The Seismic Wavefield Common Task Framework
Authors: Alexey Yermakov
Yue Zhao
Marine Denolle
Yiyu Ni
Philippe Martin Wyder
Judah Goldfeder
Stefano Riva
Jan P. Williams
David Zoro
Amy Sara Rude
Matteo Tomasetto
Joe Germany
Joseph Bakarji
Georg Maierhofer
Miles Cranmer
J. Nathan Kutz
Venue: ICLR 2026 Poster (2026)
Keywords: Seismology
Scientific Machine Learning
Common Task Framework
Seismic Wavefields
Geophysics
Primary area: datasets and benchmarks
TL;DR: A multi-metric Common Task Framework that standardizes characterization and comparison of modeling methods for seismic wavefields.

Abstract: Seismology faces fundamental challenges in state forecasting and reconstruction (e.g., earthquake early warning and ground motion prediction) and managing the parametric variability of source locations, mechanisms, and Earth models (e.g., subsurface structure and topography effects). Addressing these with simulations is hindered by their massive scale, both in synthetic data volumes and numerical complexity, while real-data efforts are constrained by models that inadequately reflect the Earth's complexity and by sparse sensor measurements from the field. Recent machine learning (ML) efforts offer promise, but progress is obscured by a lack of proper characterization, fair reporting, and rigorous comparisons. To address this, we introduce a Common Task Framework (CTF) for ML for seismic wavefields, demonstrated here on three distinct wavefield datasets. Our CTF features a curated set of datasets at various scales (global, crustal, and local) and task-specific metrics spanning forecasting, reconstruction, and generalization under realistic constraints such as noise and limited data. Inspired by CTFs in fields like natural language processing, this framework provides a structured and rigorous foundation for head-to-head algorithm evaluation. We evaluate various methods for reconstructing seismic wavefields from sparse sensor measurements, with results illustrating the CTF's utility in revealing strengths, limitations, and suitability for specific problem classes. Our vision is to replace ad hoc comparisons with standardized evaluations on hidden test sets, raising the bar for rigor and reproducibility in scientific ML.

Published as a conference paper at ICLR 2026
THE SEISMIC WAVEFIELD COMMON TASK FRAMEWORK
Alexey Yermakov!:2
Yue Zhao
Marine Denolle*,
Yiyu Ni²,
Philippe M. Wyder
Judah Goldfeder
Stefano Riva
Jan Williams,,
David Zoro',
Amy Sara Rude.,
Matteo Tomasetto
Joe Germany",
Joseph Bakarji
Georg Maierhofer"
Miles Cranmer
Nathan Kutz +
1 DDepartment of Electrical and Computer Engineering, University of Washington, Seattle, WA 98195
2 "Department of Applied Mathematics, University of Washington, Seattle, WA 98195
3 High Performance Machine Learning team, SURF, Amsterdam, the Netherlands DDepartment of Earth and Space Sciences, University of Washington, Seattle, WA 98195
"Distll AI, New York, NY 10016
Department of Computer Science, Columbia University, New York, NY 10027
7 Department of Energy, Nuclear Engincering Division. Politecnico di Milano. Milan, Italy Department of Mechanical Engineering, University of Washington, Seattle, WA 98195
9 Department of Mechanical Engineering, Politecnico di Milano, Milan, Italy 10 Department Mathematics, American University in Beirut, Beirut, Lebanon Department of Mechanical Engineering, American University in Beirut, Beirut, Lebanon 12 Department Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK
ABSTRACT
Seismology faces fundamental challenges in state forecasting and reconstruction (e.g., earthquake early warning and ground motion prediction) and managing the parametric variability of source locations, mechanisms, and Earth models (e.g.. subsurface structure and topography effects). Addressing these with simulations is hindered by their massive scale, both in synthetic data volumes and numerical complexity, while real-data efforts are constrained by models that inadequately reflect the Earth's complexity and by sparse sensor measurements from the field. Recent machine learning (ML) efforts offer promise, but progress is obscured by lack of proper characterization, fair reporting, and rigorous comparisons. To address this, we introduce a Common Task Framework (CTF) for ML for seismic wavefields, demonstrated here on three distinct wavefield datasets. Our CTF features curated set of datasets at various scales (global, crustal, and local) and task task-specific metries spanning forecasting, reconstruction, and generalization under realistic constraints such noise and limited data. Inspired by CTFs in fields like natural language processing, this framework provides a structured and rigorous foundation for head-to-heal algorithm evaluation. We evaluate various methods for reconstructing seismic wavefields from sparse sensor measurements, with results illustrating the CTF's utility in revealing strengths. limitations, and suitability for specific problem classes. Our vision is to replace ad hoc comparisons with standardized evaluations on hidden test sets, raising the bar for rigor and reproducibility in scientific ML.
"Corresponding author; kut @uw. edu

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Published as a conference paper at ICLR 2026
INTRODUCTION
Earthquake-indueee hazards, tragically illustrated by events like the 1971 M6.7 San Fernando, the 1999 M5.9 Athens (143 fatalities), and the 2011 M5.8 Virginia (up to S300M in damage) earthquakes, are among the most challenging domains for prediction due to the complexity of the dynamics of shaking and ground failures. The physies of seismic wavefields governed by the 3D elastodynamic wave equation, which must be solved in media whose elastic properties vary sharply in space, depth, and sometimes time. Even small-scale heterogeneities distort and scatter craeticce highly nonstationary, multi-frequency, and multi-path signals that are far more complex than those produced by uniform or low-dimensional dynamical systems. These wavefields matter at societal scales, where small variations in near-surface structure strongly affect shaking intensity and earthquake hazard, and at planetary scales, where subtle features of global wave propagation reveal the structure and dynamics of Earth's 's deep interior.
Computational simulations of seismic wavefields must accommodate high-dimensional heterogeneous media, with the numerical complexity increasing with the frequencies that need to be resolved for effective hazard mitigation. At the same time, the rise of distributed acoustic sensing (DAS) and related fiber-optic technologies has made it possible record dense, unaliased seismic wavefields over kilometer-scale arrays, exposing unprecedented detail in wave propagation, scattering, and mode conversions. These observational advances amplify both the opportunity and the challenge: they provide richly sampled data that can constrain models more tightly than ever before, but they also highlight the limitations of current forward simulations and inversion pipelines. Consequently, the seismological community has begun to investigate machine learning (ML) and artificial intelligence (AI) techniques to accelerate the accurate reconstruction of wavefields from simulations and data. Farly results indicate that A Al-accelerated techniques can advance the probabilistic modeling of earthquake ground motions. Recent Al methods for wavefield modeling include Neural Operators
(e.g.. Yang et al. 2023 Zou et al., 2024 Huang & Alkhalifah, [025 Kong et al. [025 Lehmann
cct al. 2024b).
Prysics-Informen Neural Networks (Moseley et al. 2023). combinations of both
(Huang et al. 2025), established general deep learning architectures (Lyu et al.
2025 Nakata et al.
2025). and reduced-order models that leverage many simulations to discover Proper Orthogonal
Decomposition and function bases for low-cost wavefield reconstruction (Rekoske et al. 2023 2025).
The rapid development and adoption of these methods on seismological data and simulations have outpaced efforts to compare them objectively. In the absence of common evaluation standard, new methods are not assessed fairly against existing approaches, resulting in weak baselines, reporting bias, and inconsistent evaluations (MeGreivy & Hakim| 2024 Wyder et al.). Few scientific and engineering domains have mitigated these problems, instead relying on self-reporting by providing both training and testing datasets to the community. While reducing the evaluation burden on the original authors, self-reporting opens the door to problematic practices such as p-hacking and implicit optimization on the test set. Only with truly withheld test set is rigorous and impartial comparison among methods possible.
1.1 RELATED WORKS
Common task frameworks (CTFs) have been pivotal in driving and quantifying progress in ML and AI
(Donhho 2017). Landmark CTFs catalyzed key advances: ImageNet (Deng et al. 2009) provided the
stage that revitalized Convolutional Neural Networks, with (Krizhevsky et al. (012) demonstrating
their superiority over classical methods in large-scale image recognition. Natural -arall-lange language challenges, ranging from code generation (Chen al. (202]) to formal reasoning and mathematies (Hendrycks ct al. 2021: Cobbe et al. (021). have been central to advancing large language models, with recent systems such as DeepSeek- RI-Zero demonstrating competitive performance alongside state-of-the-art models (DeepSeek-Al et al. (025). Competitive games have served as another CTF: matches in Go and shogi provided the testbed that led to AlphaZero (Silver et al. 2018). Video game environments such as the Arcade Learning Environment (Bellemare et al. 2013), based on Atari 2600 games, enabled the breakthrough results of deep Q-networks (Mnih et al. 2015). Finally, platforms providing models with control inputs, such as the OpenAl Gym (Brockman al. 2016) and MuJoCo(Todora methods at
jet al. 2012), have accelerated the development and testing of reinforcement-leammeee scale. Mature CTF platforms are, therefore, critical driving forces of innovation and progress.
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a)
Global Wavefields
b) Dense Observations of Geophysical Waves
2031
(onn)
time
1003
Bime
3000
c) Synthetic 3D Crustal Wavefields
t 2.75
space
Figure 1: The Scismic Wavefield CTF scores the performance of methods on (a) global wavefields from sparse sensor measurements, (b) dense observations of real geophysical wavefields from Distributed Acoustic Sensing, and (c) dense simulations of 3D crustal wavefields. These three datasets represent broad range of challenging datasets encountered frequently by seismologists and present challenges for models in forecasting and state reconstruction.
Despite these successes, many domains beyond the major areas of computer vision, natural language processing, and reinforcement learning still suffer from a lack of a community standard for fairly evaluating methods. Indeed, aside from critical assessment protein structure prediction CASP (predictioncenter.org (Donoho 2017). science and engineering have largely ignored CTFs and have instead relied self-reporting benchmarks. Recent work from Wyder et al.) has begun bridge this gap by providing a CTF for scientific MI. models on canonical nonlinear systems. But substantial work remains: discipline-specife CTF suites, standardized evaluation protocols (including withheld test sets), and community maintant leaderboards are needed to ensure fair comparison, reproducibility, and sustained progress.
1.2 THE SEISMIC WAVEFIELD COMMON TASK FRAMEWORK
We propose CTF for seismology that is, in its initial release, primarily focused evaluating ML and Al algorithms modeling the seismic wavefields shown in Fig. [ Based on the work from Wyder et al. this CTF provides training datasets with explicit tasks related to forecasting and reconstruction under various challenges, such as noisy mcasurements, limited data, and varying system parameters. The datasets are limited in volume by design, in order democratize access the exercise. Participants submit predictions for a withheld test set over specified timesteps. The predictions are evaluated and scored on diverse set of metrics by an independent referee and posted on leaderboard.
Scoring is by nature reductive-rrduccnng reducing method's s performance to single floating point value. We adopt a mellti-metric scoring scheme because single score often insufficient characterize suitability for different scientific uses. As result, we use the carefully designed twelve-score system from Wyder et al.] which maps to critical tasks for seismic wavefield data objectively. summary, or composite score, is also computed that gives the overall score for given method. Task-specific and overall rankings are highlighted this paper and displayed on a leaderboard.
For each submission, calculate the full suite of twelve task scores. These results quickly convey model strengths and weaknesses e.g., robustness to noise, performance in limited-date regimes, or parametric generalization-
choose methods suited to their needs. The composite score the mean of the task scores; using multiple task-specific metrics avoids winner-take-al outcome and promotes methods that are fit-for-purpose across diverse seismological applications.

