Models ML

Models — Analytical / ML Approaches

A rigorous, skeptical survey of the analytical and machine-learning model family relevant to a conditional probabilistic seismic forecasting product: temporal point processes (the unifying framework), neural temporal point processes (RMTPP, Neural Hawkes, Transformer/Self-Attentive Hawkes), neural-TPP models built specifically for earthquakes (RECAST, FERN), CNN spatial models and the canonical cautionary tale (DeVries 2018 vs. Mignan & Broccardo 2019), GNN/RNN-LSTM approaches, and the hard line between detection and forecasting.

The verdict, stated up front (honesty over hype). As of 2026, no machine-learning model has been shown to reliably beat a well-fit ETAS for short-term earthquake forecasting under fair, prospective, CSEP-style testing. This is not a vibe — it is the explicit conclusion of the most rigorous benchmark to date (EarthquakeNPP, §4). ML does add genuine value, but in specific, honest places: better catalogs upstream (detection), multivariate covariate ingestion, learned spatial anisotropy, and inference speed. The detail below sets out exactly where the line falls and why this product ships an ETAS-class core (see Models — Classical and Models — Employed) with any neural model gated as a challenger.

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Table of contents

1. The forecasting target: point processes and conditional intensity
2. The baseline to beat: ETAS as a point process
3. Neural Temporal Point Processes — the relevant model family
4. Neural TPPs applied to earthquakes — the real evidence
5. The canonical cautionary tale: DeVries 2018 vs. Mignan & Broccardo 2019
6. Other neural architectures (and why they underwhelm)
7. Where ML fails, where it helps — the honest synthesis
8. Forecasting is not detection
9. References

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1. The forecasting target: point processes and conditional intensity

A seismicity catalog is a realization of a marked spatio-temporal point process. The object that every method — statistical or neural — is ultimately trying to estimate is the conditional intensity function

$$\lambda^*(t, x, y, m \mid \mathcal{H}_t),$$

the instantaneous expected rate of events at time $t$, location $(x,y)$, magnitude $m$, given the history $\mathcal{H}_t = {(t_i, x_i, y_i, m_i) : t_i < t}$. The $*$ superscript denotes conditioning on history. For a purely temporal process, the conditional intensity and the log-likelihood over $[0, T]$ are

$$\lambda^*(t) = \lim_{\Delta t \to 0} \frac{\mathbb{E}!\left[N(t + \Delta t) - N(t) \mid \mathcal{H}_t\right]}{\Delta t},$$

$$\log \mathcal{L} = \sum_{i=1}^{n} \log \lambda^_(t_i) ;-; \int_0^T \lambda^_(\tau), d\tau.$$

The second term — the integral / compensator, or survival penalty — is what makes a model probabilistic and calibratable rather than a regressor. A forecasting system that does not evaluate this term is not doing point-process forecasting.

Why this matters. The entire reason to use a point-process formulation (statistical or neural) is that it yields a proper probability over the next-window event count and locations — which is exactly what CSEP consistency tests score. Classification/regression framings (e.g. "will there be an $M \ge 5$ tomorrow? yes/no") throw away the survival term and become un-calibratable. This is the structural root of most ML-forecasting failures discussed below.

Every classical model on the Models — Classical page is a special case of this formulation; ETAS is the parametric, physics-informed member, and neural temporal point processes (NTPPs) are the flexible, learned members.

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2. The baseline to beat: ETAS as a point process

ETAS (Ogata 1988, spatio-temporal extension Ogata 1998) is the field-standard short-term forecasting model — a self-exciting (Hawkes) point process with seismologically motivated kernels:

$$\lambda(t, x, y \mid \mathcal{H}_t) = \mu(x,y)

• \sum_{i:,t_i < t} g(t - t_i), h(x - x_i, y - y_i), \kappa(m_i),$$

with the Omori–Utsu temporal decay $g(t - t_i) = K/(t - t_i + c)^p$, the Utsu productivity $\kappa(m_i) = A,e^{\alpha(m_i - m_0)}$, a power-law/Gaussian spatial kernel $h(\cdot)$, a stationary background $\mu(x,y)$, and Gutenberg–Richter governing magnitudes. (Full equations and estimation are on the Models — Classical page.)

