In-Context Learning of Temporal Point Processes with Foundation Inference Models
Authors: David Berghaus
Patrick Seifner
Kostadin Cvejoski
Cesar Ojeda
Ramses J Sanchez
Venue: ICLR 2026 Poster (2026)
Keywords: temporal point processes
zero-shot inference
in-context learning
zero-shot parameter estimation
inference of point processes
foundation models
foundation inference models
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: We introduce a framework for in-context (zero-shot) inference/estimation of intensity function underlying temporal point processes from empirical data.

Abstract: Modeling multi-type event sequences with marked temporal point processes (MTPPs) provides a principled framework for uncovering governing dynamical rules and predicting future events. Current neural approaches to MTPP inference typically require training separate, specialized models for each target system. We pursue a fundamentally different strategy: leveraging amortized inference and in-context learning, we pretrain a deep neural network to infer, *in-context*, the conditional intensity functions of event histories from a context consisting of sets of event sequences. Pretraining is performed on a large synthetic dataset of MTPPs sampled from a broad distribution over point processes. Once pretrained, our Foundation Inference Model for Point Processes (FIM-PP) can estimate MTPPs from real-world data without additional training, or be rapidly finetuned to specific target systems. Experiments show that FIM-PP matches the performance of specialized models on multi-event prediction across common benchmark datasets.

Published as a conference paper at ICLR 2026
IN-CONTEXT LEARNING OF TEMPORAL POINT PRO- CESSES WITH FOUNDATION INFERENCE MODELS
David Berghaus Patrick Seifner!., Kostadin Cvejoski',
César Ojeda* & Ramsés J. Sánchez!
Lamarr Institute', Fraunhofer IAIS?, University of Bonn', JetBrains Research* University of Potsdam"
david. berghaussiaata fraunhofer.de
ABSTRACT
Modeling multi-type event sequences with marked temporal point processes (MTPPs) provides a principled framework for uncovering governing dynamical rules and predicting future events. Current neural approaches MTPP inference typically require training separate, specialized models for each target system. We pursue a fundamentally different strategy: leveraging amortized inference and in- context learning, pretrain deep neural network to infer, in-context, the con- ditional intensity functions of event histories from a context consisting of sets event sequences. Pretraining is performed on large synthetic dataset of MTPPs sampled from a broad distribution over point processes. Once pretrained, our Foundation Inference Model for Point Processes (F (FIM-PP) can estimate MTPPs from real-world data without additional training, or be rapidly finetuned to spe- cific target systems. Experiments show that FIM-PP matches the performance of specialized models on multi-event prediction across common benchmark datasets. Our pretrained model, repository, and tutorials available online'
INTRODUCTION
The mathematical modeling of asynchronous, irregular event sequences has long occupied a dis- tinetive place in the machine learning community. Temporal point processes provide the canonical framework for modeling neural dynamics (Truccolo et al., 2005; Linderman Adams, 2014), and they serve the de facto tool for describing wide range of internet phenomena, including retweet- ing, posting, and information cascades (Zhao et al., 2015; Cvejoski et al., 2020). Their ability to encode fine-grained temporal structure, together with their capacity to reveal causal interactions be- tween events in an interpretable manner, has made them indispensable not only in neuroscience and social media, but also in finance (Ait-Sahalia et al., 2015) and epidemiology (Chiang et al., 2022). Despite this centrality, the development of foundation models has followed different trajectory. Large-scale pretraining first emerged in natural language processing, enabled by massive internet corpora, and has only recently been extended to dynamical systems, with recent work addressing ODEs (d' Ascoli et al., 2024; Mauel et al., 2025; 2026), MJPs (Berghaus et al., 2024), SDEs (Seifner et al., 2025a), and even specific applications in pharmacology (Marin et al., 2025). It is therefore ironic that event data - the very modality underlying the internet activity that made text-based pretraining possible - has not yet given rise to corresponding foundation model for point pro- cesses. The present work takes a first step toward filling this gap by developing a foundation model explicitly designed for temporal point processes.
Marked Temporal Point Processes (MTPPs) (Daley & Vere-Jones, 2007; Rasmussen, 2018) are stochastic processes consisting of ordered occurrence times, each accompanied by categorical mark specifying its type. Formally, the objective is to specify the conditional distribution of the next event time and mark given the history of the process up to the current time. The extensive point process literature explores different ways of encoding event histories and specifying the stochastic mechanism that governs new arrivals and their marks (Lin et al., 2024). common approach
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to represent this distribution through conditional intensity function, which describes the instan- tancous rate at which events of different types occur given the history. Traditionally, models such as the Hawkes process (Hawkes, 1971) define this conditional intensity a superposition of self- exciting effects from past events. Forecasting is then carried out recursively using Ogata's 's thinning algorithm, applied the conditional intensity. Building extended Hawkes processes by parameterizing event histories with neural architectures (Mei & Eis- ner, 2017), including recurrent neural networks (Du et al., 2016), attention mechanisms (Zhang al., 2020), and Transformers (Zuo et al., 2021). These models are typically trained either via maximum likelihood often requiring expensive integral evaluations that bypass intensity modeling altogether and directly sample future events conditional on the past (Kerrigan et al., 2024; Zeng et al., 2024). However, a fundamental limitation of these approaches their lack of transferability: each new dataset requires retraining from scratch, forcing the model to relearn representations for each distinct dynamical regime.
this cornerstone, more recent work has
or through generative approaches
In contrast, modern approaches dynamical systems increasingly prioritize pretraining syn- thetic data, yielding general models that can learn dynamics in-context. This paradigm offers a crucial advantage: practitioners no longer need train models de novo for every dataset, but can instead obtain accurate characterizations in a zero-shot manner. Within this family, two variants have emerged: Prior-fitted Networks (PFNs) and Foundation Inference Models (FIMs). PFNs train networks to approximate predictive posterior distributions in a sequence-to-sequene
or context- to-sequence fashion (Müller et al., 2022; Hollmann et al., 2022: Müller et al., 2025). FIMs, by contrast, focus on directly estimating the infinitesimal operators of dynamical system (e.g.. drift and diffusion functions for SDEs), thereby retaining degree of interpretability (Berghaus et al., 2024; Seifner et al., 2025a;b; Mauel et al., 2026). Access to these operators enables the explicit study of physically relevant observables such as entropy production, stationary distributions, and attractors.
In the context of point processes, we adapt the FIM pretraining paradigm to MTPPs in three steps. First, we define broad family of conditional intensity functions, thereby inducing diverse prior over MTPPs. This prior captures assumptions about excitatory and inhibitory effects between events, as well as the interaction structure across event types. Second, we sample MTPPs from this prior, generate synthetic event sequences, and construct tuples consisting of context event sequences, event histories, and their corresponding intensities, thereby creating a meta-learning task that amortizes inference across heterogeneous dynamics. Third, train neural network to recover conditional intensities from observed context. We summarize our contributions as follows:
1. Introduce synthetic data generation framework for sampling event sequences from a broad prior distribution over MTPPs, with randomized base intensities, kernels, and in- teraction types (i.e., excitatory, inhibitory, neutral). We empirically demonstrate that this construction encodes strong prior, enabling models trained on it to generalize across both in-distribution processes and real-world event data.
2. Train the first transformer- based recognition model capable of estimating in-context the conditional intensity functions of MTPPs, where history representations serve as queries and the encoded sequence context provides the keys and values.
3. Show that the resulting model achieves strong zero-shot performance across synthetic benchmarks and multiple real-world datasets, and that it can be rapidly finetuned on new event data.