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(a) Noise-free Data
Xj Xg
X
e~ NO,, o.), B22
B2
Low Noise
X
Kj X2
x ==xBBe
(c)
High Noise
X
8 x
Train
Esy
Prediction
Egy
Error
EsT RMSE (short-time)
Train
Egg
Prediction
E
Efr Spectral Error (long-time)
E,Eg
EgE
Time
(b)
Train
Prediction
Eyr
(d)
Train
Prediction
E,y
Train
Train
Test
Train
Test
Train (Test for noisy data)
Esy
Prediction
L
2,
E
E,
E,E,E,
Eyz
The Seismic Wavefield CTF scores the performance of methods on seismic wavefield datasets. (a) Data is collected and organized into matrices, which then split into testing and training sets. RMSE errors are computed for reconstruction and short-time forecasting, while the spectral error computes the statistics of long-time forecasting (spatial or temporal). (b) Forecasting and reconstruction tasks are evaluated on noise-free, low-noise, and high-noise data. Methods are also evaluated when (c) only limited data is available and (d) for reconstruction of parametrically dependent data.
Figure 2:
Once the Seismic Wavefield CTF is launched on Kaggle we invite the community to evaluate their methods on the CTF by taking the following steps:
1. Sign-up and Sign-in on Kaggle
2. Train your model with our training data and generate predictions for each task
3. Submit prediction files the competition platform
4. See your score on the leaderboard
To interact with the Seismic Wavefield CTF before the competition launches, visit our GitHub repository |
install the Python package, and evaluate your method on our publicly available datasets. Our dataset and Python package don't require perfoorerfmm hardware and can be run on laptop.
DATASETS & EVALUATION METRICS
We extend the CTF with three new challenging seismic wavefield datasets and evaluate two them, the global wavefields dataset and the DAS dataset, several commonly used models in scientific machine learning. A visualization of the spatio-temporal structure of these wavefields is in Fig. All datasets exhibits complex and challenging behavior for the tasks of reconstruction and forecasting under the constraints of noise, limited data, and parametric dependence. While these datasets serve as a starting point, the CTF will evolve to include both more complex data and more challenging tasks. The Seismic Wavefield CTF is sustainable platform that evolves and grows as the community develops more sophisticated methods and algorithms and faces new challenges.
2.1 SEISMIC WAVEFIELDS AS
SYSTEMS
Seismic wavefields are the response to the elastodynamic wave equation in Earth models of elastic properties that vary in space. When the Earth models are uniform, solutions are simple spatiotemporal fields of ballistic waves: P, S, and, when the Earth's surface included as traction-free boundary, surface Rayleigh and Love waves. Spatial and depth variations in Earth properties distort and scatter the wavefields, yielding great complexity in the spatio-temporal pattern of seismic wavefields, with numerical complexity that scales with resolvable seismic frequencies and domain size, the two main bottlenecks in predicting ground motions relevant for seismic hazard analysis. The three data sets provide different in scale and scope: the first two products of numerical simulations
'ee are proposing a launch date of March 1, 2026, on Kaggle Availlble athttpa:// //i thub. coom/CCF-for-SS wtt.orrr-SSeeeeeeefaaaccccccc/e
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the global and crustal scales and target different goals and research audience (an earthquake hazard specialist or a geodynamicist), The third data set is purely observational from a sensor array and will target an audience of oceanographers and marine geophysicists.
The first dataset, for which we present results in the leaderboard, is a dataset of globally propagating seismic waves (Fig. a)). |van Driel et al. (2015) developed an efficient workflow generate and store Green's 's functions that can be reused for arbitrary source locations and receivers on the Earth's 's surface using the AxiSEM numerical solver (Nissen-Meyer et al. 2014). Such a framework has enabled the vast dissemination of complex global wavefields, for Earth and Mars structures, democratizing access of the scientific community to modeled full wavefields (Krischer al. 2017), which would be otherwise computationally intractable for most scientists. We leverage this database of Green's s function to construct a series of earthquake wavefields recorded at 2048 sensors located at the surface of a sphere. The wavefields are computed in the IASP91 Earth model that was designed to match the arrival time of P and S waves (Kennett & Engdahl 199T) and has become a standard of radially symmetric Earth models, where seismic properties (P wavespeed, S wavespeed, and Earth material density) increased as a function of depth. Our dataset is built upon these Green's 's functions, which, when convolved with source mechanism, deliver realistic seismogram. We choose 2048 sensors distributed on a Fibonacci sphere of 6371 km radius, a sampling rate of Hz (resampled from the original simulations that resolved as short as 2-second period waves), and time series up 3600 s. Each dataset file contains NumPy arrays with shape (time_steps, 2048) representing the vertical z-component of velocity seismograms. Given that realistic waveform data span many orders of magnitude, we normalize the dataset to have zero mean and unit variance per earthquake event, ensuring temporal continuity and predictability for each task.
The second dataset uses a novel geophysical sensing technology that leverages optical scattering to transform celecommunication fibers into arrays of virtual sensors. Referred to as Distributed Acoustic Sensing (DAS) (Fig. b)) this technology is revolutionizing the observations of carthquakes (e.g..
Yin et al. ) marine mammals (e.g.. Wilcook al. 2023). ocean dynamics (Lindsey et al. 2019), and structural health of offshore wind turbines. Major challenges for DAS are the massive data volumes SSica et al. 2023 Ni et al. (023) and the complexity of the wavefields (Xu ct al. 2025).
motivating Al-based compression and reconstruction (e.g.. et al. (024). We prepared dataset of ten 1-minute recordings sampled at 5 Hz and low-pass filtered up to 1 Hz. The data present an interesting overlap between earthquake and oceanographic signals multiple spatial and temporal scales. The channel spacing is 9.57 m, and we trimmed the data files to 3000 channels among the 9000 channels available on that particular fiber. Typical waves that dominate the shallow offshore DAS are the surface swell that follows the dispersion relation w² = gk tanh kh, where w is the angular frequency, g is the gravity constant, & the wavenumber, and the water depth. For the test set provided, f ~ 30m, making surface gravity waves particularly dispersive in the data. This dataset is purely observational.
The third dataset (Fig. (c) is an extension of the curated datasets from Lehmann et al. (022a)). It comprises synthetic 3D seismic wavefields in heterogeneous 3D crustal model. Earthquakes were modeled as point sources with a double-couple mechanism represented by six parameters; source location and focal mechanism (strike, dip, and rake angle) were drawn at random within the model volume. The modeled spatial and temporal scales are aligned with those used in recent Al models of wavefields (c.g.. Kong et al., [2255] Rekoske et al./ (0225). relevant for seismic analysis of crustal earthquakes that can pose substantial risk to people and infrastructure when they occur in populated areas. Ten independent simulations were produced. Each simulation yields three-component velocity seismograms 32 x 32 x 32 heterogeneous grid. Virtual sensors form a 94 x 94 grid arranged on the top of the model volume with 100 m spacing. These seismograms are sampled for 6 seconds at a 50 Hz.
2.2 METRICS
2.2.1
FORECASTING (2 SCORES)
The first set of tasks, shown in Fig. 2-b, involves the approximation of the future state of the system. Thus, given a data matrix representing the dynamics over i E [0, 47 (X1 E Rdmxa), a generated forecast is requested from the model being tested for t E [4T, 6T) (X Ipred E Rmmxn) to compare with the ground-truth (X Iest E R2mxn), with being the dimension of the system and being the
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number of forecasted time steps. The forecasting score is composed of two scores evaluating both the short-time forecast EsT, which is computed using Root Mean Square Error (RMSE) between the test set and the model's approximation, and the long-time forecast Eir, which is computed using the spectral error based upon the power spectral density, see Fig. a Short-time forecasting measures trajectory accuracy where deterministic prediction is feasible, while long-time forecasting measures statistical fidelity where only the system's 's broad statistical properties are recoverable.
For the challenge dynamics of interest, the sensitivity of initial conditions is common, making long-range forecasting to match the test set an unreasonable task, given fundamental mathematical limitations with Lyapunov times. Thus, the long-time error is computed by reast-squares fitting of the power spectrum P(X, k, k) = In([FF((X[ to model the data in the wavenumber domain and k n//2 kinox n//2 + (kmaz + 1) with kmaz = 100. This means that we look at the match in the first 100 wavenumbers of the power spectrum over a long-time simulation. Let X be the ground-truth matrix, X be the prediction matrix, and & E ()) an integer specifying how to split the matrices for the short-time time and long-time long-imme&t scores. The following two enror scores are then computed:
-1,k)) where the fftshift has been used
[ kr, :] X :ll
SST X, X)
||X[1 [1 :1]]
|P k, k) - P(X,k,)) KII
SLT X, X)
(2)
|P k, k)ll
is clear that there are many ways to evaluate the long-range forecasting capabilities. We followed in the footsteps of Wyder et al.] and chose a simple and transparent metric, fully understanding that more nuanced scoring could be used. To provide reasonable range, we then compute the two scores
E = 100(1 - SST X lpred; X
Fg = 100(1 - Sur X ppred, X Xitest)
meaning in each case a score of E, = 100 corresponds to perfect match. Note that, as a baseline, a solution guess of zeros X[1 k, :] 0 (corresponding also to P k. k) = 0) gives a score of E2 = O.
E1
Input: Xitrain I 4mxn
Output: Xipred E R 2mmx; Scores: E1, Eg.
2.2.2
NOISY DATA (4 SCORES)
The ability to handle noise is critical in all data-driven applications as sensors and measurement tech- nologies are by default embedded with varying levels of noise. Methods that work with numerically accurate data, for example, data points that are 10 accurate, may be useful for model reduction, but are rarely suitable for discovery and engineering design from real-world data, Both strong and weak noise are considered, as these represent realistic challenges to be addressed in practice.
This task is very similar to the forecasting described above, but now with noise added to the data. Specifically, the model is provided data matrices X2train E R amxn and X Xttroin E Rdmmx xx representing the evolution with low or high noise respectively. The objective is to first produce a reconstruction of the data itself, i.c. denoise the data to produce estimate of the true state of the dynamics, X2pred: Xipred E R4mxn for X2train, Xstrain respectively, and the second objective is to then forecast the future state, matrices Xapred, XSpred E 2m2mxx for X2train, X respectively. For the reconstruction objective, a least-square fit is used between the approximation of the denoised data and the truth, and for the forecasting objective, a long-time evaluation is computed, leading to the following scores:
Eg 100(1 SST 2ppreds X 22rest ));
Ea 100(1 SiT X Spreds X X3test)); Es = 100(1 - SIT (on(Xiipe (Xipred, Xstest)).
Es = 100(1 - SST (X (Xspred, X4test)),
Input:
X Xstrain E 4mmn R2mxn Scores: Fe Fes, Fs, Fs.
Output: X X Xspred E 4mmx Xspred, Xspred
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2.2.3 LIMITED DATA (4 SCORES)
Data limitations are common in real-world physical systems and often affect the success of data- driven methods. Thus, testing for model performance on low data is critically important and provides important insight to potential users.
Figure c demonstrates the nature of the task. In this case, only a limited number of snapshots . on numerically accurate data are given Xtaann RMxx xm From this limited data, a forecast must be made which is evaluated with both error metric equation I and equation
the approximated future X6pred E 2mxx The experiment is repeated with noise on the measurements using the training matrix X5train E RRMxR for which forecasting prediction matrix is produced X7pred R Rmxx The performance is evaluated on the following scores representing short and long-time metrics for both noise-free and noisy data, respectively.