ETAS is the right baseline because it is fully generative (you can simulate $N$ synthetic next-day catalogs and compute every CSEP test), it has ~5–8 interpretable parameters (little overfitting headroom), and it has decades of prospective CSEP track record — it is the CSEP baseline.

ETAS's genuine limitations are the openings for ML: fixed functional kernels (cannot learn anisotropy or fault structure unless hand-coded); single-catalog (hard to fuse geodesy, pore pressure, InSAR, or multiple networks); the separability assumption (space / time / magnitude treated independently); and expensive MLE / simulation. These are exactly the gaps neural methods claim to fill — and, as the evidence below shows, the gaps where they sometimes genuinely help.

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3. Neural Temporal Point Processes — the relevant model family

The forecasting-relevant ML family is neural TPPs: replace ETAS's hand-designed kernels with a neural network that maps history to a conditional intensity. These were validated on non-seismic event streams (social-media, retail, synthetic), and their log-likelihood wins there do not automatically transfer to seismicity (§4). Four foundational architectures:

3.1 RMTPP — Recurrent Marked Temporal Point Process (Du et al., 2016)

An RNN encodes history into a hidden state $h_j$ (after the $j$-th event); the intensity between events is

$$\lambda^*(t) = \exp!\big(\mathbf{v}^{\top} h_j + w,(t - t_j) + b\big).$$

The $w(t - t_j)$ term gives a log-linear, Omori-like decay; $h_j$ carries the marked history. Limitation: the exponential form is restrictive (monotone between events), and a single RNN state compresses all history. (Du, Dai, Trivedi, Upadhyay, Gomez-Rodriguez & Song, KDD 2016, doi:10.1145/2939672.2939875.)

3.2 Neural Hawkes Process (Mei & Eisner, 2017)

A continuous-time LSTM whose cell state decays between events; the intensity is

$$\lambda^*(t) = f!\big(\mathbf{v}^{\top} h(t)\big), \qquad f(\cdot) = \mathrm{softplus}.$$

The continuous-time decay lets past events lower future intensity (inhibition), which a classical Hawkes process cannot represent. (Mei & Eisner, NeurIPS 2017, arXiv:1612.09328.)

3.3 Self-Attentive Hawkes Process — SAHP (Zhang et al., 2020)

Replaces recurrence with multi-head self-attention over event embeddings to build the history representation, then parameterizes the intensity from it — capturing long-range dependencies without RNN memory decay. (Zhang, Lipani, Kirnap & Yilmaz, ICML 2020, arXiv:1907.07561.)

3.4 Transformer Hawkes Process — THP (Zuo et al., 2020)

A Transformer encoder produces a hidden representation $h(t_i)$ from positionally/temporally encoded history; the continuous intensity (per mark $m$) interpolates between events as

$$\lambda_m^*(t) = \mathrm{softplus}!\Big( \alpha_m, \frac{t - t_i}{t_i}

• \mathbf{w}m^{\top} h(t_i) + b_m \Big), \quad t \in (t_i, t{i+1}].$$

THP and SAHP generally report better log-likelihood than RMTPP / NHP on standard event-sequence benchmarks. (Zuo, Jiang, Li, Zhao & Zha, ICML 2020, PMLR v119.)

Key caveat carried into seismology. These models were validated on social-media, retail and synthetic streams — not earthquakes. Their log-likelihood wins on those domains do not transfer to seismic forecasting (§4).

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4. Neural TPPs applied to earthquakes — the real evidence

4.1 EarthquakeNPP — the decisive benchmark

Stockman, Lawson & Werner (accepted TMLR 2026; arXiv:2410.08226) built the field's first rigorous benchmark for neural point processes on earthquake catalogs.

• Models tested (5): NSTPP, DeepSTPP, AutoSTPP, DSTPP (a diffusion STPP), and SMASH (score-matching pseudolikelihood).
• Data: California 1971–2021 — ComCat (whole CA), SCEDC (Southern CA at $M_c$ 2.0/2.5/3.0), and several fault-zone catalogs — with strict chronological train/validation/test splits.
• Evaluation: temporal and spatial log-likelihood plus CSEP consistency tests; for testing, each model generates 24-hour forecasts via 10,000 simulated catalogs per day.
• Headline result: none of the five NPPs outperform ETAS. ETAS wins spatial log-likelihood "consistently". On the ComCat dataset, ETAS passes the consistency tests at — N-test 95.8%, Spatial 92.0%, Magnitude 93.8%, Pseudo-likelihood 97.6% — whereas the best NPP passes ~86–88% on the number / pseudo-likelihood tests but only ~68.6% on the spatial test, which is exactly where forecasting value lives.
• Verdict (the authors'): "current NPP implementations are not yet suitable for practical earthquake forecasting."

