fréchet derivative of the model ma w.r.t. the parameters @ at a point is:
dmo(y) d0(y)
8(y z).
(97)
d0(z)
d0(z)
This follows because a change in the parameter 0 (i.c., the velocity model m) at the point z directly and solely affects the model mg at the point y = z. This wave- based NTK in the model space, K,, which measures the inner product of the functional gradients of the model output, is then given by:
Ky 2) = // dmo(y). dmo(z)
dw
d0(w) d0(w)
(98)
[. ōly wy-wwolz w)dw 8(y z).
The inner product is taken over the parameter space, resulting in the Dirac delta function due to the orthogonality of the functional gradients; perturbation at affects the output only at w. The general definition of the wave-based NTK is an inner product over the parameter space 0:
[ [ 8G(x,t) [m]]]
8g(x', ,t') [m]] Km Km,y, 2; 0 dydz 8m((z)
Omk ((x, 1), (x,():
dya(y)
[( { 8g t)
[m]] 8g tt)
[m] -5(y - z)dydz
dy((y)
8m((z)
(99)
8g(x, t)
8g (x',t')
[m]]
[m]dy
Ju dm(y)
dm(y)
O ((x, t), (x)
47

---

Published as a conference paper at ICLR 2026
This completes the proof that the wave-based neural tangent kernel reduces to the standard wave kernel when the neural representation is the identity function.
Proposition E.4. Consider the wavefield evolution equation governed by the wave kemnl::
Hu D (T) -uoss)
(u)))uu
-uobal
where K(O) is an integral operator defined as K(O)[u| Sn (((,,t), .t) (d)))))uxx -u(x t') dx'.
Then, K(@)) has spectral decomposition K(@)) = Ek=1 XR @ @b. where {> is a sequence of non-negative eigenvalues and {Φx { x forms complete orthonormal system of eigenfunctions. Define the spectral projection operator Q as Qf _ (f, dz)k=1 and the diagonal matrix A diag λ1, .2....). Then the evolution of the data residual in the spectral domain satisfies:
Maryn (T) - = =
Myn (0))
D
where the absolute value is understood component-wise.
Proof. Let the data residual function e(T) = -Wsy((T) - -ubss- and the evolution equation becomes:
θe(T) = -K(O)(((T) Òr
(100)
Since K((@) is self-adjoint, compact, and non-negative definite (see Proposition E.2), by the spectral theorem for compact self-adjoint operators, there exists a complete orthonormal system {Φx ) and non-negative eigenvalues (A ) such that IC(e)= = k=1 d @ Φx. Hence, we can expand
the residual function in this eigen-basis:
E(T)
=
ag (T)
k=1
where the expansion coefficients are given by (T) = (e(T), oz). The spectral projection operator defined as Qe(T) = {ax [T)())) Substituting the expansion into equation 100
(x(T)< = -((O)
Σ axx(7 (T)
Σ ax (T)
Due to the orthonormality of fdb 1 obtain the system of ordinary differential equations:
day(T)
Asap(T) for all k = 1,
(101)
dr
The general solution to equation equation 101 is (x(T) as (0)e Aww Therefore, the solution becomes (T) = (u (0) uobs: Øz)e Au In the spectral domain, we have:
D
Qe (T) {ax (ox(T)) (T)
(1u (0) a Øx)e 207700
= - Ar ATQ(U (0) (.
Taking absolute values component-wiss
(102)
where E -Ar a diagonal matrix with non-negative entries e -rT 2 0. This completes the proof.
Next, we apply Theorem E.1 prove Theorems
4.2 5.1 and
5.2
48