2 RELATED WORK
Here we provide a brief overview of temporal point processes. For detailed surveys and benchmarks of deep TPP models, including open challenges history encoding, conditional intensity design, relational discovery, and learning strategics, see e.g., Lin et al. (2024) and Xue et al. (2024).
While the mathematical theory of point processes is extensive (Daley & Vere-Jones, 2007; Kingman, 1992), work on temporal point processes (TPPs) in machine learning has crystallized around two central questions: (i) how should representations of past events be constructed, and (ii) how should the future be modeled (Lin et al., 2024). Early approaches, epitomized by the Hawkes process, addressed both questions using linear self-exciting kernels. natural extension is the neural Hawkes process (Mei & Fisner, 2017), along with related recurrent formulations (Du et al., 2016), which
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rely neural representations of past events, trained via likelihood maximization, and model future events using the thinning algorithm. Later work introduced more expressive architectures. Attention mechanisms (Zuo al., 2021: Zhang et al., 2020; Yang et al., 2021) extend the memory horizon of TPPs, though at the cost of higher computational demand. Neural ODEs (Chen et al., 2018) have also been incorporated better capture the irregular timing of events in latent representations (Song et al., 2024; Kidger et al., 2020).
To improve predictive accuracy over long horizons, different decompositions of the likelihood for future arrivals have been proposed (Rasmussen, 2018; Panos, 2024; Deshpande et al., 2021; Draxler al., 2025). These works highlight the limitations of intensity- based inference, particularly when relying on thinning algorithms. Such limitations have motivated a shift toward generative models, which typically sample entire sequences. Approaches include optimal transport (Xiao et al., 2017), diffusion models (Zeng et al., 2024; Lüdke et al., 2023), and flow-matching methods ( (Kerrigan et al., 2024), often trading accuracy for interpretability. In contrast, traditional machine learning methods et al., 2022) emphasize interpretability from the outset. A
(Rasmussen, 2013; Malem-Shinitse key advantage of the Hawkes process is that its excitation graph makes causal structure explicit (Xu al., 2016; Wu al., 2024), which has been especially relevant in neuroscience (Linderman & Adams, 2014; Truccolo et al., 2005) and in finance, where Hawkes models often serve as hidden drivers of observed activity (Ait-Sahalia et al., 2015). Applications extend more broadly, for instance to dynamics of text (Cvejoski et al., 2020; 2021), social online activity (Zhao et al., 2015) and operations rescarch (Ojeda et al., 2021).
While Liu & Quan (2024) fine-tuned pre-trained LLM for point-processes (with modest success), the best of our knowledge, no prior work has presented a foundation model for point-processes that can operate zero-shot without further training.
3 PRELIMINARIES
In this section, recall the definition and basic properties of marked temporal point processes (Daley & Vere-Jones, 2007) and Hawkes processes (Hawkes, 1971; Laub et al., 2015). Additionally, define the inference problem our proposed approach tackles.
Marked Temporal Point Processes: We consider marked temporal point processes (marked TPPs, or MTPPs) as simple point processes R x K, where KC is discrete and finite set of marks. The density f of a sequence of events S = {(1,, {=) in the interval [0, T), w.l.o.g. ordered by their time component t; E R,, factors into conditional densities
n
(((ti,
II f ((1,, Ho.) II H4,) 44, H.,))))
where Hi = {(12,83) 14 1 C S the history strictly preceding L. By the last equality of equation 1, MTPPs may be characterized by dependent densities of the next-event time J(L H.) and event mark [(x] I, H.). MTPPs are commonly represented by their piece-wise wise continuous conditional intensity function
f(t H4)
A(t, K. Ho) 1- 5t f(s (H, dss
f(K t, H, X(t H.) |t, H,,.
(2)
where e' is the last event time in He, o' 0 if, Ø. The conditional intensity function may be interpreted of the instantaneous rate of mark & occurring at f, conditioned of the history up to time t, Reversely, any such function satisfying some mild conditions, defines the density of an MTPP on set of events in an interval [0, T] by
II X(a, Ks He,) exp (-1," A(s H.)ds
f (i
Collection of TPPs: A TPP is just MTPP with single mark. An MTPP be viewed collection of TPPs per mark, interdependent through a joined history. Indeed, given an MTPP as above, the conditional intensity Axe( | He) X(t H4)((s He) defines the marginal TPP
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for mark K E K. that may depend other marks via H4. Conversely, collection of TPPs with conditional intensity Ay per A. E I can be joined to an MTPP. Using
A. Hc) A(t H4)
A(t H4)
' le H4) and f(E t, H₂) NEK
(4)
in Eq. 2 defines the conditional intensity function A(t, H2) of an MTPP. fact, A(1, K. H.) =
(t He). In contrast some other neural methods (Du al., 2016), which estimate X(t H.) and
f(x t, He), we design our model to parametrize TPPs per mark, conditioned on the joined history of all marks.
Hawkes Processes: A Hawkes MTPP with marks KC is defined by the conditional intensity
Xt, R H4) = max (0,uu(t)+ +
E Yow (tx (t))
(5)
(C,R))E
where {JKbKer is a set of time-dependent base intensity functions, and {Tixe TR,eeE is a set of interaction kernels, specifying the influence of mark k' on mark If Tie is positive, the influence of n' on k is called excitatory or exciling, otherwise it is called inhibitory or limiting.
Simulation: We Ogata's 's modified thimming algorithm (Ogata, 1981) generate synthetic train- ing data from MTPP, and to simulate processes inferred by our model.
Let C = {S) J3=1 be a collection of m event sequences S = ((1) m²) ) observed from some system. Our objective is to predict simulate the next event and estimate the likelihood of a (previously unseen) sequence S, assuming an MTPP model. Previous neural methods train an autoregressivv encoding network on C that compresses the history H. of S into some embedding h, for a neural estimate '(t, k. h₂) of the conditional intensity. In contrast, we pretrain a deep neural network model to estimate in-context Å, given a history of events, from a collection of context event sequences. Once trained, the model be applied to any C and Hi. without any further training.
Inference Problem:
4 FOUNDATION INFERENCE MODELS FOR POINT PROCESSES
In this section, we present novel in-context learning method for the MTPP intensity inference problem. In two-step approach, we first generate large of marked event sequences from parametrized MTPPs, sampled from a broad distribution over MTPPs. This yields train data for neural network recognition model, trained to estimate the underlying known, ground-truth intensity functions. Such pretrained inference model can be applied directly to real-world problems, or swiftly finetuned for improved performance.
4.1 SYNTHETIC DATASET GENER GTIORATION
To construct synthetic dataset of MTPPs, define distribution over MTPPs via distribution over conditional intensity functions of the form
A(1, K Hc) = max 0, (,)
Σ (F(x'))EH
Yose - 13) ())
(6)
Given sample from this distribution, simulate large set of marked event sequences and record the corresponding conditional intensities for training.
We Eq. parameterize several classes of processes:
The classical Hawkes process, for which μı is constant in time and Tou (t) is exponential function;
The Poisson process, for which μu constant in time and Toxe
Periodic processes, for which (t) is positive sinusoidal function;
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Context Sequences
Transformer
Intensity Estimate
S
V
K
V
S
Q
enc
K
L/
dec
Ô Hr.
Br
He
+ A(t, K HH))
H
S
K
Time
History
Mark
Figure 1: Schematic representation FIM-PP. context of marked event sequences St is encoded by self-attentive transformer encoder. The result is further processed by transformer decoder, using a history H, of marked events before time as queries. The final embedding is joined with an encoding of mark K. The results is projected to set of parameters that determine the value of the conditional intensity function Ả evaluated at (t, xc).
Processes with high initial excitation, for which we choose yax (() to follow Gamma dis- tribution;
Processes with non-monotonic, shifted kernels, for which we choose Toxe (t) to follow a Rayleigh distribution.