Why this benchmark matters more than prior "NPP beats X" papers: it explicitly repaired a data-leakage flaw in earlier neural-TPP-for-earthquakes work — non-chronological / alternating splits that artificially inflate performance measures due to the nature of earthquake triggering, and the exclusion of the 2011 Tohoku $M_w$ 9.0 sequence. With temporal splits and the big sequences restored, the apparent neural advantage evaporated. This is the evaluation lesson.

Scope caveat. This "NPPs do not beat ETAS" result is established on the California-only benchmark to date (1971–2021), not proven globally. It justifies shipping ETAS-class only for v0 and gating any neural model behind a CSEP win — but it is stated as "on the benchmark to date," not as an unconditional law. We do not claim ML can never add skill.

4.2 FERN — the strongest honest pro-neural result

Zlydenko et al. (2023), A neural encoder for earthquake rate forecasting, Scientific Reports 13 — an encoder–decoder that generalizes ETAS: a "Recent Earthquakes Encoder" replaces ETAS's fixed response function with an MLP over event features (magnitude, elapsed time, distance, depth), plus a long-term seismicity encoder and a learned location encoder, with permutation- invariant summation (Deep-Sets-style; no RNN/transformer). It keeps the same intensity skeleton as ETAS:

$$\lambda(x, y, t \mid \mathcal{H}_t) = \mu(x,y) + \sum_i T(t - t_i), S(x - x_i, y - y_i;, M_i).$$

• Claimed gain: a 4–12% improvement in Information Gain Per Earthquake (IGPE) when sub-completeness-magnitude events are added (the FERN+ variant) — and it learned anisotropic spatial structure aligned with fault traces without fault inputs. (The reported region endpoints, e.g. 2.278→2.395 and 1.395→1.803, are ETAS→FERN+ values; plain FERN is lower.)
• Runtime: ~1 s vs. ~10 h for ETAS simulation (≈1000×).
• Critical caveats — stated by the authors themselves: it was not tested under CSEP; it provides no uncertainty quantification at all; its test period ends before Tohoku $M_w$ 9.0; its parameters are not interpretable; and performance varies by region.

The honest reading. FERN's gain is real but modest, retrospective, region-specific and uncalibrated. The value came largely from ingesting more data (sub-$M_c$ events) and learning anisotropy — exactly the two ETAS gaps — not from deep-learning magic. This is the good news: neural value is concentrated in covariate ingestion + spatial flexibility, which can be added incrementally on top of an ETAS skeleton.

4.3 RECAST and other catalog NPP / hybrid results

• RECAST (Dascher-Cousineau et al., 2023, GRL 50, e2023GL103909) — a GRU-based (recurrent) encoder–decoder neural TPP. It improves on temporal ETAS only when the training catalog is large ($\gtrsim 10^4$ events); on smaller catalogs it merely matches ETAS. An honest, data-hungry result.
• Schultz (2026), Forecasting the Rate of Induced Seismicity as a Neural Temporal Point Process, JGR: Machine Learning and Computation — a neural TPP for induced seismicity performs on par with prior statistical benchmarks (not clearly better).
• Hybrid DL + ETAS approaches (e.g. GJI 239(3):1545, arXiv:2310.02574) combine a DL component with ETAS; gains are incremental and not yet prospectively validated.

Synthesis of §4. No pure ML / NPP has robustly beaten a well-fit ETAS in prospective CSEP as of 2026. Where neural models match or slightly beat ETAS, it is in narrow settings (large catalogs, sub-$M_c$ augmentation, induced seismicity), and the gain traces to data ingestion and spatial flexibility, not architecture depth.

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5. The canonical cautionary tale: DeVries 2018 vs. Mignan & Broccardo 2019

This is the single most important evaluation lesson in ML-for-seismicity, and it must shape any forecasting product's testing discipline.