---

Published as a conference paper at ICLR 2026
E.4.1 PROOF OF THEOREM 4.2
Proof of Theorem 4.2
Recall the definitions of the operators:
Twave = FmK&Fmm
(wave kernel operator)
(103)
Twane Fm (wave-based NTK operator)
where Fm (.) (22 T) is the Fréchet derivative operator defined in the main text. The operator Ks (U) + (U)) denotes the identity kernel operator, which is equivalent to the identity operator on (U). Under proper neural network initialization, the NTK operator KT (TT) + (T) satisfies |Ko I 1. This implies the operator inequality Ks > Kr, i.e.,
K3 K- positive semi-definite operator. Applying the Spectral Comparison Theorem E. with K1 K6 and K2 Kr, we immediately obtain the eigenvalue comparison:
X (Twave) kgF%) 2 (,(mm
(T
(104)
for j 2, This establishes the desired eigenvalue decay relationship between the standard wave kernel and the wave-based NTK.
E.4.2 PROOF OF THEOREM 5.1
Proof of Theorem 5.1
Recall the operator definitions for the two learning paradigms:
TINR = FmKINR mn
(operator of INR-FWI)
(105)
TMPM FM.K
(operator of MPE-FWI)
where KINR and KMPE the NTK operators for the INR and MPE models, respectively. From Audia et al. (2055), we have the operator decomposition:
MME KINR + K*
(106)
where K+ is a positive semi-definite operator. This decomposition implies the operator inequality K MME = KINR, as WMPE KINR = K+ > 0. Applying Theorem E.1 with K₃ = Mee and
K2 = KINR, we directly obtain the spectral enhancement property:
Xy (TMPE) Ky 2 Am X (TINk)
(107)
for all j 1,2, This completes the proof that the MPE approach exhibits enhanced spectral properties compared to the standard INR approach.
E.4.3 PROOF OF THEOREM
Proof. We consider three CR-FWI methods: INR, MPE, and IG (INR-Grid). Let their respective model representations be:
MINR (x) = MLP (
(
INR
(x)):
01
MAPE (x) MLPP (
(
MPE
(x)),
(108)
02
mgG (x) MLP ( 1G (x)),
where diG Va Po, (x) + (1-oa) o (x) denotes the concatenation of feature vectors with weighting factor [0, 1]. key observation is that the NTK for each method admits a natural decomposition into two components: one from the MIP parameters and one from the feature encoding parameters. For any method, have:
@me(y) 0m0(z)
K(y, 2)
20i
00:
Jmo(y) @ml(2)
Oma(y) 0mf(z)
(109)
00i
00i
d0i
00i
Ksar
Kexoli
49

---

Published as a conference paper at ICLR 2026
Crucially, assume that all three methods share the same MLP architecture, and thus have identical MMLP components in the infinite-width limit. The differences arise only in the encoding components:
KINR - KMLP +
INR
Ficode:
KMPE = -KMLP + MPE Fincdde
(110)
K1G =KMLP + K
From Audia al. 2025), we have the operator inequality K we assume the INR and MPE feature gradients are orthogonal, and we obtain:
Encode 3 K KPE For the IG method,
Encode
KIG
Encade (x, x') (Vā
Po, (x)
v1
MPE 082
(x), Va Po, (x') e a:)
900
(M))(()))
MPE
(00) (x),
+ (1 a) (00
MPPE (xx, 902
MMPE (x))
a: K K-NN
Fmerde [x, x')
(1 a.) KMPE
(x, x')
(111)
This linear combination preserves the ordering:
K
INR Fncnde
K-G
MPE Fineule
(112)
Encode
Since all methods share the same K MLP component, this ordering extends to the full NTK:
KINR Kig = KMPE
(113)
Defining the following wave-based NTK operators:
TIN =m
Ti FiGK
(114)
TMPE
and applying Theorem E.I twice yields the desired spectral ordering:
(TIRR) (TTi)) $ (TTMP)) for all 1.2....
(115)
This completes the proof.
50