A complete overview of the process families and their hyperparameters is provided in Table 6 in the Appendix. Furthermore, Figure in the Appendix reports summary statistics of our simulated pretraining distribution, and confirms that it is broad enough cover the real-world datasets depicted in Figures 8 to 12, also in the Appendix.
For each process, we randomly sample Znn' {- for k' E K. to cover excitatory (Zxx! 1), inhibitory (2nm' = 1) and non-inftuencing (Exx² = 0) interactions.
4.2 FOUNDATION INFERENCE MODEL ARCHITECTURE
We now present the architecture of IP, a pretrained deep neural network for inference of MTPPs from sets {8) yym of marked event sequences S =
FIM-PP processes the context sequences and estimates the conditional intensity function of MTPP that describes the observed dynamics. Following previous dieesstt-bbssd methods (Zhang et al., 2020; Zuo et al., 2021), Å is implemented by flexible parametrized function family A(. H.) for all к K. FIM-PP estimates its parameters by encoding the history Hr = {( xx) > before time I > hist subject to the processed context sequences. Figure depicts schematic representation of this "Mai"" approach.
To cover applications in different time scales, FIM-PP instance normalizes its inputs and renormal- izes Å accordingly. Appendix C provides the details. Once trained, F FIM-PP can be applied for all counts of marks K.| up to some fixed upper bound, similar to in-context methods in other domains (d' Ascoli et al., 2024; Berghaus et al., 2024).
We denote linear projections by c, feed-forward neural networks by 4 attention layers with residual connections by v/, transformer encoders by V Vene and decoders by V Let E denote the model's embedding dimension.
Context Encoding: To encode C, we combine encodings of individual sequences S), Recognizing the importance of inter-observation times for the inference problem, we consider At! = 1. 1 additional feature, identifying t/ O. To encode 8), first embed the features (t/ R/ AP of the i-th event in sequence j into embeddings
uf (1) die (x/) + (st( R
Sinusoidal output activations from Shukla & Marlin (2020) enhance the networks di and dat- Let S jup...., n E RR,, denote the matrix of embedding of sequence $1. We extract
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context sequence embedding ç followed by fixed-query attention
R by applying transformer encoder S
(S
R
C = (q S S²) E E
(8)
where q.th! ER is learnable query.
We emphasize that we use an hierarchical approach where every sequence gets processed inde- pendently by You and encoded into a single embedding с₁. This much more scalable com- pared to combining all events into single long sequence. The embeddings of all sequences C= -=.c.,...,. Cm] E RmxE are finally combined by another transformer encoder Č = Vreab (reab(C) RmxE.
Context-aware History Encoding: To encode the history H. of events prior to time t >> thiss embed each tuple (e) bog Affii) into feature vectors H [u] " E R x reusing the networks from Eq. 7. These history embeddings serve as the queries transformer decoder
Vrhise which attends the context representation C (used as keys and values), yielding unified encoding h that integrates both history and context:
h
hist
hi (H, CR.
Intensity Parametrization: To extract intensity functions for all marks from h we concatenate [hiin, ohiu (w/) E R22,, and project it to non-
h with a (linear) encoding of nt to form negative parameter estimates
Mr. t.
2 E IR+,
3 = ( E IR+ and
2h = $ He
E
R.
(10)
We enforce non- negativitt via softplus output activations. These parameters define our neural con- ditional intensity estimate:?
Å(t, (H:)) = H. + (a H. exp (-9 ((mmma)
(t-maat
(11)
This parametrization both flexible and interpretable. Indeed, immediately after incorporating new event into the history, the intensity jumps to a Over long inter-event intervals, the intensity relaxes toward p The relaxation rate is governed by 87
Although this parametrization may appear tailored to Hawkes-type dynamics, its parameters 2t aHt and 3 are themselves history- and mark deeendennn.. As result, the model can represent rich local intensity behaviors, including localized triggering patterns (e.g.. Rayleigh- or power-law-like kernels) and time-dependent baseline intensities. We highlight this empirically in Section 5.
Training: To train FIM-PP on set of sequences CΛ from our synthetic pretraining data, we select
target sequence T C2 to provide history of events and remaining sequences Cy (T) as context. We subsample Cx, truncate sequences and vary the number of marks throughout training, which enables us to apply pretrained FIM-PP in a wide range of (real-world) settings. Our train objective is the next-event negative log likelihood of the target sequence:
-E(' i(s K H.)ds Σ (t,K)TT
LNLL = REK
Å(t, H.).
(12)
We remark that also experimented with supervised sMAPE loss, the results of which can be found in the Appendix Tables, starting at Table 3. We found that this approach performs similarly well but is more computationally expensive and less flexible, which is why use the NLL model in the main text. Appendix D discusses the training of FIM-PP in greater detail.
FIM-PP can be finetuned on the train split C of an evaluation dataset, minimizing LNLL. For each iteration, a random sequence T E C in the train split is selected as the target sequence. The remaining sequences \ (T) serve as context. Finetuning progress is monitored by processing target sequences from the validation split, given the train split context.
Finetuning:
Note that the functional form of is similar the conditional intensity in Zhang et al. (2020).

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FIM 2P
Ground
Syuthetic Hawkes
mmma
1:54
4.30
226
8005
Time
Figure 2: Example intensity estimates FIM- PP on a synthetic Hawkes process with three marks, constant base intensity and exponential decaying kernels (left) and a real-world RETWEET dataset (right). Each row contains the intensity for one mark. Events of the same mark are colored ma- genta, while events for other marks are gray. For the Hawkes process, the model (blue line) esti- mate matches the ground-truth intensity level (black dashed line) closely. For the RETWEET data, FIM-PP estimates mixture of many excitatory and few inhibitory interactions.
EXPERIMENTS
In this section, repeat the experiments by Zeng al. (2024), who introduced CDiff, re- cent state-of-the-art diffusion-based marked event sequence forecasting model. They compare their method against range of intensity-baset and intensity-free baselines, on common benchmark datasets, evaluated on standard metrics. Moreover, they made their code and exact evaluation dataset splits available', which allows us to directly compare against their results. In the following, we re- call their experimental setup, describe the pretraining and application of F IM-PP, before presenting and analyzing our results.
5.1 EXPERIMENTAL. SETUP
Given sequence of events S = k()4, the task is to predict the next N E N events following S, where is the prediction horizon length. We denote the ground-truth continuation by S.N and the model prediction by S.N- We refer to the case N = 1 as next-event prediction and to N > 1 as multi-event prediction.
Prediction Task:
Evaluation Metrics:
We evaluate predictions by comparing SN with Siul across five metries. For N > 1, we report the Optimal Transport Distance (OTD) (Mei et al., 2019); event count error (RMSE,), comparing the number of predicted and true events per mark; and standard regression metrics on waiting times: RMSEAt and sMAPE For the special case N = 1, OTD and RMSE, are not applicable, and we instead report next-event mark prediction accuracy (Acc). Formal defini- tions of all metrics are provided in Appendix F.
We benchmark on five widely used real-world datasets: TAXI, TAOBAO, STACK- OVERFLOW, AMAZON, and RETWEET. These datasets vary in the number of marks, sequence lengths, and event counts, making them a strong testbed for evaluating the broad applicability of FIM-PP. We use the preprocessing and train/test/validtiion splits of Zeng et al. (2024), Appendix E contains further details, including dataset statistics and original sources.