5.1 The original claim

DeVries, Viegas, Wattenberg & Meade (2018), Deep learning of aftershock patterns following large earthquakes, Nature 560:632–634, trained a deep neural net (6 hidden layers, 50 nodes per layer, 12 stress-tensor input features) on ~131,000 aftershocks from ~199 mainshocks to predict the binary spatial pattern of aftershock occurrence per grid cell. It reported AUC = 0.85, vs. AUC = 0.58 for the classical Coulomb-failure-stress criterion, and suggested the net had discovered new physical triggering quantities (e.g. functions related to the second invariant of the deviatoric stress tensor / von Mises stress).

5.2 The rebuttal

Mignan & Broccardo (2019), One neuron versus deep learning in aftershock prediction, Nature 575:E1–E3, showed that a two-free-parameter logistic regression ("one neuron") —

$$\sigma(\mathbf{x}) = \frac{1}{1 + e^{-(w x + b)}}, \qquad \text{two parameters } w, b,$$

using essentially a single physically simple feature (the sum of the absolute values of the stress-change components) — matches or exceeds the deep net's AUC. Their critique:

• The DNN had ~13,451 free parameters for a tabular problem with only 12 inputs and effectively ~199 independent mainshocks — massively over-parameterized, with overfitting masked by the evaluation setup.
• The "computer-vision-like" per-cell classification framing (131k samples from 199 mainshocks) inflated the apparent sample size and broke the independence assumption underlying the AUC.
• DeVries et al. gave no physical justification for the chosen absolute-stress inputs.

5.3 The reply

DeVries et al. (2019), Reply to: One neuron versus deep learning in aftershock prediction, Nature 575:E4–E5, acknowledged that the simpler model performs comparably, while maintaining the DNN as a useful exploratory tool for surfacing candidate physical quantities.

5.4 The evaluation lessons (baked into this product)

1. Always include a trivially simple baseline (logistic regression / single physical feature / ETAS). If a 2-parameter model ties your deep net, the deep net learned nothing generalizable.
2. Count effective sample size, not row count. Spatially/temporally correlated cells from a few mainshocks are not independent samples; AUC / log-likelihood computed as if they were is misleading.
3. Parameters ≫ effective samples ⇒ assume overfitting until proven otherwise on held-out time.
4. AUC / classification is the wrong metric for forecasting. It is invariant to monotone rescaling, hence blind to the miscalibration of the rates/probabilities a forecast publishes, and on a rare per-cell-per-day task it degenerates into a region classifier (it largely measures between-region rate differences). AUC / accuracy are banned as primary forecasting metrics in this product. Use proper scoring + CSEP consistency tests on the generative forecast.
5. Retrospective ≠ prospective. Nothing counts until it runs forward-in-time on data the model never touched at training.

(This is the deep-learning analogue of the selection-bias trap that discredited Accelerating Moment Release — see Models — Classical §10.)

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6. Other neural architectures (and why they underwhelm)

6.1 RNN / LSTM seismicity-rate regression

Many papers feed binned seismicity-rate / magnitude time series into LSTMs to "predict the next bin." The pitfalls:

• Predicting a binned rate drops the point-process survival term → un-calibratable, no honest probability.
• Heavy class imbalance (large events are rare) → models default to "no large event," score well on accuracy, and fail on the events that matter.
• Random shuffling of train/test (the EarthquakeNPP leakage problem) → inflated metrics that vanish under temporal splits.
• LSTM memory decay biases toward the most recent bin; there is no built-in Omori / GR physics.

6.2 CNNs for spatial seismicity

CNNs over gridded seismicity images (DeVries-style, or "seismicity image → next-window heatmap") can learn spatial texture but inherit the DeVries evaluation traps — correlated pixels, AUC on per-cell labels, overfitting on a handful of effective sequences. No CNN forecaster has passed prospective CSEP testing against ETAS. This is precisely why, in this product, a CNN is used only as a spatial-context encoder feeding a point-process intensity, never as a standalone classifier — see Models — Employed.

6.3 Graph Neural Networks (GNNs)

Active 2024–2025 research direction. GNNs are genuinely strong for source characterization, phase association and location (where the graph is the station network), but for forecasting magnitude/occurrence the literature repeatedly reports that depth and magnitude prediction "remain weak for all tested models." GNNs help on the detection side, not where you need a calibrated future rate.