Reviews:
[{"review_id": "yBdZOWHlXY", "reviewer": "Reviewer_sJvk", "summary": "The paper develops a wave-based Neural Tangent Kernel (NTK) analysis for continuous-representation FWI (CR-FWI), explains why INRs help (and why they converge slowly at high frequency), and proposes practical CR-FWI variants (including a hybrid INR + multiresolution grid, IG-FWI). The submission claims strong empirical results on standard FWI benchmarks (Marmousi, SEG/EAGE Salt & Overthrust, 2004 BP, 2014 Chevron).", "questions": "see weekness.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Interdisciplinary theoretical contribution. The wave-based NTK framework is a thoughtful specialization of NTK tools to FWI that helps explain observed phenomena (robustness to initialization, spectral bias / slow high-freq convergence). This new perspective can guide architecture and sampling choices.\n\n- Use of standard, realistic benchmarks. The authors report results on well-known FWI models (Marmousi, SEG/EAGE Salt & Overthrust, BP, Chevron). Using these makes the claims more credible to applied communities and shows practical intent.\n\n- Actionable design proposal (IG-FWI). The hybrid INR + multi-resolution grid is a practical idea that directly follows from the kernel eigenvalue arguments and aims to trade off robustness vs convergence speed.", "weaknesses": "- Ablation & failure modes. The paper claims IG-FWI is an “optimal trade-off” — I’d expect ablations that vary eigenvalue decay (e.g., different INR frequency bases, different grid scales) and show how performance changes. Please also show failure cases.\n\n- The paper does not compare CR-FWI or IG-FWI with data-driven inversion methods (e.g., supervised CNN-based methods) on synthetic datasets such as OpenFWI. These baselines are now standard in the ML-driven FWI literature. The omission limits the ability to judge the practical competitiveness of the proposed framework relative to current deep-learning-based inversion systems.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "bHiUVd7fgB", "reviewer": "Reviewer_4aqd", "summary": "This paper focuses on the problem of full-waveform Inversion (FWI) and investigates the mechanisms that cause the differences between conventional FWI and continuous representation FWI (CR-FWI) in robustness and convergence. Through a series of theoretical analyses from the perspective of the neural tangent kernel (NTK), the authors identified that these differences arise from different eigenvalue decay behaviors. Based on these observations, the authors further proposed LR-FWI, MPE-FWI, and IG-FWI, which aim to achieve a trade-off between robustness and convergence. Experimental results on several datasets and scenarios demonstrate the superior performance of the proposed methods.", "questions": "1. The authors proposed to integrate a tiny INR and MPE in IG-FWI. Can the tiny INR be replaced with other INR-based methods such as LR-FWI?\n2. In IG-FWI, the features from the tiny INR and MPE are concatenated with a weighting factor $\\alpha$. What are the values of $\\alpha$ in different experiments? Is the algorithm sensitive to the value of $\\alpha$?\n3. In Table 1, given smooth initial models, MPE-FWI performs worse than CR-FWI baselines except for Marmousi, but it should have slower eigenvalue decay. Could you please provide an explanation or analysis of these results? \n4. [minor] In Equation 4 and Equation 7, there is $(\\mathbf{x}’, t’)$ before $dt’d\\mathbf{x}’$. What’s the meaning of this notation? From my understanding, they are redundant.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. I think the insights regarding eigenvalue decay behaviors are valuable to the community for future algorithm development.\n2. The authors provided solid, step-by-step derivations of the theories proposed in the paper. \n3. The proposed methods (MPE-FWI and IG-FWI) have clear theories to explain their performance in terms of accuracy, robustness and convergence. \n4. The proposed methods achieved superior performance on the public benchmark datasets, which is convincing.", "weaknesses": "1. Among the proposed methods, LR-FWI lacks an analysis of its eigenvalue decay behavior and is discussed much less than the other two methods (MPE-FWI and IG-FWI). I think it is important as LR-FWI yields promising performance, and it even outperforms IG-FWI in some cases.\n2. Although data-driven and physics-informed FWI methods are covered in the related works section, they are missing in the experiments. The experiments can be more solid by including a few of them as the additional baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ctuOMNtXB4", "reviewer": "Reviewer_GUpL", "summary": "The authors study full waveform inversion (FWI), in particular the case where the material field is parameterized with a neural network (the \"continuous representation FWI\"). Previous work has identified challenges such as slow convergence for high frequencies, but good performance and fast convergence for smooth fields. Two main questions are adressed: how to explain the differences in robustness and convergence between conventional and network-based FWI, and how to find a continuous representation (e.g., neural network) that can achieve a good trade-off between accuracy in the high frequency domain and convergence speed.\nThe authors use the neural tangent kernel perspective to study the frequency domain properties, and introduce a combination of representations tailored to FWI problems, including multigrid approaches, to resolve the frequency issues.