Evaluation Data:
We compare FIM-PP against methods falling into two categories: models that learn joint distributions over multiple events, and autoregressive approaches such as FIM-PP. The first category includes Dual- TPP (Deshpande et al., 2021), HYPRO (Xue et al., 2022), and the Cross- diffusion Model (CDi ff) (Zeng et al., 2024). The second category further splits into intensity-basee and intensity-free approaches. Intensity -naeed basedd baselines are the Neural Hawkes Process (NHP) (Mei
Baselines:
httpps: comme//eeooottttekeeeeeeeee.e ab/c

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CDNI DEal-TP
FIM (inestrne) FIM lawrowhok)
1P TCDDM
True Predicted
Mark 3
IWPRO
Mark 8
114
2.4
Step (k-k+1)
9-10F9
RMSE.
SMAPE,
100
1.0
10-11
102
1.2
11-12
96
0.6
10
10
12-13
Horizon
Horizon
Event time
(a) Mean performance metrics across five different datasets for different horizon lengths.
(b) Next event prediction on Taxi after fine-tume tue-tunnng
Figure 3: shows that FTM-PP (zs) is competitive but slightly worse than the baseline models.
FIM-PP (f) however performs best among all horizon lengths. (b) shows that FIM-PP PP (f) also reliably captures patterns in the Taxi dataset.
& Eisner, 2017), and the Attentive Neural Hawkes Process (A-NHP) (Mei et al., 2022). Intensity- free baselines are the Intensity- Temporal Point Process (IFTPP) (Shchur et al., 2020), and the Temporal Conditional Diffusion Denoising Model (TCDDM) (Lin et al., 2022).
5.2 PRETRAINING, FINE-TUNING AND INTENSITY PREDICTIONS
Pretraining. We pretrain a single FIM-PP on a synthetic dataset containing 14.4M events, simu- lated from 72K point processes of diverse kernels, and sparsity levels, and varying number of marks, sequences, and events. Appendix B contains the details. The model has 16M parameters and sup- ports up to KC| 22 marks cover all evaluation datasets. Further pretraining details are provided in Appendix D.
Finetuning. In zero-shot mode, we apply the pretrained model directly all evaluation datasets, and label the results by FIM-PP (zs). We also experiment with finemming F FIM-PP on the train split of all evaluation datasets, and label these results by FIM-PP (f). FIM-PP utilizes up to 2000 sequences from the train split of an evaluation dataset as context, limited only by the maximum number of sequences seen during training. The ablations in Figure also indicate that typically much less than 2000 context paths would be sufficient. The sequences in the test split are given as context to F IM-PP. Finally, for multi-event prediction, FIM-PP simulates events autoregressively. similar to other basee)) baselines.
Due to the relatively small parameter size of FIM-PP and the pre-trained prior, fine-tuning can be performed quickly. For all datasets, fine-tuning achieved in few minutes, requiring not more than GB memory. This means that fine-tuning FIM-PP does not take longer than training the baseline models from scratch, which reportedly takes up to 4 hours (Xue et al., 2024, Sec. D.1). Fine-tuning is therefore feasible for users with small computational resources. Further speed-ups might be possible with LoRA (Hu al., 2022).
Intensity Predictions. Figure shows intensities inferred by FIM-PP in zero-shot mode, both from an unseen synthetic Hawkes process (in-distribution generalization). and from the RETWEET dataset (out-of-distributiio generalization). Morcover, Figure in the Appendix illustrates the per- formance of FIM-PP (zs) on various synthetic datasets and emphasizes that FIM-PP (zs) also generalizes well for processes with powerlaw kernels, kernel type that was not present in the pre- training dataset. This confirms that our parameterization general enough. In what follows, we quantitatively evaluate PIM-PP.
5.3 EVENI PREDICTION
Table reports OTD and sMAPEA results for N 20 and four datasets. Remarkably, FIM-PP achieves competitive performance in zero-shot mode, matching or surpassing specialized basclines on TAXI and RETWERT data. This shows that, solely from pretraining synthetic data, the model
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Table I: Performance on four real-world datasets, predicting N 20 events. Results for baseline methods were extracted from Zeng et al. (2024). We report mean and standard deviation over 10 trials for two metrics. Best results are bold.
TAXI
STACKOVERFLOW
AMAZON
RETWEET
Method
OTD
sMAPEac
OTD
SMAPEA
OTD
sMAPE&c
OTD
sMAPEA:
HYPRO
21.60=0.20
93.8+0.4
42.40+0.20
111.00+0.60
38.6
82.5+0.8
61.0320.00
10..1
Dual-TPP
24.48+0038
95.2+0.2
41.7540.25 20 117.58 42.6+0.7
86.5+2.0
61.10
106.90 1122
A-NHP
24.76+0.22
97.4+0.4
42.59 41 108.54+0.58
39.5+0.3
84.3+1.8
60.63+0.11
107 +1.29
NHP
25.11+0.27 96.5+0.s
43.79+0.15
116.95++.40
42.6+0.3
92.1+1.6
60.95
107 008
IFTPP
24.05:0.61
95.720.8
46.2840.8
115.
43.8
90.9#1.6
61.7210.15 106.71
TCDDM
22.15:0.53
90.620.6
42.1340.5
107.66
42.2:0.2
83.8+1.5
60.5010.00
106.05
CDiff
11.01:0.16
88.010.2
41.25
106. 1810.34 37.720.2
82.0+1.9
60.6610.1 10 106.1821.2
FIM-PP (zs) 23.15:0.07
76.8
49.26410
96.3611.00
46.2.0.1
128.610.4
60.2441.11
99.07
FIM-PP (f) 17.91=0.12 76.8 39 39.80=0.04
88 20 00. 81.2 91.110.1 $9.44.10.00 87. 17.19
can translate contextual patterns into accurate multi-event predictions, without any further training or supervision.
The same table also demonstrates the effectiveness of finetuning. The finetuned FIM-P2 PP (f) con- sistently outperforms both FIM-PP (zs) and all baselines, across the four datasets and nearly all metries. Additional experiments with shorter horizons (N 10, 5) and alternative metrics (RMSE,, RMSE&a) are reported in Appendix A, providing complementary view.
Figure summarizes performance across horizon lengths by averaging results over all datasets. In aggregate, FIM-PP (zs) performs on par with the baselines, whereas F FIM-PP (f) consistently outperforms them, agreement with our previous analysis. We attribute the effectiveness of fine- tuning to two factors: (i) the strong prior encoded in the model weights through pretraining synthetic distribution, which provides favorable initialization for finetuning; and (ii) the flexibility of the foundation model architecture, which enables direct access patterns in the training split at evaluation time, since these patterns can be retrieved from the context. The first point is supported by Figure 5 in the Appendix, which shows that the pretrained model converges faster and achieves better final performance than the same architecture trained from scratch.
5.4 NEXT-EVENT PREDICTION
The next-event prediction task is special case of multi-event prediction, but differs in nature. Whereas multi-event prediction requires estimating the distribution over set of future events, next- event prediction focuses on accurately forecasting a single event. This distinction is also reflected in the evaluation metrics (see Appendix F)".
Table reports next-event prediction results (N 1) on two real-world datasets. FIM-PP (zs) performs well on event-time prediction for TAXI, but struggles with mark accuracy on TAXI and with both event-time and mark accuracy on TAOBAO. These difficulties can be explained by dataset- specific patterns: sequences in TAXI often alternate consistently between two marks, pattern un- likely to appear in our point process prior. In contrast, TAOBAO is heavily dominated by a single mark and occasionally exhibits long waiting times. Again, patterns not covered by our pretraining distribution.
Compared to the baselines. which easily incorporate such patterns during training, FIM-PP (zs) cannot reproduce them accurately from the context alone. Appendix G further analyzes these out- of-distribution patterns.