F 100 Ser X 6pred; X Xotest)), Eg = 100(1 (ar((Tpred)) Xttest)), Eio = 100(1 - S (X ?pred: Xttest))
Es = 100(1 StT [ Spred: X X });
Two error scores (analogous to E and F22) are produced for the noise-free and noisy limited data. These scores are Fr (short-time forecast) and Egg (long-time forecast) for the noise free case and Eg (short-time forecast) and Eio (long-time time forecast) for the noisy case.
Input: Xatrain Xstrain MM Output: Xepreds Xipred 2mxn Scores: E7, Eg, Eg, E10-
2.2.4 PARAMETRIC GENERALIZATION (2 SCORES)
Finally, the ability of model to generalize different parameter values is evaluated. For this case, the model's s ability to interpolate and extrapolate to new parameter regimes is considered with noise-free data. The interpolation and extrapolation are each their own score, resulting in two scores that evaluate parametric dependence.
Figure 2ld shows the basic architecture the task. Three training datasets are provided with three different (unknown) parameter values Xetrain; Xitrain: Xstrain Rimxn Construction of the dynamics in parametric regimes that are interpolatory Xspred 2mxm and extrapolatory X R22xn are required. For both of the tasks, burn-in matrix of size M x n (where M m) is given (Xgtrain and X X10train respectively) and the performance is evaluated using the short-time
metric equation [
E1 100(1 - SST X Xxpreds X Steest)) E12 100(1 SST (X XXpredd X X
Input: Xetrain: X X7train: Xstrain E Rmxx Xetrain Xiutrain E RRM Mxn,
Output: Xspred: Xspred E R 2mmxn Scores: E11, E12-
2.2.5 COMPOSITE SCORE
We compute composite score (AugScore) per dataset from metrics ES through F12 by averaging the resulting scores for each method. This score is evaluated per method, not per model. Thus, each method can fit model for each task and produce the best possible score. All scores clipped such that E; E [-100, 100), thus AugScore E [-100, 100). Methods that cannot produce result for given task receive the minimum score 1100
2.3 DATASET DETAILS
In Table 2 we provide a detailed breakdown of the matrices used for training and testing each metric. The specific matrix shapes for the global wavefields dataset, the DAS dataset, and 3D crustal dataset provided in Table 3 Note that
500 and M 500 for the global wavefield and DAS dataset, and m = 250 and M = 200 for the 3D crustal wavefield dataset. the 3D crustal wavefield dataset, the data parameterized by the Earth velocity model and the carthquake source (location and mechanism). Such information is not provided in the parametric generalization tasks (En and E,2), therefore making matrices 66,7,89 8.9, 10tain and X Stess smaller compared to matrices in the other tasks.
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Table 1: Model Scores on the global wavefields and DAS datasets. The evaluated models are:
Chronos (nsari et al. 2024b). DeepONet (Lu et al. ( FNO (Li et al.
2021). Higher Order
DMD LL Clainche & Vegal 2017a), KAN (Liu et al. )025b), LLMeime Gruver al. (2023).
LSTM (Hchreiter&& & Schmidhuber 1997). Moirai (Liu et al. 2024a), NeuralODE (Ruthotto 2024
ODE-LSTM (Coelho et al.,
2024),
Opt DMD (Askham & Kutz (2018a), Panda Lai et al. 2025))
PyKoopman (Brunton et al.
2022
Pan et 2024), Reservoir (Jaeger 2200 Maass & Markram
2004 Pathak et al. 2018),
S Brunton et al. 2016; Fasel et al., 2022). Spacetime (Zhang et al.
2023
Sundial (Liu et al. 2025a).
and TabPFN Hoo al. (2025)
Global Wavefiekds Datasel
Model
Avg Score
E2
E3
E4
E3
E6
E7
ETO
E11
E12
1.STM
13.18
-1.11
13.8
69.7
19.84
48.83
24.32
-40.07
11.34
18.18
30.97
4.09
-5.22
ODE-LSTM
5.71
-0.07
6.67
65.78
-20.36
41.1
24.1
-67.04
13.16
-0.29
5,31
-0.38
-5.23
Baseline Average
0,16
0.0
0.0
0.0
0.02
3.59
0.03
-1.73
0.01
-0.02
0.02
-0.03
-0.02
Baseline Zeros
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
HigherOrder DMD
-0.01
0.0
-0.02
0.0
0.0
0.02
Reservoir
-14.76
-0.96
6.73
75.37
100.0
-100.0
33.63
6.03
1.61
1.06
-100,0
1.15
-1.75
Opt DMD
22.76
22.65
9.12
2.81
91.47
-36.52
110.00
-00.3
-2.221
-0.55
.0.03
SINDY
-26.44
0.0
0.02
-10..0
3.89
100.00
0.13
-19.12
2.69
-1.65
-10..0
-3.12
-0.08
FNO
-30.92
4.91
-100.0
80.82
100.00
75.81
-100.0
-23.72
-9.36
-28.82
-100.0
-35.29
-35.4
PyKoopman
-34.48
-0,16
5.08
4,51
100.0
-0.21
-100.0
2.2
0.21
-9.95
-100,0
-15.44
-100.0
NeuralODE
-40,89
-58.12
17.81
-100.0
-25.08
-100.0
9.49
-100.0
9.95
-63.55
30.73
-57.47
-54.39
Spacetime
-45.19
-11.2
1100.0
-100.0
-48.9
3.26
-19.18
-100.0
-34.2 -26.83
DeepONet
.5011
110.00
110..0
-100.0
0.08
-100.0
100.00
6.95
5.73
0.0
0.0
.0.08
-100.0
KAN
63.55
-0.43
3.51
3.3
100.0
-7.51
100..0
100.00
100.00
.100.0
100.0
-100.0
Moirai
-100.0
100,0
-100.0
-100.0
-100.0
-100.0
-100.0
-100.0
100.00
-100,0
-100.0
1000,0
-100.0
Punda
-1000
100.00
-100.0
-100.0
1000,0
-100.0
-100.0
-100.0
100.00
-100.0
-100.0
1000.0
-100.0
Chrones
-10..0
100.0
100.00
-100.0
100.0
-100.0
100..0
-100.0
100.0
10000
-000.0
100.0
-100.0
TabPEN
-100.0
100.0
100.00
-100.0
110..0
-100.0
-100.0
-100.0
100.00
1000.0
-100.0-
100.00
-100.0
LLMTime
-10.0
-100.0
-100.0
100.0
-1000
100.00
1000
-100.0
-100.0
Sundial
100.0
100.0
100.0
100.0
100.0
100.0
100.0 100.0
100.0
100.0
100.0
100.0
100.00
Distributed Aconstic Sensing Dataset
Model
Avg Score
E1
E2
E3
E4
E5
E6
E7
E8
E9
E10
E11
E12
PyKoopman
12.7
7.11
14.5
33.38
0.42
71.52
3.0
7.35
25.06
2.94
2.55
-15.41
-0.01
ODE-LSTM
11.58
-1.2
0.39
36.65
7.24
82.88
14.39
15.76
-16.0
14.79
-27.93
8.79
Baseline Average
0.21
-0.04
0.01
0,02
0,2
0.0
0.0
2.79
1.67
-0.22
3.5
-2.99
-2.45
Baseline
0.0
0.0
0.0
Sundial
-0.57
49.57
0.82
-63.34
3.77
-49.28
7.09
33.24
23:33
13.83
20.43
-30.52
-15.81
TabPEN
-1.68
-7.09
-3.32
82.28
-4.55
89.83
0.16
-5.94
100.00
-11.66
-2.03
-5588
-1.3
HigherOrder DMD
-3.59
-0.01
0.0
-0.02
0.0
0.0
0.0
0.04
-1.14
-0.19
0.0
.41.64
-0.07
LSTM
-8.0
-21.7
19.39
38.47
-17.39
88.96
15.87
-26.4
-10.61
-36.66
-60.52
-95.21
9.81
SINDY
-18.55
18.84
16.81
-38.93
0.17
FNO
-26.6
-24.15
100.0
77,11
100.0
88.54
40.97
-44.24
110..0
1.36
-70.49
-30.52
-15.81
KAN
-2895
-13.88
4.77
13.17
100.0
65.16
100.0
-100.0
-1.99
100.0
.0.17
NeumlODE
-33.44
-55,37
21,16
-100.0
-41.64
-100.0
10.37
-57,64 57,66
18.58
-100,0
-36.08
24.68
14.63
Opt DMD
-44.08
-00.4
-100.0
81.66
-100.0
91.49
-100.0
-34.38
-49.86
-6.97
-100.0
-100.0
-100.0
DeepONet
-45.11
100.0
100..0
0.05
-42.97
-100.0
-0.05
110..0
0.03
1.61
-100,0
0.0
Spacetime
-56.04
-40,51
-100.0
-36.97
-100.0
60.73
-100.0
-27.47
-100.0
-100.0
-100.0
-4.35
-23.91
Moirai
-61.56
-20.33
100.0
100.0
-5.81
-100.
-30.22
-1581
Chrenes
87.19
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
30.52
-1581
-98.58
100.0
100.0
-10.00
100.0
-1000
100.0
100.0
100.0
100.0
1000.0
-82.98
-100.0
LLMTime
-100.0
100.0
100.00
-100.0
100.0
-100.0
-00.00
-000.0
100.0
100,00
-100.0
-100,0
-100.0
Our GitHub repository provides the global wavefields, DAS, and 3D crustal wavefields datasets in npz files. The repository contains all the necessary code for data loading, validation split creation, hyperparameter tuning, model inference, and final scoring. For hyperparameter tuning metrics E, through F1o, we split the training matrix into two parts: the first 80% is the training data, and the last 20% is the testing data. For hyperparameter tuning metric En we use matrices X6, for the training data and X for the testing data, and the burn-in matrix comes from X as well. For hyperparameter tuning metric E12 we matrices X6,7umii for the training data and X for the testing data, and the burn-in matrix comes from Xstrain as well. More details on hyperparameter tuning can be found in the Appendix subsubsecttin A.2.1
Below, we highlight our results on the global wavefields and DAS datasets while reserving the 3D crustal wavefields dataset for the upcoming Kaggle competition mentioned in the introduction.
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3 METHODS, BASELINES AND RESULTS
We characterize eighteen highly-cited modeling methods the global wavefields and DAS datasets. Table I (shwss all scored methods and their resulting performance scores. The Seismic Wavefield CTF includes two naive baseline methods: predicting zero and predicting the average. In evaluations, we use the zero prediction as the reference baseline for both datasets. Due to the dataset having a zero mean after normalization, the average and zero baseline report similar scores.
While some models score high on specific tasks, no model scores high across all tasks. Overall, the results demonstrate that the dataset and specific tasks are challenging enough to produce distribution of scores that characterizes the methods. complete overview all models' performance metrics on the global wavefields dataset can be found in [Table I
3.1 OBSERVATIONS
The results in Table demonstrate that many complex ML architectures fail to outperform the baseline of predicting zeros the global wavefields dataset. Our multi-score evaluation scheme provides deeper insight into the difficulty of the dataset as well as the strengths and weaknesses of the evaluated models. Notably, many commonly used MI.AI models perform very poorly on the assigned tasks, Although the current CTF provides limited data, the difficulty most methods have in exceeding the zero bascline indicates that considerable research and development are still needed before ML/AI models make a meaningful impact in the field. The overall best models are the RNN- based architectures the LSTM and ODE-LSTM (Coelho et al. [2224] Hochreiter & Schmidhuber 1997) scoring in the top two for both the global wavefields dataset and the DAS dataset. The RNNs perform exceptionally well on tasks Eg and Es, corresponding to the denoising of low- and high-noise data. Their advantage likely stems from relatively modest parameter count combined with relatively strong expressive power, allowing them to act as implicit regularizers when training on the limited data available for rugr--egressvve forecasting, In contrast, statistical approaches such as DMD tend to overfit the noise in these datasets, whereas the RNNs, trained with an MSE loss, are more robust. Furthermore, the Sundial foundation model performed best in short-term forecasting for the DAS dataset on metrics E1, E7, and E9, Due to none of the models scoring above 50 on any of the forecasting metrics, we omit the prediction plots of all models since they show nothing of interest.