\nThe theoretical results are supported by several computational experiments in seismic imaging, where the approach works well.", "questions": "1) Why do the authors refer to the wave equation being nonlinear, and then state it as a linear PDE? There are nonlinear wave equations, but eq. 1 is not.\n2) What is the time and memory complexity of the method introduced in the manuscript? FWI is used beyond seismic imaging (e.g. in nondestructive testing), in 3D domains, where the computational challenge is much larger than in 2d domains.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1) The authors employ NTK theory to study the spectral bias problem when using neural networks in an FWI setting. This is a very well defined problem, and has not been addressed in a lot of previous studies; even though the idea of using neural netowkrs for the material field is used in several works. This makes the study particularly relevant, not just for seismic imaging, but also nondestructive testing.\n2) The authors not only analyze the issues of FWI in this setting, but also propose a new method using multigrid, and demonstrate its performance in 2D examples. This makes this work a significant advancement in the field.\n3) The appendix includes a thorough explanation of the proof, as well as details on the experiments.", "weaknesses": "* The manuscript contains the term \"nonlinear partial differential equation\" multiple times (l011, l046, l104, l107, l232), but equation (1) is a linear PDE (the classical wave equation). The linearity of the wave equation does not matter for the inverse problem being nonlinear, but it is strange that the authors refer to the nonlinearity so often.\n * More literature on FWI with neural networks material fields should be cited.\n\n - Rasht‐Behesht, Majid, Christian Huber, Khemraj Shukla, and George Em Karniadakis. 2022. “Physics‐Informed Neural Networks (PINNs) for Wave Propagation and Full Waveform Inversions.” Journal of Geophysical Research: Solid Earth 127 (5). https://doi.org/10.1029/2021JB023120.\n - Herrmann, Leon, Tim Bürchner, Felix Dietrich, and Stefan Kollmannsberger. 2023. “On the Use of Neural Networks for Full Waveform Inversion.” Computer Methods in Applied Mechanics and Engineering 415 (October): 116278. https://doi.org/10.1016/j.cma.2023.116278.\n\n * Only 2D examples are considered, not 3D. There is a significant computational challenge in FWI going from 2D to 3D, and it is not clear if the proposed method can still be used there.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UrjTJqXbNt", "reviewer": "Reviewer_Lw1m", "summary": "In this paper, the authors propose a unified theoretical analysis framework based on the wave kernel and\nwave-based NTK to investigate the convergence and robustness of both conventional and continuous representation FWI. Their analysis reveals the distinct eigenvalue decay patterns, which explain why traditional FWI has fast convergence but lacks robustness, while INR-based CRFWI has improved robustness but with slower convergence. Motivated by theoretical analysis, they introduce a hybrid\nrepresentation that integrates INR with a multigrid strategy for FWI, aiming to balance robustness\nand convergence. Experiments demonstrate the superior performance of the proposed methods.", "questions": "1. Missing experiments about $\\alpha$ and convergence. Please refer to Weakness for more details.\n\n2. The introduction of LR-FWI seems insufficiently motivated and conceptually detached from the main theoretical analysis. Its role within the paper is unclear. Could you please clarify its motivation and the connection of the LR-FWI with other parts of the paper?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper proposes a unified wave-based Neural Tangent Kernel (NTK) framework that connects conventional FWI and CR-FWI. It provides a solid theoretical explanation for observed differences in convergence and robustness across FWI variants.\n\n2. Across multiple benchmarks, the proposed methods show strong performance and robustness under different conditions (e.g., noisy data, poor initial models, missing low-frequency content, etc).\n\n3. For the most part, the manuscript is generally well written, clearly organized, and easy to follow.", "weaknesses": "1. Missing discussion of weighting factor $\\alpha$. The weighting factor α in IG-FWI is introduced but never discussed or ablated. Since it determines the balance between Robustness and Convergence, its selection strategy and sensitivity analysis should be provided.\n\n2. Lack of convergence evidence. Although the convergence rate is repeatedly discussed in the paper, the paper does not show any related results, e.g., convergence curves, generated velocity visualizations corresponding to different iterations, or the number of iterations required to converge.\n\n3. The claim that “IG-FWI achieves an optimal trade-off between robustness and convergence rate” is overclaimed. The paper provides no formal definition or proof of optimality.\n\n4. The multi-grid parametric encoding module is largely adapted from the existing INR paper. While this integration is effective, it somewhat limits the originality of the methodological contribution.\n\n5. The Figures 10-15 in the Appendix are in very low resolution. Replacing them with high-resolution or vector graphics would improve readability.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)