When finetuned, however, FIM-PP can adapt to (some) of these characteristics. Table shows substantial improvements in mark prediction accuracy (Acc) after finetuning, and Figure illustrates that FIM-PP (f) successfully recovers the alternating pattern in the TAXI dataset. Nevertheless,
For instance, mark accuracy (Acc) targets the correctness of one event, while RMSF, compares histograms over multiple events.

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Table 2: Next-event prediction performance on two real-world datasets, displayed with mean and standard deviation over 10 trials. Results for baseline methods were extracted from Zeng et al. (2024) (we remark that they did not report next-event prediction results for the other datasets). Best results are bold.
TAXI
TAOBAO
Method
RMSEA:
Ace
sMAPE&!
RMSE&
Acc
SMAPEA
A A-NHP
0.32 #0.00
0.91
=0.01
85. 13 =0.26
0.53 #0.00
0.47 20.01
129. 13 11.35
Dual-TPP
0.34 20.01
0.91
20.01
89.12
=0.75
0.53 #0.01
0.47 #0.02
131.43
11.99
NHP
0.34
0.91
4.0.1
90.63
#0.61
0.53
0.46 40.01
133.69
IFTPP
0.38 +0001
0.90
+00.1
90.03
#0.47
0.53 #0.01
0.45 +0.01
126.01 +1148
CDiFE
0.34 +0.01
0.91
+00000
87.12
#0.61
0.52 #0.01
0.48 +0.00
127.12
+1.36
FIM-PP (2s)
0.34 +0.01
0.47
0.00
98.03
#0.92
0.13 +00.00
0.52
#d..11
112.89
#6066
FIM-PP (f)
0.45
+0.03
0.91
+:0.00
85.91
#0.86
0.16 $0.00
0.60
+0000
115.98
+0.72
remark that released source of CDIFF Zeng al. (2024) only predicts event per sequence for next event prediction. We believe that this is bug because it also produced very unreasonable results. We hence report the performance for all events, not just the last.
FIM-PP PP (f) still lags behind the baselines mark prediction accuracy (Acc). Appendix G, we suggest broadening the synthetic pretraining distribution to better capture such distinctive patterns upfront.
CONCLUSIONS
In this work, we introduced F TM-PP, the first Foundation Inference Model for inferring marked tem- poral point processes (MTPPs) from real-world data. Our experiments show that a single F FTM-PP, pretrained only on synthetic MTPP data, can already match the predictive performance of existing intensity-based MTPP methods in zero-shot mode, i.e. without any further training. Finetuning target data further improves results within only a few iterations, enabling FIM-PP to outperform competing methods the majority of evaluated tasks.
Limitations: Although our pretraining distribution is broad, it is not universal and therefore cannot capture many real-world patterns. As result, zero-shot performance may degrade under distribution shift. FTM-PP is further constrained by the maximum number of marks [K] and the maximum sequence length used during training. When these limits are exceeded, the model may not fully exploit the available context.
An important direction for future work is broaden the pretraining distribution beyond the parametrization in Eq. 6, with the goal of capturing a wider range of data patterns in zero- shot mode, and providing an even stronger initialization for finetuning. In addition, intensity-free methods have recently shown strong predictive performance (Panos, 2024). We plan to investigate how such methods can be incorporated into our amortized in-context learning framework.
Future Work:
REPRODUCIBILITY STATEMENT
Our core methodology consists of two parts: synthetically generated training data and founda- tion inference model for marked temporal point process. Data generation is described extensively in Section 4.1 and complemented by Appendix B, which covers the exact hyperparameters and de- sign choices required to reproduce our training dataset. Section 4.2 describes the architecture of FIM-PP. Training details, including hyperparameter choices and submodule sizes, are described in Appendix D. Our pretrained model, repository, and tutorials are available online".
The real-world datasets used our experiments are described in Appendix E, including dataset sizes and numbers of marks. For data sourcing and preprocessing, we follow Zeng al. (2024), as discussed in Appendix Finally, the evaluation metrics used in all experiments are described in Appendix F.
https::/ misciencee githtttthhh. 10clb..////// op wwt.itere../.....ttttot
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ACKNOWLEDGMENTS
This research has been funded by the Federal Ministry of Education and Research of Germany and the state North-Rhine Westphalia as part of the Lamarr Institute for Machine Learning and Arti- ficial Intelligence. Additionally, César Ojeda was supported by Deutsche Forschungsgemeinschaatt (DFG) Project-ID 318763901 SFB1294
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A ADDITIONAL RESULTS
A.1 PERFORMANCE ON VARIOUS SYNTHETIC DATASETS
In this section we highlight the performance of F IM-PP (zs) on various synthetic datasets coming from different processes. Figure 4 compares the estimated intensity the ground-truth some synthetic processes, including the powerlaw kernel. The model captures the essence all processes, including the oft-of-distributton powerlaw kernel.
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FIM-PP (zero-sho()
Ground Truth
Mark
Mark 2
Mark
Gamma Base Intensity
Poisson
0.00
0.55
Avss
0.50
0.45
10
15
20
25
30
20
60
60
80
022
2021
8818
0.15
2048
0.00
Powerlaw Kernel
Rayleigh Kernel
845
L.S
0.3
20
40
60
80
100
120
140
2.5
5.0
75
10.0
12.5
15.0
17.5
20
40
100
140
5.0
10.0
15.0
17.5
0.24
Intreaty
4.5
0.20
0.160
Figure 4: Intensity predictions of FIM-PP (zs) on synthetic datasets of four different process types. We remark that the model has not been trained on powerlaw kernels but still predicts them with decent accuracy.
A.2 LONG-HORIZON PREDICTION
The experimental setup defined by Zeng et (2024) covers four metrics (OTD, RMSE, RMSEA sMAPEA() and five real-world datasets (TAXI, TAOBAO, STACKOVERFLOW, AMAZON and RETWEET). Table 3, Table 4 and Table 5 contain the long horizon results for these datasets and metrics for horizon sizes N 20, N 10 and N 5, respectively.
A.3 TUN-TUNING
Due to the relatively small parameter count of 16.1M, finetuning takes just few minutes and less than 1IGB of GPU memory for all datasets. In Figure we compare the finetuning speed against training a F FIM-PP from scratch, i.e. with randomly initialized parameters. The results indicate that pre-training yields good initialization for finetuning, which converges much faster than training from randomly initialized set of parameters. Moreover, the finetuned model generally archives lower errors on the test split.
A.4 PERFORMANCE WITH VARYING CONTEXT SIZE
We also investigated the sensitivity of FIM-PP to the number of context paths passed to the model. We studied this behavior on data from three synthetic processes and evaluated the results based on the sMAPE metric. Figure 6 contains the results of this experiment. Initially, providing more context paths to the model improves the accuracy the estimated process. After a few hundred context paths, the performance saturates; more paths only improve marginally improve the estimate. Crucially, the performance does not degrade with more paths, highlighting the robustness of our model.
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(pretrained)
FIM trandom intalizatiio
Amazon
Stack Overflow
Taobao
Taxi
NLL (validation)
2,0
(validation)
ULL (alddation)
(valldation)
1.6
2,7
3.2
2.4
2.8
0.0
2.1
5000
10000
15000
5000
10000
15000
1000
2000
3000
500
1000
1500
2000
Epoch
Epech
Epoch
Epoch
Figure 5: Comparison of the fine-tuning loss curves of a pre-trained FIM-PP model versus random initialization. Note that one epoch corresponds to just one inference-path prediction and is therefore very fast. Our results indicate that the pre-training achieves faster convergence well higher loglikelihood when converged.