This result also demonstrates how the multi-metric scoring in the CTF is better than a single score. While the RNNs achieve the highest average score, they perform exceptionally poorly on tasks Fr and Eg. corresponding to short-term prediction on limited data (noiseless and noisy, respectively). This disparity underscores that no single model dominates across all regimes and that task-specific evaluation is essential. We encourage practitioners to consider what properties they value most in model and to match those with the metrics we provide to identify the model(s) that best serve their purposes. As we've demonstrated, there is no true one-size-fite fits-all model that performs well across the board.
We present these findings to stimulate discussion within the seismology community about the appropriate choice of ML models for different problem settings and to demonstrate the value of comprehensive CTF framework. As Table I shows, every task the global seismic wavefields dataset offers room for improvement that can result in real-world benefit. We hope that, as the CTF grows, more models will emerge that reliably forecast seismic data, perform state reconstruction, and interpolate across diverse parameter regimes.
LIMITATIONS & FUTURE WORK
The globally propagating seismic wavefield datasets used here are generated from an axisymmetric Earth model (van Driel et al.) 2015), which captures the increase of seismic velocities and densities with depth but omits the heterogeneous geological structures that drive real-world geodynamics. Future datasets, such those simulated from the REVEAL project (Thrastarson et al.) 2024), or others generated for other planets (e.g.. Mars Stahler et al. 2021), could expand the parameter space and enable more rigorous assessment of model generalization. Furthermore, distributed acoustic sensing (DAS) recordings with earthquake wavefields as more dominant features would add another layer of complexity. Such data contain higher frequency wavefields that exhibit extreme

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Published as a conference paper at ICLR 2026
scattering due to real, near-surface structure heterogeneity (e.g., features highlighted in Shi et al. 2025), challenging Al models to capture earthquake ground motion with societal relevance for natural hazard mitigation. Additional datasets may also come from laboratory experiments of earthquake behavior, which exhibit a particularly complex spatio-temporal pattern relevant to dynamical system studies (e.g.. |Corbi et al. 2022 2025). Incorporating such experimental data would broaden the scope of our CTF, increasing the relevance of the models for broader tectonic implications and for scientists conducting laboratory experiments.
Additional limitations are inherent to our provided evaluations. While many models were tested in this work, there are many more that should be implemented and scored in the future. We hope that the Kaggle launch will inspire the scientific community to test many more models than what we have here, with the hope that community-drivn vastly outperform our tested models across all tasks. There is also room for improvement in the tasks provided. While we start the CTF with a set of twelve measurements on the global seismic wavefields dataset, more tasks can be provided, yielding deeper understanding of a model's capabilities. Finally, for the Kaggle competition, we will add a larger number of training data sets to potentially unlock the approximation capabilities of the models. With significantly more training data, there is the potential to improve the overall performance of many of the methods and beat the zero baseline. For instance, will provide P 100 simulations with varying initial data and parametrizations with the goal of predicting new initial data (Q= = 10) with different parameter settings.
engagement results model improvements that
Input: X Jtrain E Rlmxx for J 1, .., PP
Output: X.pred E Rmxn for J = 1,2, ... ". ,Q: Scores: E13 Ez2-
5
CONCLUSION
CTFs have been critical to the tremendous advancements the big three ML fields of computer vision, natural language processing, and reinforcement learning. They have been proven to be catalysts for key model advancements by providing a clear objective measurement in a competitive environment. This work marks the beginning of incubating similar environment in seismology by providing a challenging dataset and an objective quantification of model performance on tasks that notoriously challenging. Our aim is and their usefulness in seismology as well as fostering an environment where novel methods are developed that excel for specific problem classes.
our platform to evaluate the current state of methods
In this work, we introduce the Seismic Wavefield CTF, which quantifies the performance of modeling approaches on an amalgam of diverse tasks in seismology. As first step. we provide the challenging global seismic wavefields dataset and evaluate 18 different models on the data to understand the usefulness of current MI. models in forecasting, state reconstruction, and parametric variability. Our work demonstrates that the current state of ML is far behind any meaningful use on challenging seismic datasets. We hope to inspire researchers and engineers to identify and develop models that will advance the current modeling capabilities of seismic wavefields, ultimately leading to more accurate and reliable tools for earthquake science and subsurface characterization. By establishing CTF, we aim to shift the research focus from incremental improvements simplified problems to substantive breakthroughs on complex, realistic challenges. The CTF is designed to be a living framework, with plans expand the datasets and tasks to continuously push the boundaries of what is possible in computational seismology.
REPRODUCIBILITY STATEMENT
In this work, we introduce the Seismic Wavefield CTF. With the goal of evaluating ML and AI algorithms on seismic wavefields, we provide publicly available GitHub repository containing the training datasets, hyperparameter tuning scripts, visualization notebooks, and all relevant documenta- tion needed to reproduce and extend our results.
ETHICS STATEMENT
The datasets in this CTF do not contain private or sensitive information. We release this CTF to encourage rigorous, reproducible research and urge the community to use it responsibly. To our best
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understanding, the datasets and the source code do not provide harm to the scientific and global communities. Throughout this work, we adhered to the ICLR Code of Ethics
ACKNOWLEDGMENTS
The authors acknowledge support from the National Science Foundation AT Institute in Dynamic Systems (grant number 21 12085). This work is also partially supported by the Seismic Computational Platform for Empowering Discovery (SCOPED) project under the National Science Foundation (award numbers OAC-2103701) GM acknowledges support from the EPSRC programme grant in 'The Mathematics of Deep Learning' (project EP/V026259/1), AY is supported by the NSF Graduate Research Fellowship Program under Grant No. DGE-2140004. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsors.
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A APPENDIX
A.1 DATASET FILES AND EVALUATION METRICS
The dataset files and evaluation metrics are provided in Table 2 and Table
Table 2: Files and corresponding evaluation metrics (E₁ E12) for benchmark datasets.
Score
Test
Task
Train / Burn-in File(s) Ground Truth File
E1
Forocasting
Short-time
Iran
E2
Forccasting
Long-time
X Izin
X
Es
Noisy (medium)
Reconstruction (denoising)
X Xrran
X X
E4
Noisy (medium)
Furecast (long time)
2min
X
Fs
Noicy (high)
Reconstraction (denoising)
X Suain
X 4bi
E6
Noisy (high)
Forecast (long-time)
X
Siol
Fr
Limited Data (clean)
Forecnst (short-time) time)
X 4rii
X Gast
Es
Limited Data (clean)
Forecast (long-time)
X Aran
X Kiol
Limited Data (noisy)
Forecast (short time)
Xaren
X Taol
B
Limited Data (noisy)
Forecast (long-time)
Xomin
E
Parametric Generalization
Interpolation forecast
X6.7 B.nit Xanain
Brstt
Parametric Generalization
Extripolation forecast
X Suuis
X 10mis
X Dool
Table 3: Matrix shapes and indices for the global wavefields dataset (left), DAS dataset (center), and 3D crustal wavefield dataset (right). Start and end index refer to relative time-steps in the simulation used generate the dataset matrices. Each successive index represents one Al time-step. Matrix shapes follow the [Timesteps, Dimension] format.
Global Wavelields Dataset
Distribuled Acoustic Sensing Dutaset
3D Crustal Wavefield Dataset
Matrix
Shape Start Index
End Index
Shape Start Index
End Index
Shape Start Index
End Index
2000, 2048
2000
2000,, 3000
0
2000
500, 62451
500
X 2vain
2000., 2048
2000
2000, 3000
0
2000
5000 62451
500
X Suan
(2000, 2045
2000
22000, 3000
0
2000
500, 6245
500
Xtean
[500,20:1)
1500
2000
[500, 3000
1500
2000
2000, 62151
300
500
X
[500, 20-15
1500
2000
[500, 3000
1500
2000
2000, 62451
300
500
Xavain
(2000, 20-18
2000
[2000. 3000
0
2000
[000, 26508
500
X
[2000, 20-18
2000
2000, 3000
0
2000
5500. 26508
500
X Smain
2000,, 2048
2000
22000, 3000
0
2000
5000, 26508
500
X Ouain
2048
1500
2000
[500, 3000
1500
2000
200. 26508
300
500
X
2048 2048
1500
2000
[500, 3000
1500
2000
200, 26508
300
500
Dost
11000, 2048
2000
3000
[1000, 3000
2000
3000
1000, 62451
500
X X20i
[2000,2045
2000
22000, 3000
0
2000
[500. 62451
500
X Shat
1000, 2048
2000
3000
11000, 3000
2000
3000
1000, 62451
500
600
X 4
2048
2000
22000. 3000
0
2000
[500, 6245
500
X Suat
11000, 2018
2000
3000
[1000, 3000
2000
3000
[000, 62151
500
600
Xovai
[1000,200]
3000
1000, 3000
2000
1100, 62451
X
11000, 2048
2000
3000
11000, 3000
2000
3000
1000, 62451
500
600
(1000, 2048
2000
3000
11000, 3000
2000
3000
1000, 26508
500
600
Xoxi
1000, 2048
2000
3000
1000, 3000
2000
3000
100, 26508
500
600
A.2 EVALUATIONS
A.2.1 HYPERPARAMETER OPTIMIZATION
Hyperparameter optimization is performed using out publicly available GitHub repository using the tune tune_module,
www.moouee...mmmm e. py script. We employ Ray Tune Liaw et al. (2018) for systematic hyperparameter optimization across all models. Hyperparameters are defindd in YAML configuration files specifying parameter types, bounds, and sampling distributions. Multiple parameter types are supported, including continuous distributions (uniform, log-uniform), discrete distributions (random integer, log-random integer), and categorical choices.
The optimization follows trial-based approach where each trial randomly samples hyperparameter configuration from the defined search space. Each trial trains the model using a train/ main/valiattion split of the original training dataset. The tune_ tune_modu wwl.modullee py script splits the training data into train and validation sets, using the latter exclusively for evaluation. Thus, the test set remains unseen during hyperparameter tuning.
For hyperparameter tuning metrics E, through Eio we split the training matrix into two parts: the first 80% is the training data and the last 20% is the testing data. For hyperparameter tuning metric
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E11 matrices X6, for the training data and Fiaan for the testing data, and the burn-in matrix comes from X as well. For hyperparameter tuning metric E12 we use matrices 7 for the training data and X for the testing data, and the burn-in matrix comes from Xstrain as well.
Optimization terminates when either predefined number of trials time budget is reached. We employ ASHA (Asynchronous Successive Halving Algorithm) scheduling Li al.) (2020) for early stopping of poorly performing trials. Resource allocation is automatically managed, distributing trials across available computational resources.
For our results, each combination of model, dataset, and pair_ _id is allocated hours of tuning time on dedicated nodes equipped with I NVIDIA A100 GPU with 40 GiB VRAM and 18 CPU cores from an Intel Xeon Platinum 8360Y processor with 120GiB RAM. Some models complete tuning in less than the alotted time.