6.4 Transformers over catalogs

THP / SAHP-style transformers over event sequences are the most principled neural-forecasting option (they are point processes), and their attention can in principle model long-range / cross-fault dependencies. But in EarthquakeNPP-style testing they do not beat ETAS (§4.1). They are an open research direction, not a shipping capability.

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7. Where ML fails, where it helps — the honest synthesis

7.1 Systematic failure modes documented in the literature

The ML-for-earthquakes literature and the broader CSEP literature converge on a recurring failure list:

• Data leakage via non-temporal splits (the #1 sin; see EarthquakeNPP §4.1).
• Class imbalance for large events → degenerate "never predict a big one" models that score well.
• Improper metrics (accuracy / AUC instead of proper scoring + CSEP consistency).
• No prospective testing — almost all "AI beats X" claims are retrospective.
• Magnitude-prediction hype — deterministically predicting the magnitude of the next event is not scientifically supported (GR says magnitude is ~memoryless given that an event occurs).
• Overfitting of over-parameterized nets to a handful of effective sequences (DeVries §5).

7.2 Where ML genuinely adds value (the defensible roadmap)

1. Multivariate covariate ingestion. ETAS uses one catalog. A neural conditional intensity can fuse geodesy / InSAR, pore-pressure / injection data (induced seismicity), multiple catalogs, and sub-$M_c$ events. FERN's gain came largely from sub-$M_c$ events — the clearest, most honest win.
2. Declustering-free, end-to-end conditional intensity. The network learns triggering directly from raw history, avoiding arbitrary declustering choices.
3. Richer spatial structure. Learned anisotropy aligned with fault traces (FERN) without hand-coding fault geometry.
4. Inference speed. ~1000× faster simulation enables a cheap daily public app.
5. Catalog quality upstream (the biggest real lever). Better ML detection (§8) → more complete catalogs → better ETAS and better neural forecasts. Most of the realizable near-term gain is here, not in the forecaster.

7.3 The blunt verdict

ML does not yet beat ETAS prospectively for short-term forecasting (2026). It overfits and fails to generalize when (a) parameters ≫ effective samples, (b) splits are not temporal, (c) it abandons the point-process survival term for classification/regression. It adds value mainly as an ETAS-generalizing conditional intensity that ingests covariates and learns spatial flexibility, and as upstream catalog-building detection. For a product that must be calibrated and testable, ETAS is the floor you must clear, not the ceiling you can ignore. This directly motivates the gated-challenger architecture on Models — Employed.

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8. Forecasting is not detection

A hard line the product must communicate. Detection / phase-picking / association ML is mature and excellent — but it is NOT forecasting. It tells you what already happened (better catalogs); forecasting estimates what will happen.

Tool	Task	Maturity	Forecasting?
PhaseNet (Zhu & Beroza, 2019)	P/S phase picking, U-Net CNN (~268k params)	Production	No — detection
EQTransformer (Mousavi et al., 2020)	Joint detection + P/S picking; CNN + BiLSTM + attention	Production	No — detection
PhaseNO (2023)	Multi-station picking via Fourier/Graph Neural Operator	Maturing	No — detection/association
SeisBench (Woollam et al., 2022)	Benchmark/toolbox + datasets (ETHZ, GEOFON, STEAD)	Production	No — infrastructure
SeisLM (Liu et al., 2024)	Self-supervised Wav2Vec2-style foundation model; detection, P/S, onset regression, foreshock/aftershock classification	Research	No — explicitly retrospective characterization

SeisLM (arXiv:2410.15765; SeisLM-base 11.4M / SeisLM-large 90.7M params; masked-reconstruction contrastive pretraining) is positioned for detection / phase-picking / characterization, not forecasting: its "foreshock–aftershock classification" labels existing waveforms relative to a known mainshock — retrospective, not predictive.

These tools build better, more complete catalogs (a lower, more stable $M_c$), which helps both ETAS and any neural forecaster — the single biggest realizable near-term lever. But they do not forecast.

Product copy rule. Never let detection-model branding imply prediction. Use these tools to build the input catalog, then forecast with the point-process model. Saying otherwise is the kind of overclaim that gets seismology papers rebutted in Nature.