0.6
SMAPE
Hawkes
Sin Base Intensity
Poisson
0.4
0.2
100
500
1000
1500
2000
Context Size
Figure 6: sMAPE error against the ground truth intensity for varying number of context paths. We used 100 points per path. Our results indicate that the performance of the model does not noticeable improve for more than 500 paths, least for the synthetic datasets tested.
B DATA GENERATION
To train our Foundation Inference Model, we generate comprehensive synthetic dataset of marked temporal point processes. Each process is instance of multivariate Hawkes process. The con- ditional intensity function X(t, x H4) at time e R_. for a mark x E IC given a history marked events H, ([t,,)) ta t} is defined
(0, (. (t)++
Σ Yese (t - t2)
X(e, K H4) = max
O.
(13)
(P,R))EH
where Me is the time-dependent base intensity for mark K, "ikee the interaction kernel between and k E K. and 2 E are sampled fre-factors of the interaction, varying the interaction behavior further.
To the best our knowledge, no ooun-source solution for sampling such Hawkes processes with time-dependent base intensity functions exists. Hence, implemented efficient custom sam-
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pling library for such processes in C++. We will release the source code of this library in the supplementary material of our work.
B.1 DATASET CONFIGURATIONS.
We sample Hawkes processe instances over a of marks in equation 13 in two stages.
At first, the functional forms for the base intensities μı and interaction kernels "nne" are drawn from library of parametric functions. The parameters for these functions are then sampled from specified prior distributions. Our used functional forms and their parameters are summarized in Table 6. The parameter ranges were chosen more or less arbitrarily so that the paths look realistic. We keep these choices fixed and did not modify them based on the performance on our evaluation sets in order to prevent overfitting to those, which would be contrary to the concept of foundation model.
The pre-factors further diversify the sampled processes by introducing sparse connectivity and in- hibitory effects. For each process with interactions, we choose one of two pre-factor distributions Zsrong and Zsuarse {-1,0, 0, 1}, which differ by their induced connectivity:
Zuwong Categorical (-1 : 0.06, 0 0.4, 1 0.54)
(14)
Zyuase Categorical (-1: 0.01, 0 0.9, 1: 0.09)
(15)
In other words, for Zanong only 40% of interactions will be non-influencing, while for Zppase. 90% of interactions will be non-influencing. For influencing interactions, 90% will be excitatory, while 10% will be inhibitory.
Once the full intensity function for a process is defined, event sequences are generated using Ogata's modified thinning algorithm.
Figure 7 contains summarizing statistics of our train distribution. Aggregated, this prior is very broad. Importantly, it covers distributions of the real-world datasets in our experiments. The corre- sponding statistics for these datasets are depicted in Figures 8 to 12.
B.2 DATASET SIZE
We sample instances of every Hawkes process configuration in Table 6, and simulate them for differ- ent number of marks, sequences and events, detailed in Table 7. In total, training data consisted of 72k processes and 14.4M events.
C INSTANCE NORMALIZATION
To ensure that FIM-PP can generalize across datasets with vastly different time scales, we introduce an instance normalization scheme that makes the model agnostic to the absolute units of time. Let C (S) -ー denote context of F FIM-PP, iee set of marked event sequences S {(t), k/) 1 Identifying to = O, we define the inter-event times as At! t -t/_ and the maximum inter-event time in the context
pcont
max
max
max by...
(16)
All reletrelated inputs to the model, including context event times t/, inter-event times features Ati. and history event times (c.g. from target sequence during training) t, are scaled by the maximum inter-event time:
(17)
A
fcom
max
This transformation maps all temporal information canonical scale where the largest inter-event gap becomes 1.
This change of time variable also transforms the intensity function. To preserve the number of expected events within differential interval, the intensities must be related by X(t)dt = (('))dtt. Since dt Atset dt', it follows that the intensity in the normalized time domain, X X(t'), is scaled
version of the original:
X((()
A cont X(t)-
(18)
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Synthetic Hawkes Process Dataset Analysis (22D, train split)
Aggregate Event Rate (train split)
Aug (Trjectory
Time tho m time, since, start!
Aeg. Erents (tagetoryy 1imme
per split)
Time tho time sinoe_ start
Raster Plot Sampled Trajectories from train split)
12
Evert Na 55
time dinco_startttt
My
Average Event Count Heatmap (train split)
My
Sequence Length
Sequence Length
Event Type Distributtinn
1000
700
Figure 7: Distribution of our synthetic training data for Hawkes processes with 22 marks. The distribution is almost uniform and very broad and therefore also captures real world phenomena.
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EasyTPP Dataset Analysis Amazon (16D, train split)
Aggregate Event Rate (trein split)
Ang
Time thom time, since, start)
per split)
lvg Evensy Trojector Time
Mak-1
Mah
Mah.1
Mak.6
11
Mark 14
Rester Plot Sampled Trejectories from train split)
Event
Mars
Sampled Deajettryy
My'
Time (fro Bine Sinoo Start]
My
Average Event Count Heatmap (train split)
time_sinc_sttart
Sequence Length
Sequenoe
Event Type Distribution
300
Evere 3ype
Figure 8: Amazon dataset statistics.
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EasyTPP Dataset Analysis Taxi (10D, train split)
Aggregate Event Rate (train split)
Avg. Events Trajectory Time
sinne start)
Event Rate per Mark (train split)
Time
Time (from time, since start
Rester Plot (8 Sampled Trajectories from train split)
III
III
B1
Evert
Sampled Treectrry index
IBI
1:111
B1
Pano
1101 I IHI1R 11:l1
I1III
111
Paxc
per
Time (from time www.enenceeeee 22art
Average Event Count Heatmap (train split)
Mar
marh
Bits Eimee Sinc slar]
Sequence Length Distribution
PreQucTiy
11.25
32.3
Secuence Lergth
2000
Th01 Coust
Caent Type
Figure 9: Taxi dataset statistics.
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EasyTPP Dataset Analysis Taobao (17D, train split)
Aggregate Event Rate (trein split)
Event Rate per Mark (train split)
Maro
223
Mak.5
Avc.
ffom time,
Raster Plot Sampled Trajectories from train split)
Event 95
11101
110
1111
II
11
Mar
11:08
0101
Myt
111
time,
Ma
Average Count Heatmap (train split)
160
Hart
Time Eins (iram time, since start)
Sequence Length Distribution
SequeinoR Length
Type
Evert 3ypo
Figure 10: Taobao dataset statistics.
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EasyTPP Dataset Analysis Stackoverflow (22D, train split)
Aggregate Event Rate (train spiit)
Treectoryy
time sinco stort
Avg. bvenss Treectrry Time Unit
Event Mark (train split)
Mar.1
Mar-
Time thom time, since, start]
Sampled Tejectory Eniex.
Sampled Trajectories train split)
IEI 11
B1B
III!
Har 10
1111
11 BI
118
E1l
11
Mar
11
18
Mat
Timm (fram tima sinio, slart]
Mar
Mas
12
Mat
Average Event Count Heatmap (train split)
Ang tam
Mat
Met
Ma
Sequence Length Distribution
Sequencc Length
Type
Ev0re: 7pp0
Figure 11: Stackoverflow datasct statistics.
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EasyTPP Dataset Analysis Retweet (3D, train split)
Aggregate Event Rate (train split)
Everds (Bajectorry Tirun
1.00
02
sirre start)
Event Rate per Mark (train split)
Mark
Evends ("hjectory TTma
Time (from time since start
Rester Plot (8 Sampled Trajectories from train split)
18
Event
Sampled Treectrry Index
1011
III
1811
10
III
Parc
Time ffrm time sincesinceeeee start
Average Event Count Heatmap (train split)
Bits Hime since slar]
Sequence Length Distribution
154
20
Sessenoe Lergth
Eatnt Type
Figure 12: Retweet dataset statistics,
23

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Consequently, the model is trained to predict this normalized intensity (tt)). During inference, to obtain the intensity in the original, woaldworld time scale, the model's output is simply denormalized by dividing by the same constant Arae. This entire process allows the FIM to learn scale-invarians temporal dynamics, a key requirement for effective zero-shot inference on unseen data.