A.3 MODELS
A.3.1 BASELINES
We implement two baseline models. One of the baselines predicts all zeros. The other baseline predicts the average of the input data per spatial dimension. We not perform hyperparameter optimization for either of these models.
A.3.2 LSTM/ODE-LSTM
LSTM networks are specialized type of recurrent neural network (RNN) designed to address the vanishing gradient problem inherent traditional RNNs Hochreiter & Schmidhuber (197).. They achieve this through unique architecture featuring memory cells and gating mechanisms (input, forget, and output gates), which regulate the flow information over time. These gates enable LSTMs selectively retain discard historical data, making them particularly adept at capturing long-term dependencies in sequential data. time-series forecasting. L.STMs excel at modeling temporal patterns, such trends, seasonality, and irregular fluctuations, by leveraging past observations to predict future values. Their ability to handle complex, non-linear relationships and variable-length input sequences makes them robust choice for tasks like stock prediction, energy load forecasting, or weather modeling, where historical context is critical to accurate predictions.
ODE-LSTMs are flavor LSTMs that try to further tackle the vanishing gradient problem by using an ODE solver to model the hidden state of the LSTM CCelho et al. (2024). They show that traditional L.TTMs can still suffer from a vanishing or exploding gradient and provide theory demonstrating ODE-LSTMs do not suffer from either of these problems.
We evaluate both classical LSTM and the ODE-LSTM by searching over the following hyperpa- rameters: hidden_state_sin state_stttteeeeeee (dimension of the latent space), seq_length (input sequence length), and (learning rate).
hyperparameter hidden_sttan state size seq lengt
type randint randint
min (or options)
max (or none)
8
256
5
512
Ir
log uniform
10-5
10
Table 4: Hyperparameter search space for the ODE-LSTM and LSTM models on metrics E, through Eg for the tested datasets. We train with batch size of 128 for 200 epochs.
A.3.3 SPACETIME
State-Space Models (SSMs) are mathematical frameworks that describe systems using latent (hidden) states evolving over time, observed through measurable outputs. They are widely used in control theory, signal processing, and time-series analysis model dynamic systems. Modern adaptations like S4 (Structured State Space for Sequence Modeling) and SpaceTime Time are deep learning variants of SSMs tailored for sequential data. These models parameterize state transitions with structured
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hyperparameter hidden_ state_ size seq lengh Ir
type
min (or options)
max (or none)
randint
8
256
randint
5
74
log uniform
10
10-2 2
Table 5: Hyperparameter search space for the ODE-LSTM and LSTT models metrics E7 through E1g for the tested datasets. We train with a batch size of 5 for E, through E10 and a batch size of 128 for En1 and E12 for 200 epochs.
matrices efficiently capture long-range dependencies while remaining computationally tractable. Unlike LSTMs, SSMs are particularly effective at time-series forecasting of long-range dependencies with minimal memory overhead.
SpaceTime Zhang et al. (2023) is one such SSM that claims to be a state-of-the-art model on time- series forecasting and classification tasks. The authors claim that their model captures "complex, long-range, and autoregressive" dependencies, can forecast over long horizons, and is efficient during training and inference. They demonstrate improved performance over the popular S4 SSM and NLinear.
Based on the hyperparameter optimization described in the original paper and the hyperparameters which can be adjusted in the publicly available code, we do a hyperparameter search over the following values: lag (input sequence length), horizon (output sequence length), n blockkk (number of SpaceTimme layers in the model encoder), dropout, weight_decay, sernel_dim (dimension of SSM kernel each block), and (learning rate).
hyperparameter
type
min (or options)
max (or none)
lag
randint randint
32
512
horizon n blocks
32
512
choice
(3,4.5,6)
dropout
choice
[0, 0.25)
weight decay
choice
{0, 0.0001]
kernel dim
(32,64.128) choice
log uniform
10-5
10
Table Hyperparameter search space for the Space model on metrics E₁ through Eg for the tested datasets. We train with batch size of 128 for 200 epochs.
hyperparameter
type
min (or options)
max (or none)
lag horizon
randint randint
T0
45
10
45
n blocks
choice
(3,4,5.6)
dropout
choice
[0, 0.25)
weight_ decay
choice
{0, 0.0001)
kernel_dim
choice
(32.64.128)
log uniform
10
10
Table Hyperparameter search space for the Space model on metrics E, through Eno for tested datasets. We train with batch size of 5 for 200 epochs.
A.3.4 DEEP OPERATOR NETWORKS
Deep Operator Networks (DeepONets) Lu al. 202 recently emerged as a powerful tool designed to efficiently model high-dimensional physical systems and complex input-output relationships, as well as to solve challenging problems in scientific machine learning and engineering, such partial differential equations. Specifically, DeepONets decompose an operator G V + li between two cooperating sub-networks, namely branch and trunk net. The trunk encodes the input function
class of neural operators which functional spaces and 2/ into
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hyperparameter
type randint
min (or options)
max (or none)
lag
10
45
horizon
randint
10
blocks
choice
(3,4.5.6)
dropout
choice
[0, 0.25)
weight decay
choice
{0, 0.0001]
kernel dim
choice
(32,64.128) 10-5
log__nnnn _e__unffrm
10
Table Hyperparameter search space for the Space model on metrics E11 through E for the tested datasets. We train with batch size of 128 for 200 epochs.
INPUT FUNCTION AT FIXED SENSORS
0
V
BRANCH
b
G(V) b(v) t(E)
OUTPUT COORDINATES
E
Figure Architecture of the Deep Operator Network. The target field the evaluation point is approximated by the inner product of the outputs of the branch net, which takes as input the measurements of the input function e E V and returns set of coefficients b(v), and the trunk net, which encodes the coordinates & into vector t()).
which is typically sampled at finite set of fixed sensors, resulting in the measurement vector v E R" into p coefficients b (v) E RP, Instead, the branch net provides the evaluation of neural learnable p-dimensional basis (E) E IR at the spatial coordinates & in the domain Ω C IRd, Doing so, the value of the output function u E U > I= at the evaluation point approximated through the basis expansion
E : Ω -d R
u(E) (v) (E) x (v) (E).
See Lu et al. (2021a) Chen & Chen 1995); Lu et al. (2022) a complete presentation of Deep- ONets, including also universal approximation theorems for operators. A graphical summary of the DeepONet architecture is available Figure
DeepONets for dynamical systems DeepONets versatile neural architectures designed to learn mappings between functional spaces. DeepONets are traditionally exploited for inferring the space- time evolution of physical variables. such the solution of partial differential equations, starting from known quantities, such as forcing terms, initial conditions, parameters or control variables Lu al. [022aa] 2022 Wang al. (202T); Kontolati al. (202)). However, it is possible to adapt and employ DeepONets in the proposed CTF in order to model and forecast time-series data and c.g.,Chen
dynamical systems, proposed by, e.g., Chen al. (2024a b): Lin et al. (023): He al. (2024a b);
Michalowska et al. (2024). Specifically, we consider the operator
Ut (E) G (u.-1,..., u.... (E) ~ b (u,-1, u, (()
where U, Ω Th and u, E _ are, respectively, the solution of the dynamical system under investigation at time and the corresponding spatial discretization, E E Ω Rd are the spatial coordinates where to predict the evolution of the dynamics. Along with the evaluation point ç, the trunk imput may be enlarged with the time instance or the time-step At, proposed by Lu et al. (2022) [in et al. (2023).
the lag parameter and
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DeepONets implementation The implementation of DeepONets within the proposed CTF is based on the DeepXDE library [Lu et al. (2021b). In particular, when dealing with forecasting tasks, we predict the state evolution in autoregressive manner, and we enlarge the trunk input with the time-step At, as it results in better performance. As proposed by Lu et al. (2022), we consider a scaler to normalize the data before training. Moreover, we employ branch and trunk networks with the same number of neurons per hidden layer, so as to reduce the number of hyperparameters.
The tested datasets deal with one-dimensional scalar-valued functions, that is d 7u The datasets are discretized and evaluated over uniformly spaced domains. Notice that we take into account the same locations across all the input-output pairs, resulting in a lower computational cost.
Hyperparameters The DeepONet hyperparameters mainly concern the neural network architec- tures and the corresponding training procedure. In addition, the lag parameter determines the length of the past state history fed into the branch input for forecasting. Notice that the lag value cannot be larger than the dimension of burn-in data, and it is set equal to zero when dealing with reconstruction tasks. [Table 9 provides a summary of the hyperparameters in play, along with the corresponding search spaces explored for hyperparameters tuning.
hyperparameter
type
min (or options)
max (or none)
lag branch layers
integer
99
integer
Trunk_layer
integer
5
neurons
integer
512
activation
choice
("tanb", "relu", "elu") ("Glorot normal", "He normal")
initialization
choice
optimizer learning rate
choice
{ "adam", "L-BFGS" 10-5 10000
loguniform integer
10
epochs
10000
Table 9: Hyperparameter search space for DeepONet.
A.3.5 SPARSE IDENTIFICATION OF NONLINEAR DYNAMICS
Sparse Identification of Nonlinear Dynamics (SINDy) Brunton et al. (2016) a powerful algorithm designed to discover interpretable and parsimonious governing equations from time-series data. Given the data matrices
x (t1)
(12)
x1 (tm)
(ti)
i (t2)
(1(lm)
X =
; x =
Inn xn,(12)
nn [mm)
(11()) (x2(
stn (mm)
collecting, respectively, the state vector x(t) = (t), 2n (t)] and the corresponding time deriva- tives x() E (t), .... 2,, (t)] at the time instances .,...., tm. we aim at identifying the (possibly nonlinear) underlying governing equation x(t) = f(x(t)). To this aim, SINDY considers the follow- ing approximation
x = [(XE
where ( is a library of candidate regression terms, such as polynomials or trigonometric functions, while E\t are the corresponding regression coefficients. Sparsity promoting strategies are crucial to identify simple and interpretable dynamical systems, capable avoiding overfitting and accurately extrapolating beyond training data. In particular, the regression coefficients E are determined through sparse regression strategics, such as Least Absolute Shrinkage and Selection Operator (LASSO) or Sequentially Thresholded Least SQuares (STLSQ). See Figure for scheme of the SINDy algorithm on the Lorenz system.
SINDy can easily handle parametric dependencies: indeed, augmenting the state vector with the (possibly time-dependent) parameter values μu and adding u-dependent terms in the library O(X, pt), it is possible to identify parametric sparse dynamical systems.
Identifying sparse dynamical systems from high-dimensional data may be computationally expensive. possible workaround is given by dimensionality reduction techniques, such as Proper Orthogonal
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1. True Lorenz System
8
Si & 61.
01
(V
[-3.9956]
9.9990
127
[-0,
9580
9197]
x(p
2)
y
Data
0]
0]
01
[-2.6655)
zy
Bz.
xy
0]
L.0900
01
01
36820
0]
01
25
Spanse Coefficients of Dynamics
Model
50
X
(
III. Identified System
4x
# O
50
420
25
50
Spanse Regression to Solve for Active Terms in the Dynamics
30
Figure 4: Schematic the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm from Brunton et al. (2016), demonstrated on the Lorenz equations. The temporal evolution of the state variable and its derivative are collected in the data matrices X and X. The dynamical system X x(X) = is then identified through sparsity promoting algorithms.
Decomposition (POD) Brunton et al. (2016) autoencoders Champion al. (201). which project state snapshots onto low-dimensional manifold. SINDy can thus be applied on the low-dimensional latent variables, allowing for efficient and accurate forecasting of the high-dimensional state evolution.