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References

1. Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. JASA 83(401), 9–27. doi:10.1080/01621459.1988.10478560
2. Ogata, Y. (1998). Space-time point-process models for earthquake occurrences. Ann. Inst. Statist. Math. 50(2), 379–402. doi:10.1023/A:1003403601725
3. Du, N., Dai, H., Trivedi, R., Upadhyay, U., Gomez-Rodriguez, M. & Song, L. (2016). Recurrent Marked Temporal Point Processes. KDD 2016. doi:10.1145/2939672.2939875
4. Mei, H. & Eisner, J. (2017). The Neural Hawkes Process: A Neurally Self-Modulating Multivariate Point Process. NeurIPS 2017. arXiv:1612.09328
5. Zhang, Q., Lipani, A., Kirnap, O. & Yilmaz, E. (2020). Self-Attentive Hawkes Process. ICML 2020. arXiv:1907.07561
6. Zuo, S., Jiang, H., Li, Z., Zhao, T. & Zha, H. (2020). Transformer Hawkes Process. ICML 2020. PMLR v119.
7. DeVries, P.M.R., Viegas, F., Wattenberg, M. & Meade, B.J. (2018). Deep learning of aftershock patterns following large earthquakes. Nature 560:632–634. doi:10.1038/s41586-018-0438-y
8. Mignan, A. & Broccardo, M. (2019). One neuron versus deep learning in aftershock prediction. Nature 575:E1–E3. doi:10.1038/s41586-019-1582-8 · preprint arXiv:1904.01983
9. DeVries, P.M.R., Viegas, F., Wattenberg, M. & Meade, B.J. (2019). Reply to: One neuron versus deep learning in aftershock prediction. Nature 575:E4–E5. doi:10.1038/s41586-019-1583-7
10. Zlydenko, O., Elidan, G., Hassidim, A., Kukliansky, D., Matias, Y., Meade, B., Molchanov, A., Nevo, A. & Bar-Sinai, Y. (2023). A neural encoder for earthquake rate forecasting. Scientific Reports 13. doi:10.1038/s41598-023-38033-9
11. Dascher-Cousineau, K., Shchur, O., Brodsky, E.E. & Günnemann, S. (2023). Using deep learning for flexible and scalable earthquake forecasting (RECAST). GRL 50, e2023GL103909. doi:10.1029/2023GL103909
12. Stockman, S., Lawson, D. & Werner, M.J. (2026, accepted). EarthquakeNPP: A Benchmark for Earthquake Forecasting with Neural Point Processes. TMLR. arXiv:2410.08226
13. Schultz, R. (2026). Forecasting the Rate of Induced Seismicity as a Neural Temporal Point Process. JGR: Machine Learning and Computation. doi:10.1029/2025JH001052
14. A combining earthquake forecasting model between deep learning and ETAS. Geophys. J. Int. 239(3):1545. arXiv:2310.02574
15. Serafini, F., Bayona, J.A., Silva, F., Savran, W., Stockman, S., Maechling, P.J. & Werner, M.J. (2025). A benchmark database of ten years of prospective next-day earthquake forecasts in California from CSEP. Scientific Data 12, 1501. doi:10.1038/s41597-025-05766-3
16. Zhu, W. & Beroza, G.C. (2019). PhaseNet: a deep-neural-network-based seismic arrival-time picking method. Geophys. J. Int. 216:261–273. doi:10.1093/gji/ggy423
17. Mousavi, S.M., Ellsworth, W.L., Zhu, W., Chuang, L.Y. & Beroza, G.C. (2020). Earthquake Transformer (EQTransformer). Nature Communications 11:3952. doi:10.1038/s41467-020-17591-w
18. Woollam, J. et al. (2022). SeisBench — A Toolbox for Machine Learning in Seismology. SRL 93:1695–1709. doi:10.1785/0220210324
19. Liu, T., Münchmeyer, J., Laurenti, L., Marone, C., de Hoop, M.V. & Dokmanić, I. (2024). SeisLM: a Foundation Model for Seismic Waveforms. arXiv:2410.15765
20. CSEP / pyCSEP — Collaboratory for the Study of Earthquake Predictability. https://cseptesting.org · https://github.com/SCECcode/pycsep

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See also: Models — Classical · Models — Employed · Evaluation · Methodology.