D TRAINING DETAILS
Our Foundation Inference Model was trained on the comprehensive synthetic datasets described in Appendix B. The training took about days on single NVIDIA A100-80GB.
D.1 DATA HANDLING AND BATCHING
Each sample in our dataset represents single underlying process, comprising up 2000 distinet time series paths. During training, we dynamically partition these paths into context and inference sets for each batch.
On-the-fly Path Selection For each sample, randomly select single path (Pinfrencce to serve as the inference target. The remaining paths are designated as the context set. To train a model that is robust varying amounts of contextual information, the number of context paths presented in each training step is randomized. Specifically, for each sample in a batch, we uniformly sample number of context paths between minimum of 400 and a maximum of 2000.
Variable Sequence Lengths As form of data augmentation, also vary the length of the his- torical sequences. For 90% of the training batches, all sequences (both context and inference) are truncated random length chosen uniformly from the interval [15, 100]. For the remaining 10% of batches, the full sequence length of 100 events is used. This strategy encourages the model to fixed,
make reliable predictions from both short and long historical contexts. For validation, full-length sequences to ensure consistent and comparable evaluation metrics.
D.2 HYPERPARAMETERR AND OPTIMIZATION
The model architecture is based on the Transformer Vaswani et al. (2017). Context sequences are processed by 4-layer Transformer encoder, and the resulting path summaries further refined by 2-layer Transformer encoder. The history of the target sequence is processed by 4-layer Transformer decoder, which attends to the context summary as memory. Both encoders and the decoder use 4 attention heads and a hidden dimension of 256. The final intensity parameters (J, a. 33) are predicted by three separate Multi-Layer Perceptrons (MLPs), each with two hidden layers of 256 units.
In total, our model has 16.1 million trainable parameters.
We trained the model using the Adam W optimizer Loshchilov & Hutter (2019) with learning rate of 5 x 10-5 and a weight decay of 10-4. To accelerate computation, we utilized b mixed- precision training.
D.3 TRAIN OBJECTIVE
We use the standard negative likelikel (NLL) for a marked temporal point process as the train objective for TIM-PP. By Section 3, the MTPP density at a sequence of events S {(t,, kL) in the interval [0, 7"
II A(t,, ki H4,) exp (-/(- A(s Ha)ds
(19)
Thus, the NLL of target sequence under the distribution induced the model's predicted intensity function A is
LNLL Taa==A(R Σ((T,K)
Σ
Á(t. He)
(20)
KEK
(C.))T
24

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where Á(T, Jo A(s H.sds is the predicted integrated intensity. We approximate the integral using Monte Carlo integration
Nuc
T NMC
A(T,R)
Σ
A(s Had)
(21)
with NMc 100 samples and 8; ((O,T))
E EVALUATION DATASETS
To evaluate the inference capabilities of FI-PPP, we use five widely-recognizee real-world datasets that were not seen during training:
This dataset comprises sequences of product reviews from users on the Amazon plat- form, collected over a ten-year period from 2008 to 2018 Ni et al. (2019). Each sequence represents the review history of a single user. event defined by the timestamp corresponds to one of 16 distinct product categories. The analysis is performed on subset of 5,200 of the most active users to ensure sequences are sufficiently long for meaningful analysis.
AMAZON
review, and its mark
TAXI Derived from New York City's public taxi trip records, this dataset captures the operational patterns of taxi drivers. Each sequence corresponds to the activity log of individual driver. Events are either pick-ups or drop-offs, and the event marks are defined by the combination of the event type (pick-up/drop-off) and the borough where occurred, resulting in 10 unique marks. The dataset consists of sequences from random sample of 2,000 drivers.
TAOBAO This dataset originates from the 2018 Tianchi Big Data Competition and contains logs of user interactions on the Taobao e-commerce platform over a period late 2017 Zhu al. (2018). The sequences track the behavior of anonymous users, including actions like browsing and pur- chasing. The event types correspond different product category groups. For the evaluation, sequences from the 2,000 most active users utilized.
STACKOVERFLOW Sourced from the popular question-and -aw-an--ndd-a---iinnnn website StackOverflow, this dataset tracks the awarding of achievement badges to users over a two-year span Leskovec & Krevl (2014). Each sequence represents user's history of earned badges. The events the timestamps when badges were awarded, and the marks are the 22 different types of badges available on the platform. The evaluation subset includes 2.200 active users.
RETWEET This dataset tracks the dynamics information spread through time-stamped user retweet sequences Zhou et al. (2013). Each sequence corresponds the retweet history of an in- dividual user. An event defined by the timestamp of a retweet, and its mark categorized into one of three types based on the influence of the original poster: "small" (fewer than 120 followers), "medium" (fewer than 1,363 followers), and "large" (all other users). The analysis is performed on subset of 5,200 active users.
For all real-world datasets, we use the pre-processing and splits from Zeng et al. (2024).
To compare against the other models, FIM-PP uses the sequences which the other models used for training as context and used the same inference sequences for evaluation.
F EVALUATION METRICS
Following Zeng et al. (2024), we adopt a comprehensive set of metrics evaluate both the temporal
and categorical aspects of the predicted sequences. Let Stuture = {(t,, N be ground truth sequence of N future events, and let Sunure be the corresponding predicted sequence. The metrics defined based on the sequence of inter-arrival times At [At..... At (where At, = t, - 1) and the sequence of marks.
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Optimal Transport Distance (OTD) We use the Optimal Transport Distance (OTD) to provide holistic measure of similarity between the predicted and ground truth event sequences (Mei et al., 2019). OTD calculates the minimum cost required to transform the predicted sequence Siunue into the ground truth sequence Siutre through series of operations (insertions, deletions, and substitutions), each associated with a cost. This metric effectively captures discrepancies in timing, marks, and the total number of events.
RMSE on Event Counts (RMSE.) This metric evaluates how well the model captures the distri- bution of event types in the predicted sequence. For each event type &: E K, we count its occurrences the ground truth sequence (C,)) and the predicted sequence (C₂). The RMSE, the root mean squared error over the vector of these counts, averaged across all m test sequences:
1 m
RMSE,
(Cj,- Cj
(22)
J-IKER
Event Type Accuracy (Ace) This metric directly measures the model's ability to predict the cor- rect event type each position in the sequence. It is calculated as the fraction of events for which the predicted mark & matches the ground truth mark Kü, averaged over all test sequences. This provides a strict, position- wise evaluation of the categorical predictions.
N
Acc
Σ
I(K, = R5,))
(23)
m
N
where I(-) is the indicator function. Unlike RMSE,, which assesses the overall distribution of event types, accuracy penalizes mispredictions at specific positions, making a more challenging metric for sequential order. A higher accuracy indicates better performance.
Time-series Forecasting Metrics To specifically assess the accuracy of the predicted inter-arrival times At. we report two standard time-series forecasting metrics.
RMSE Inter-arrival Times (RMSEA)): The standard root mean squared error between the predicted and true vectors of inter-arrival times.
N
1
RMSEA
(Atss Âtz,)²
(24)
N
111
Symmetric Mean Absolute Percentage Error (sMAPEAd): A normalized version of MAPE that less sensitive to outliers and zero values.
sMAPE 100
1
2| Adi
(25)
At + Atr.