SINDy implementation The implementation of SINDy is based on the PySINDy library de Silva et al. (2020). After collecting the data and approximating the time derivatives through numerical schemes, the SINDy algorithm is applied to identify sparse dynamical system describing the data evolution over time. The integrator solve_ing by scipy Virtanen et al. (2020) is considered to simulate the system and to predict future state values. Notice that, whenever the identified model very complex and the integrator fails, the static dynamical system x is employed.
Proper Orthogonal Decomposition (POD) is thus exploited to compress the temporal data of the seismic wavefields data, and SINDy is applied to identify the dynamics of the most energetic POD coefficients. Therefore, the predictions are retrieved by integrating the SINDy model and projecting the POD coefficients onto the original high-dimensional state space.
Parametric SINDy models are considered when testing the ability of the model to generalize to different parameter values. Since the parameter values employed for data generation not publicly available, we take into account fictitious values mimicking the interpolatory and extrapolatory regimes.
The SINDY algorithm can exploit different differentiation methods to approx- imate time derivatives, different terms in the library ( such as, e.g., polynomials and/or trigonometric functions up to chosen order well different sparse regression techniques. Table 10 provides a summary of the hyperparameters in play, along with the corresponding search spaces explored for hyperparameter tuning.
Hyperparameters
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hyperparameter POD modes infferentiation_melleee method
type integer choice
min (or options)
max (or none)
1 "finiite difference", "spline", "savitzky _oolay" "spectral",
50
"kalman")
differentiation method order @eature_lbraay
integer choice
10
"polynomial", "Fourier", "mixed")
feature_ Ceature_library. optimizer
integer choice
1 ("STLSQ". "SR3",
10
"SSR". "FROLS")
threshold
choice
{ "adam", "L-BFGS" } 10-3
learning_rate alpha
loguniform
10a
loguniform
10-3
10a
Table 10: Hyperparameter search space for SINDy.
A.3.6 DYNAMIC MODE DECOMPOSITION
The Dynamic Mode Decomposition (DMD) is a data-driven method developed by Schmid (2010) in the fluid dynamies community to identify spatio-temporal coherent structures from high-dimensional data. The DMD algorithm is based on the Singular Value Decomposition (SVD) of a data matrix; in particular, DMD is able to provide a modal decomposition where each mode consists of spatially correlated structures that have the same linear behaviour in time. The DMD method is found to have a significant connection with the Koopman operator theory Rowley et al. (2009): in particular, the algorithm able to learn the best-fit linear dynamical system to advance
DMD can be formulated in time (Figure 5)
There many variants of DMD, connected to existing techniques from system identification and modal extraction Brunton & Kutz (022). Here, we will provide a brief overview of the underlying idea of the original DMD algorithm, from which all the other variants can be derived. The first step to collect a set of snapshots of the system at different time steps. The data matrix is then constructed by stacking the snapshots in columns, i.e., X [ X2, ........ XN.] E C.A where X E C.V. is the k-th snapshot at time fg and N₂ is the number of snapshots. The original formulation from Schmid (2010); Rowley et al. (2009) supposed uniform sampling in time, i.e. tx k&tt where At is the time step and [x+11 l At. Overall, the DMD algorithm seeks the leading spectral decomposition of the best-fit linear operator & E C, x.s, that advances the system time, i.e.
Xk+1 AXE
X(2.N1 RJ AX
As we said above, the DMD algorithm is based on the SVD of the data matrix X of rank r. which can be written as X ~ UEV* U C represents the left singular vectors and are also known as modes, describing the dominant spatial structures extracted from the data; the diagonal matrix Σ I contains the singular values, which are related to the energy/information retained by the modes; in the end, V* E CrxN, represents the right singular vectors, which are related to the temporal dynamics of the modes. This compression operation allows to compute the dynamical matrix A in a more efficient way J. Nathan Kutz et al. (2016); Brunton & Kutz (2022), avoiding the direct inversion of the -inenssian snapshot matnx.
Indeed, in the literature different variants of DMD have been proposed: in this context, the High- Order DMD (HODMD) Le Clainche & Vega (017b), which exploits time delay embedding to the optimal Koopman Operator, and the Optimised DMD (OptDMD) & Kutz (2018b); (ashidhar Kutz (022), which is a variant of DMD that can also use the Bagging algorithm to improve the robustness of the DMD algorithm against noise. This latter variant has been shown to be the most robust and stable algorithm for real-world applications Faraji et al. (2024). The implementation of the DMD algorithm is available in the DYDMD package Demo al. (2018); Ichinaga et al. (2024). which is a Python library for DMD and its variants. The library is designed to be easy to use and flexible, allowing users customise the algorithm for their specific needs.
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Experiment
Collect Data
DMD
a) Diagnostics
past future
Xm
Dynamic modes
Time dynamics
X=
X1 X2
Xm
W
A-X
M
Data
Regression
X
X2 X3
Xm
b) Future state prediction
X3
X2
x
X1
Figure 5: Scheme of the Dynamic Mode Decomposition algorithm from J. Nathan Kutz et al. 2016)). The data matrix X is constructed by stacking the snapshots in columns. The SVD of the data matrix computed, and the dynamical matrix fitted to the data. This allows to compute the state the system for future time instances.
Parametric DMD The extension of DMD to parametric systems is a recent development in the field of system identification. Different approaches have been proposed in the literature; in this work. the implementation of Andreuzzi et al. AAdreuuzi et al.] (2023b) within PyDMD is adopted. Up to now, the package does not support the OptDMD algorithm directly, we have implemented wrapper to use the OptDMD algorithm with the parametric DMD following the same approach of the package, based on the interpolation of the forecasted reduced dynamics. We appreciate that further work and rigorous testing of this implementation are planned for future work. Similar SINDy, since the parameter values employed for data generation are not publicly available, fictitious values mimicking the interpolatory and extrapolatory regimes have been used.
Hyperparameter tuning The hyperparameters of the DMD algorithm depend the specific variant adopted. Every DMD algorithm has a set of hyperparameters that can be tuned to improve the performance of the algorithm; however, the rank of the SVD is common all of them and plays crucial role in the reduction process. The HODMD algorithm also includes the delay embedding, defining the size of the lagging window use. The OptDMD algorithm can also put constraints on the DMD eigenvalues to ensure that the dynamics follow a certain behaviour, In the end, the parametric DMD can operate two different modes: partitioned and monolithie. The hyperparameters of both DMD algorithms are listed in Tables 11 and T2
hyperparameter
type
min (or options)
max (or none)
rank
randint
20
200
delay parametric
randint
50
choice
("partitioned", "maeeedd"""""nmii" "monolithic"]
Table 11: Hyperparameter search space for the HODMD algorithm for the tested datasets (the parametric hyperparameter has effect only for metrics En and E12).
A.3.7 KOOPMAN OPERATOR-BASED DYNAMIC SYSTEM PREDICTION
The Koopman operator Koopman operator theory is useful tool that has found increasing attention the data-driven scientific computing community and can essentially be seen as an extension of dynamic mode decomposition viewing the statespace of the dynamic system through the lens nonlinear observables. This of dates back to early work by Koopman (193I); Koopman & von Neumann (932) and modern review can be found in Brunton al. 20022. We
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hyperparameter
type randint randint
min (or options)
max (or none)
rank delay parametric eig_co cos_sstttaansss
50
200
choice
{ "partitioned". "monolithic"]
choice
"nnone", "stable", "conjugate_pairs"
Table 12:
Hyperparameter search space for the OptDMM algorithm for the tested datasets (the parametric hyperparameter has an effect only for metrics E1 and E12).
outline the method briefly before describing the set-up for the chosen implementation and our testing on the CTF Consider dynamical system (either an ODE or cemiidiscretiiiaatonn of PDE) of the form:
dx dt
f(x),
where f: RNN N may be a nonlinear forcing. The central idea in Koopman operator theory is then to learn coordinate transform (i.e. set of nonlinear observables) Φ ENN - IR under which dynamics becomes (approximately) linear, i.e.
dz
Az,
ztt) (x(()))
In this new coordinate system, the exact solution of the linear dynamics is straightforward. The inference of & and A can be formulated as regression problem.
Numerical implementation and parameter choices In the Seismic Wavefield CTF we use the PyKoopman Python library the main reference point for the Koopman method for dynamic system prediction Pan et al. (2024). The Python package serves as a good reference since it is regularly maintained and has framework. As mentioned above there are two central parameters that affect the performance of the Koopman method: the observables and the regression method. Exploiting the existing implementation in PyKoopman we allowed in our CTF testing the variation of the following set of parameters:
tote implementation of several central features of the Koopman operator
Type of observable: Options include the identity, polynomials of variable degree, time delay (of variable depth), radial basis functions (of variable number) and random Fourier features, as well as the concatation of all of the aforementioned observables with the identity;
Type regressor: DMD, EDMD, HAVOK and KDMD;
Regressor rank;
seaatesquares regularisation and rank of the regularisation (this option implemented only in EDMD and KDMD).
Note that in principle a neural network-based DMD is also implemented in the PyKoopan package, but in our fine-tuning found that this lead consistently to worse performance than the above four types of regressors thus we did exclude it from the hyperparameter tuning.
Parametric PyKoopman Out-of-the-box PyKoopman does not have parametric implementation, thus in order to test the Koopman method task (d), loosely follow Andreuzzi al. (2023a); Guo et al. 2025) and implement a custom parametric version of PyRoopman by spline interpolation of the learned Koopman operator and corresponding eigenfunctions. We acknowledge that further work and rigorous testing of various parametric versions of the Koopman method are required to identify the best performing implementation for task (d).
Hyperparameter tuning Based the available choices implemented in the PyKoopman package and the examples described in the documentation Pan et al. (202)). we performed a hyperparameter search over the following parameters: type of observable and potential concatenation with the identity, observables integer parameter (representing the polynomial degree case of polynomial observables, the number of time delay steps in the case of time delay observables and the parameter D in the random
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Fourier feature case), the number of centers for the radial basis function observables, observables float parameter (representing the radial basis function kernel width and the parameter "y in the radial basis function case respectively). regressor type, regressor rank, TLSQ rank (the regularisation rank called only when the regressor is EDMD and KDMD). The details the parameter space explored
shown in Table 13
hyperparameter observables
min (or options)
max (or none)
(Identity, Polynomial, TimelDelay,
KadialBasisFuncee Random Feaurr/)a (true, false)
Identity concatenation
choice
Integer parameter
randint
10
# RBF centers
randint
10
1000
Float parameter
uniform
0.5
2.0
regressor type
choice
(DMD,EDMD, HAVOK, KDMD)
regressor rank
randint
200
TLSQ rank
randint
200
Table 13: Hyperparameter search space for the PyKoopman model.
A.3.8 RESERVOIR COMPUTING
In its broadest sense, reservoir computing (RC) is general machine learning framework for pro- cessing sequential data. RC functions by projecting data into a high-dimensional dynamical system and training simple readout from these dynamics back to quantity or signal of interest. Although there exists a large and ever-growing body of literature on leveraging physical systems to act as high-dimensional "reservoirs" Tanaka al. (2019). the most common form of RC remains an echo state network (ESN) Jaeger (2001a); Maass et al. (2002). ESNs are a form of recurrent neural network (RNN) that have been demonstrated achieve state-of-ther
performance in the forecasting of chaotic dynamical systems Platt et al. (2022): Vlachas et al. (2020). We now introduce the spe- cific form of ESN we use evaluating performance on the CTF datasets, following many of the conventions presented in Platt et al. (2022).
ESNs for Now-Dimensional Systems. Given time series uo: a randomly instantiated, high-dimensional dynamical system is evolved according to
h (1 at) + a tanh Whnhh + Wha + +,1))
where a is the so-called leak rate hyperparameter, WAA and Wis are fixed, random matrices, σ is a bias hyperparameter and denotes a vector of ones. WyA N.NNN (N₂, denotes the number of entries in h) is taken to be random, sparse matrix with density z 2% and non-zero entries sampled from (d(11,) 1) and then scaled such that the spectral radius of Wm p. Who E M, X (N₂ denotes the number of entries in u) is random matrix with each entry drawn independently from u(-o,, .). Initializing ho as ho 0, we generate a sequence training reservoir states ho.....,rr We discard the initial Napin training states an initial transient and then perform Ridge regression (with Tikhonov regularization 3) to learn a linear map Wuh such that
Wang(h:) R
g: Nv. N₁, often taken be the identity map or simply squaring every odd indexed entry of h4 We assume the latter convention, following the work of Pathak et all Pathak et al. (2018). Once
trained the reservoir dynamics can be run autonomously as
h = (1 1 ar) + a tanh (Wathy + Wha Whe. (heg(he) + a,11)
(6)
to obtain forecast of arbitrary length.
ESNs for High-Dimensione Systems. RC approaches typically rely on the latent dimension Nr. >> N.. However, the computational cost of the previous algorithm scales roughly quadratically with N₁. Thus, while the above approach works well for relatively small systems, without modifica-
tion it does not scale well to large states such those encountered in PDE simulations. Pathak et al.
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introduced a parallel reservoir approach address this issue by dividing high- digh-dimensional input into g lower dimensional "chunks" Pathak et al. (2018). A single reservoir then accepts as input only Nu/9 + 2L values, where L is locality parameter that dictates the overlap of imput for two adjacent reservoirs. The output of the single reservoir is only g entries the state. Since computational cost grows linearly in the number of reservoirs, this parallel approach allows for the application of RC to higher dimensional systems. Each individual reservoir is trained exactly as for Low-Dimensional systems; there now just g reservoirs representing different regions of the domain.
Since we introduce two new hyperparameters in the parallel setup (L and g), when we perform our hyperparameter tuning for the the tested seismic wavefield systems we a and o, following the work of Pathak et al. The complete hyperparameter search space for the tested datasets are given in Table 14
hyperparameter
type choice loguniform
min (or options)
max (or none)
(16, 32, 64, 128)
0.0001
1.0
randint
10
p
uniform
0.02
1
3
loguniform 10-10
10-1
N₂
randint
500
3000
Table 14: Hyperparameter search space for the reservoir model on the tested datasets.
A.3.9 FOURIER NEURAL OPERATOR
Neural operators are class of machine learning models designed to learn mappings between function spaces, in contrast to the finte-admensionnl Euclidean spaces typically used in conventional neural networks. Although the inputs and outputs are discretized in practice, neural operators aim to generalize across discretizations and treat functions as the primary objects of learning.
The Fourier Neural Operator (FNO), particular, a neural operator architecture that replaces the kernel integral operator with a convolution operator defined in Fourier space, which allows for learning of operators the frequency domain. It maps the input to the frequency domain using the Fourier transform, applies spectral convolution by multiplying learnable weights with the lower Fourier modes, and maps the result back to the physical domain via the inverse Fourier transform. This allows the model to learn families of PDEs, rather than solving individual instances. Without the high cost of evaluating integral operators, it maintains competitive computational efficiency.
Let D I be bounded domain. We consider learning an operator G that maps between function spaces:
G: A-
where A L3(D; Rd)) is the input function space and U L2(D, Rd)) is the output function space.
Given an input function a E A. the FNO approximates the operator through kernel integral operator:
G(a)(z) = Wa(z)++ b+ <(2, y)a(y) dy
(8)
where W Rd. is linear transformation, b E dRd is bias term, D D Rd x is learnable kernel function, and o d. + dd is pointwise non-linear activation function.
The kernel is parameterized in Fourier space
<(x, v)
Σ 2 kEzd
where R(k) are the Fourier coefficients of the kernel. The translation invariian kernel <(2, y) <(x y)) enables efficient convolution. This leads to the implementation:
G(a)(x) = [a(x)++b) +b+ Σ
R(k)@(k)e 2rikkn 2mmkk
(10)
kEZ*
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a(x))
Fourier layer
Fourier layer
Fourier layer"
u(x)
Fourier layer
v(x)
W
Figure 6: Architecture of the Fourier Neural Operator from Li et al. 2021
where a(k) represent the Fourier coefficients of the input function a. In practice, the sum over & E Zd truncated finite number of (ow-frequency modes.
Model Architecture The architecture (Figure 66 begins with an initial fully connected multilayer perceptron (MLP) that projects the input to a sigher-dimenionnal space, followed by four Fourier layers, and concludes with two fully connected MLPs that project the output to the desired dimensions.
Each Fourier layer performs spectral convolution by first transforming the data into the frequency domain using Fast Fourier Transform (FFT), then multiplying the Fourier coefficients with learable weights the frequency space, and finally transforming back to physical space using inverse FFT. The Fourier layer only keeps limited number of the lower Fourier modes, with high modes being filtered out. Additionally, each layer adds a linearly transformed version of input to the output of the spectral convolution, which helps preserve local features and adds flexibility to the layer's s expressiveness. Every Fourier layer is followed by a GELU activation function to introduce non- linearity.
Hy pyprpaarameters Based on our implementation of the FNO model, which closely follows that of the original paper, test the hyperparameters shown in Table 15 tuned separately for each mode.
The number of Fourier modes
hyperparameter Fourier modes
type
range or options
integer
[8,32]
Network width
integer
[32, 128]
Batch size
choice
16, 32, 64, 128 [0.0001, 0.01]
Learning rate (Ir)
loguniform
Table 15: Hyperparameter search space for the FNO model.
A.3.10 KOLMOGOROV-ARNOLD NETWORKS
Kolmogorov-Arnold Networks (KANs) are recently proposed alternative to traditional Multi-Layer Perceptrons (MLPs) Liu et al. (002c)). With learnable activation functions placed on edges that replace linear weights, KANs have been shown to provide improved accuracy and greater inter- pretability compared to traditional methods.
KANs were inspired by the Kolmogrov-Armold representation theorem which posits that any mul- tivariate continuous function ff on a bounded domain can be expressed as finite composition and addition of univariate continuous functions Kolmogorov (1957). In other words, for smooth function
f: [0, 1)m RR
2n+1 Σ &
f(x) = f(x1, ..., xn,) =
Σ p=1
@ap(Tpp))
(11)
9=1
where 0 [0, 1] + and 4 IR R.
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Model Architecture While the Kolmogrov-Amoll representation theorem is restricted to small number terms and only two hidden layers, this theorem can be generalized to increase the width and depth of the network. A single KAN layer is defined as matrix of ID functions thus the inner and outer functions Equation 11 Ωg. and 4 each represent a single KAN layer. deeper network can be constructed by adding more KAN layers. A general KAN network with L. layers can be represented as a composition of L functions:
no Σ
f(x) Σ 11-1=1
PL-1in
Σ
Σ
Φ 2g
Σ
0o,i so (Rig)
where n4 is the number of nodes in the 1th layer and 01.j. k is the activation function that connects the k neuron in the 1t layer to the jth neuron in the + 1 layer. The network architecture is better illustrated Figure which was adapted from Figure 2.2 in Liu et al. (2024c).
2,,
(x) = (x) twc
P1.11
L12
1
X1,1
1,2
1
0.1..
Φ
9o,3,1
Po.12
Noaz
0,,322
X
X
Figure 7: Sample architecture of a Kolmogorov- Amolld Network with three layers of size [2, 1]. Activation functions á are placed the edges and are parametrized spline. Each output of node a sum of its inputs.
Each activation function comprised of a basis function b(x) and a spline function: where
(xx) = wyb(x:) + s,spinc(())
b(x) silu(z) 1 spline(x) = CB)(((
x
Initially, w. is set to 1 and spline(x) 2 0. The weights of the basis function is initialized according Xavier initializations.
KAN Implementation Although KANs have primarily been applied science-related tasks such as function approximation and PDE solving, in this work, we adapted the model to address the reconstruction and forecasting tasks posed in the Common Task Framework. The data was processed and loaded similarly to example 14 in the pykan repository.
For forecasting tasks, the onput-output pairs were constructed in an autoregressive manner, where each input consisted of lagged observations used to predict future values. The input and output dimensions depend on both the number of spatial dimensions in the dataset and the chosen lag (I). By implementing the model in this way, the KAN network approximates the optimal 1-step numerical scheme to solve the given dynamical system.
For both the global wavefields and the DAS dataset, if the dataset dimension is d and with lag of 1, the input dimension was set to dm = d and the output dimension to d = d.
For reconstruction tasks, the model was trained in an autoencoding fashion, where each input was mapped directly to itself as the target output: the input and output dimensions were both set to din duu d.
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Hyperparameters Based the hyperparameter settings provided in the pykan package and the results reported in the original paper Liu et al. (2024c), the hyperparameters outlined Tables [6 were selected and tuned for this model. Broadly, the hyperparameters fall into two categories: (I) model architecture and (2) training.
Architecture- relatte hyperparameters include the number of layers, dimensions of hidden layers, grid resolution, the polynomial degree of the spline basis (k), and the lag. Training-related hyperparameters include the number of training steps (epochs), learning rate, overall regularization strength (A), and the regularization coefficient for the spline parameters (Ncoef).
hyperparameter
type
min (or options)
max (or none)
steps lag' Ir
randint randint
400
10a
loguniform
10-5
10-1
num sane_ayers___
randint
1
(one-five) loletiieediiii dim
randint
I
grid
randint
1
100
k
randint
À
loguniform
10-7
10
Acce
loguniform 10-7
10
Table 16: Hyperparameter search space for the KAN model on the tested datasets. NOTE: The lag
parameter is set to zero for reconstruction tasks (pair id or 4)". The dimension of each layer defined separately. For example the number of nodes in layer two would be defined as two two_dim"*
A.3.11 NEURAL-ODE
Nerual-ODEs are a type of neural network that uses an ODE solver to model the hidden state of neural network Chen al. 2018). This is very similar ODE-LSTMs, another model evaluated in this work, except it makes use of a vanilla MLP instead of LSTM.
We search over the following hyperparameters: hidden state size (dimension of the latent space),
seq lengh (input sequence length), batch size, and Ir (learning rate).
hyperparameter
type randint randint
min (or options)
max (or none)
hidden state size
8
1024
sen_length
5
74
batch_size
randint
5
120
Ir
log__niigl uniform
10-5
10-2
Table 17: Hyperparameter search space for Neural-ODE models. We train for 100 epochs.
A.3.12 LLMTIME
LLMTime Gruver al. (2023) is time-series foundation model that pre-trained LLMs perform zero-shot forecasting of time-series data. Their approach is to modify the tokenization of each model so that time-series forecasting is casted next-token prediction text problem. For our evaluation, we used the 1 1lama-ali 7b as LLMTime's 's base LLM and used the default temperature of 1.0, an alpha of 0.99, and beta 0.3. We also used L.LMTime's 's default Llama tokenizer. L.MMTime is only able to forecast univariate time-series, so we auto-regressively forecast each dimension with a context of 200 tokens and a prediction length of 100 tokens at time. Once each dimension has been forecasted, they are concatenated and evaluated on the test set. For reconstruction tasks, take the first 10 time-steps of the training data and forecast each dimension until we have a vector containing the same number of timesteps as in the testing dataset and then concatenate and caleulate our metrics as before. On all of our datasets LLMTime was unable to generate a complete prediction due to timing the 8 hour time limit, thus -100 score assigned across all tasks.
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A.3.13 CHRONOS
Chronos Ansari et al. (024a) is pre-trained probabilistic time-series foundation model from Amazon. The model is informed by the success of transformers and LLMs, and as such tokenizes time series values using scaling and quantization and trains using the cross-entropy loss function. The model is only capable of doing univariate time-series forecasting. For our evaluation, we use the pre-trained chronos -tt---t-baseeee model and do one-shot forecast of each dimension of each dataset independently and concatenate them when calculating our metrics. For reconstruction tasks, we take the first 10 time-steps of the training data and forecast each dimension until we have a vector containing the same number of timesteps as in the testing dataset and then concatenate and calculate our metrics as before. Chronos has much smaller context length than LLMTime due to requiring more VRAM for inference.
A.3.14 MOIRAI
Moirai_ MMBB Liu et al. (2024b) is time-series forecasting foundation model from Salesforce AI Research. The model uses sparse mixture-of-experts transformer architecture and is able to do one-shot multivariate time-series forecasting arbitrary time-series datasets. For evaluation, used the pre-trained base model and predicted 10 time-steps at a time with a context length of 20. For reconstruction tasks, we take the first 10 time-steps of the training data and forecast until we have a matrix containing the same number of timesteps as the testing dataset. Moirai Moirai_MoE. MoE has much smaller context length than LLMTime due to requiring more VRAM for inference.
A.3.15 SUNDIAL
Sundial Liu et al. (2025a) is family of native, flexible and scalable time-series foundation models from Tsinghua University, tailored specifically for time series analysis. It is pre-trained on TimeBench (about one trillion time points), adopting flow-mal matching approach rather than fixed parametric densities. Sundial directly models the distribution of next-patch values in continuous time-series without discrete tokenisation; it is built decoder-only Transformer architecture. For evaluation, we used the pre-trained sundial al base- 12 8m model; the model can handle multivariate time- series forecasting directly. For datasets tested, due RAM limitations, we have split the "spatial" dimension into batches, forecasting each batch independently and concatenating the results. For reconstruction tasks, we take some of the first time-steps of the training data (around 10%) and forecast until we have matrix containing the same number of timesteps as the testing dataset.
A.3.16 PANDA
Panda Lai et al. (0255) is foundation model for nonlinear dynamical systems based on Patched Attention for Nonlinear DynAmics. Panda is motivated by dynamical systems theory and adopts an encoder-only architecture with a fixed prediction horizon. It is pre-trained purely on a synthetic dataset of 104 chaotic dynamical systems, discovered using a structured algorithm for dynamic systems discovery introduced in the same work. For our evaluation, we used the pretrained model weights provided on the official code repository associated with Lai et al. (2025). The main free parameter in the forecasts with Panda is the context length. Due to RAM limitations for the datasets tested, we have to limit the context to 512 observations.
A.3.17 TABPFN-TS
TabPFN for Time Series (TabPFN-TS) Hoo al. (2025) is based on the tabular foundation model TabPFNv2/Hollmann et al, (202)), adapted to the task of time series forecasting. We use the pretrained model weights, leaving the only remaining parameter as the amount of data for each specific system that the model is exposed to before performing zero-short forecasting. For the global wavefields and DAS datasets, we restrict to at most 500 time steps to be used for context. This restriction was introduced result of limited available memory, and is similar to the restriction placed Panda.
34


Reviews:
[{"review_id": "Qatrc76wfa", "reviewer": "Reviewer_BaxP", "summary": "This work presents a new common task framework for seismic wavefield estimation tasks. The framework seems to be hosted on github, and is proposed to be hosted on Kaggle platform in future. The framework currently seems to incorporate three global scale datasets from seismology domains. The tasks are defined with multiple perspectives with 12 subtasks and scoring metrics. Surprisingly, modern models tend to perform worse than zero-knowledge baseline model of predicting only zeros of all tasks.", "questions": "Are the twelve tasks equally important for seismic wavefield research? The current proposal seems to suggest a simple averaging of all subscores --- but I suspect that depending on the research direction some tasks will have greater relevance than others. Is this something left for seismologists? As it is presented in the current form, it seems like a call for ML practitioners to play with this new set of tasks.\n\nIs there a publicly available repository to share the baseline codes and/or solicit collaboration, as alluded by the future directions given in the text? Would the proposed CTF extend well to incorporate wide range of distinct data-driven tasks in seismology (not only global but also local and/or heterogeneous)? This question may be the most important one if the authors plan to claim the audacious name of 'ctf4seismology'.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "This manuscript is presenting a new aggregated datasets and unifying evaluation system for tasks related to seismic wavefields. Seismology contains diverse tasks and domains, yet the proposed task has very large scale (i.e. planetwise), and immediate extensibility from Earth to Mars. The three datasets proposed seem to be well accepted and analyzed by domain experts. As an ML expert with seismological application research experience, I see the value underlying this work as a domain-expert-curated dataset-evaluation combination which takes care of two key issues: 1) is the data trustworthy, and 2) is the metric meaningful. These two questions had been and will be of greatest concern for a serious AI/ML driven interdisciplinary research.", "weaknesses": "Seismology contains lots of distinctive tasks from various aspects of solid earth (be it from the Earth, Moon, or Mars). Planet-wide seismic wavefield tasks, which are the primary focus of this manuscript, represent a portion of the wide field. The claimed name 'ctf4seismology' may be what the authors want to reach, but as of current form, the name is too assuming and overstating what the current dataset-evaluation-task proposal contains. I think one more spoonful of factual representation would be a good addition.\n\nAlso, perhaps due to the page limit, the materials presented in the 9 page limit did not provide sufficient information for me to gauge what would be a reasonable method and how to use the framework if I have a suitable method. This is the perspective of ML practitioner, who'd like to get an idea of 'how I can use it'. If there is a public repository, it would be much better to have a link to it explicitly shared in the manuscript.", "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": "UZeOcTACQ5", "reviewer": "Reviewer_6Sog", "summary": "This paper proposed a Common Task Framework (CTF) to evaluate machine learning algorithms for seismic wavefields. Three distinct datasets across various scales are included. The authors also established a series of tasks and proposed their corresponding evaluation metrics. They further provided the benchmarking results of 16 methods on the global wavefield dataset.", "questions": "1. For the DAS dataset, could the authors provide a description of the models used for the generation of wavefields?\n2. In the experiment section, could the authors provide some visualizations of the predictions of various methods to give readers more intuition about tasks and model performance?  \n3. [Minor] Figure 1 is organzied in a different order from the introduction of three datasets in Section 2.1.", "limitations": "", "rating": 2, "confidence": 3, "soundness": 1, "presentation": 2, "contribution": 1, "strengths": "The authors provided helpful descriptions of the models and the seismic wavefields in the datasets.", "weaknesses": "While the idea of establishing a CTF is interesting, I feel that the current work has limited novelty and the experimental evaluation appears incomplete. My details comments are as follows.\n1. Although three datasets are included in the proposed CTF, the authors did not clearly explain how these datasets relate to one another or why they should be integrated into a single framework.\n2. The authors didn’t highlight the challenges involved in generating these datasets. For instance, the global wavefield dataset is generated using a public Earth model, and the crustal dataset is an extension of the existing work. Besides, the volume of each dataset is also relatively small.\n3. In the experiment section, the authors presented benchmarking results only for the global wavefield dataset, and the results on the other two datasets are missing.\n4. The related works section is missing.", "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": "FR4ApjyLXr", "reviewer": "Reviewer_97i9", "summary": "This paper introduces a Common Task Framework (CTF) for machine learning in seismology, specifically targeting the modeling and reconstruction of seismic wavefields. The authors present three curated datasets (global, crustal, and local scales) and 12 task-specific metrics, aiming to standardize evaluation and comparison of ML methods in this domain. The framework is inspired by successful CTFs in other fields (e.g., ImageNet) and is demonstrated through benchmarking a variety of ML models on the global wavefields dataset.", "questions": "N/A", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Meaningful Contribution:\nSeismology is a domain where ML adoption is accelerating, but progress is hampered by inconsistent evaluation, weak baselines, and reporting bias. \n2. Well-Designed Evaluation Protocol:\nThe authors propose a multi-metric scoring system that captures different aspects of model performance (forecasting, reconstruction, noise robustness, limited data, parametric generalization). \n3. Benchmarking and Transparency:\nThe paper benchmarks a diverse set of ML models (RNNs, neural operators, DMD, etc.) and provides detailed results, including cases where complex models fail to outperform simple baselines. This transparency is valuable for the community and highlights the challenges ahead.\n4. Open Platform and Reproducibility:\nThe datasets, code, and evaluation scripts are made publicly available, with plans for a Kaggle competition. This openness will foster community engagement and accelerate progress.", "weaknesses": "Performance and Impact:\nThe results show that most ML models struggle to outperform trivial baselines on the provided tasks. While this highlights the difficulty and the need for better methods, it also suggests that the immediate impact of the platform may be limited until more sophisticated models or richer datasets are available. It will be helpful to classify the dataset based on the difficulty level, so that the ML area can design solutions from easy to difficult task gradually.", "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": "QGuZ01CfzY", "reviewer": "Reviewer_heXL", "summary": "This paper introduces a Common Task Framework (CTF) for benchmarking machine learning methods on seismic wavefield modeling tasks. The framework aims to provide standardized evaluation and fair comparison in seismology with forecasting, reconstruction, and generalization problems. It includes three datasets (global wavefields, crustal simulations, and fiber-based DAS recordings), defines twelve evaluation metrics, and compares 14 different highly-cited ML models.", "questions": "Please refer to the Weakness", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 3, "strengths": "1. The paper introduces a benchmark suite for seismology, trying to address reproducibility and comparability challenges in scientific problems.\n\n2. It offers a diverse range of datasets, covering different scales and difficulties.\n\n3. It provides a comprehensive metric design with 12 evaluation metrics, which reflect multiple facets of the modeling: forecasting accuracy, robustness to noise, data limitation, and parametric generalization.\n\n4. The paper provides a benchmark including 14 different methods.", "weaknesses": "1. The paper uses inconsistent names: “ctf4seismology,” “Seismic Wavefield CTF,” and “seismo dataset.” This inconsistency is confusing. \n\n2. The details of the datasets are unclear. A summary table should formally describe each dataset's size, format, train/validation/test splits, etc.\n\n3. Important details are missing for each task, such as noise levels, limited data size (M), and number of forecasted time steps (m).\n\n4. The paper would benefit from data visualizations and formal descriptions to help readers better understand the data and tasks. In particular, a more detailed explanation of the underlying physics is needed to highlight what makes seismic wavefield data different from general time-series data. Section 2.1 (called “Seismic Wavefields as Spatio-Temporal Systems”) does not clearly show what the system is and how spatial and temporal structures look like. To encourage broader community participation in the proposed CTF, the description should be more helpful to those non-seismologists in understanding the physics and challenges of the data.\n\n5. No training hyperparameters and model settings are provided for the evaluated methods.\n\n6. Table 1, Figures 3 and Figure 4 show almost the same information in different formats, consuming space without adding insight.\n\nOverall, I appreciate the work to propose a CTF for a scientific problem. However, the paper currently reads more like an introduction or announcement of the upcoming Kaggle competition rather than a formal scientific paper. Many details are missing, making it hard to fully assess the paper.", "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)