G CHALLENGES IN NEXT EVENT PREDICTION
Our evaluations in table 2 reveal that FIM-PP in zero-shot mode already performs well for next event time prediction. It however gets a noticeably worse error in the next event type prediction. Up on investigating this, found that many of the real-world datasets have specific patterns (such as oscillations between two marks) that F FIM-PP (zs) struggles to capture (see fig 13). After fine- tuning, it is however able to spot these patterns well (see fig 14). This might also explain why F FIM-PP performs better on long-horizon tasks: The specific order of the events does not matter here.
We hypothesize that the underlying reason for this shortcoming is that our synthetic dataset distribu- tions does not capture these patterns well. We are planning to investigate this further and update our synthetic distribution to include such patterns and provide an updated version of FIM-PP.
26

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Published as a conference paper at ICLR 2026
FLM-PP (zeco-sbot) Taxi Dataset Next Event Prediction
1-2
Trise
Preeliztid
3-4
Mark
Mario
4-
509-0-k11
6-9
7-8
9-10
10-11
12-13
1.2
1.4
Figure 13: FIM-PP in zero-shot mode struggles to predict the next event type right the dataset has alternating patterns such as here for the Taxi dataset.
FIM-PP (fine-tuned) Texi Dalasel Nexl Eveal Prediction
1-2
True
Predicted
3-44
Marits
Marta
4-5
5top 00-6-11
6-
7-8
9-10
10-11
12-13
1.8
Figure 14: After fine-tuning, FIM-PP is able to spot the alternating pattern between mark 3 and mark 8 in the Taxi dataset.
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Table 3: Prediction of N 20 events in test sequences of five real-world datasets. Error-bars indicate the standard deviation over 10 trials. Results for the baseline methods were extracted from Zeng et al. (2024). Best results are bold.
Datasct
Method
OTD
RMSE.
RMSEA
sMAPEM
HYPRO
21.65320.16
1.231
0.37210.000
93. 80.
Dual-TPP
24.483:0.388
1.353:10.037
0.40211.0006
95.211 15.2114H 18.
A-NHP
21.762=0.217
1.276
0.430#00000
17.388
NHP
25.11420.266
1.297
0.399
96.459
IFTPP
21.053=0.600
1.364
0.384++0.005
95.719
TAXI
TCDDM
12.1148
1.309 0.000
0.382 1:0.019
90.596 0.5374
CDi:
21.013
1.131
0.351::0.004
87 87.993
FIM-PP (ZS) (NLL)
23.145
1.421 40.014
0.277 +00000
76. 4-0.386
FIM-PP (zs) (S-MAPE)
41.543 =0.026
3.986 10.008
0.299 #0.000
90.149 10.366
FIM-PP (NLL)
17.914 10.117
0.705
20.006
0.314 10.004
76.828 #0.540
FIM-PP (f) (S-MAPE)
16.493 20.034
0. 707 20.004
0.278 #0.001
76.538 #0.214
HYPRO
44.336=0.122
2.710::0.0 0.021
0.5941100030
134.92 044.922200475
Dual-TPP
47.324
3.237 10.044
1..00s
141. 441.687:1.44
A-NHP
45.55520.344
2.73710.00
0.70810.010
134,582:0 0,90
NHP
18.131
3.355
0.837+0.000
137. 0.
IFTPP
20.757=0.297
3.193
0.575+00.012
127. 2.7
TAOBAO
TCDDM
15.563=0.880
2.850+0.055
0.569+0.015
126.512#0.401
CDif
2.653
0.55110.002
125. 10.151
FIM-PP (ZS) (NLL)
64.281
3.949
1.988 40.006
169.687 687 4:0.089
FIM-PP (zs) (S-MAPE)
65.088 +0.037
3,251
40.007
4.042 40.025
174.814 4:0.102
FIM-PP (NLL)
60. 60.106 40.464
2.428
20.005
16.068 10.109
152.528 #0.377
FIM-PP (f) (S-MAPE)
58.900 20.436
2.347
20.014
14.734 10.120
152.547 #0.406
HYPRO
42.359
L.140+0.014
L.544+0.010
110.988 +0,339
Dual-TPP
41.752=0. 200
1.13440.01g
1.514:0.011
117 117.582 58220.422
A-NHP
42.591::0 0..911:0.0000
1.142:10001
1.340
108. 118.542 0.531
NHP
43.791=0.110
1.244 030
1.487:0.004
116,952:10 052-104401
IFTPP
46. 26.280=00894
1.447
1.669+00.00
115.122 12..222
STACKOVERELOW
TCDDM
12.128=00690
40.074
L.15++0.004
107. 17.55599000 +0.94
CDifE
41.245=1.400
1.141
1.199
106.175+0.340
FIM-PP (ZS) (NLL)
49.259
2.393
10.015
1,068 1:0002
96.364 1:0046
FIM-PP (zs) (S-MAPE)
43.630 10.242
1.488
10.010
1,050 1:0000
93. 58.544 1:0485
FIM-PP (0) (NLL)
39.79 792 20.042
1,336 10.000
1.018 4:0000
88 248 1:0.189
FIM-PP (f) (S-MAPE)
39.245 20.056
1.407 20.022
1.008 20.003
88.277 #0.231
HYPRO
38.6 05.61330.5500
2.007 +0.054
0..070000010
82.506 +00800
Dual-TPP
42.646 22.646=... 60.732
2.562 000202
0.482+9.011
86. 06.453+-2.00 +-2.044
A-NHP
39.480=0.350
2.166.20.00
0.476=200089
84.323
42.571
2.561 060
0.519+0.023
12.05331.555
IFTPP
43.820=0.232
3. 2..55010.266 286
0.481 14:
90.910#1.611
AMAZON
TCDDM
12.245=00114
2.998+t0.110
0,476++0..11
83. 15.886++1500
CDifE
37. 37.728=00199
2.091
0.44-449.086
81. 81.987
FIM-PP (2S) (NLL)
46.219 10.108
2.073 10.012
0.464
128.635 +0.305
FIM-PP (zs) (S-MAPE)
45.153 20.092
2.106
110.009
0.453 10.002
124.476 10.494
FIM-PP (0) (NLL)
37.208 10.098
2,030 10.019
0.366 10.001
81.188 10.144
FIM-PP (f) (S-MAPE)
37.454 10.060
2.116 110.002
0.361 1.0002
87.257 1:0.196
HYPRO
81.031#0.0m
2.623 00.008
30,100
106.110
Dual-IPE
61.095=00101
2.67940..02
28.914
106.900
A-NHP
60.634+0.00
2.56140.00
28.812++0227
107. 1.299
60.953=0.om
2.651 045
27.130++0.24
107. 107.075 1.339
IFTPP
61.715=0.152
2.776:10.044 0443
27.58241..11
(06.711#1.615
RETWEET
TCDDM
60.501=0.08
2.387 0.0005
27. 27.303++..152
(06.048 10.600
CDif
60.661 =0.101
2.293 00.044
27.101+0.118
106.184+1.121
FIM-PP (2S) (NLL)
60.238
4.172 +0.064
24.057
99.069 +0.390
FIM-PP (28) (S-MAPE)
59.392 10.149
4.323
40.042
24.804 40.014
106.317 10.187
FIM-PP (f) (NLL)
59.437 20.082
2.703
10.012
21.985 1.0004
87.585 10.171
FIM-PP (f) (S-MAPE)
59.150 10.061
3.081
10.020
21.800
1.0025
87. 10.109
28

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Table 4: Prediction of 10 events in test sequences of five real-world datasets. Error-bars indicate the standard deviation over 10 trials. Results for the baseline methods were extracted from Zeng et al. (2024). Best results bold.
Datasct
Method
OTD
RMSE.
RMSEA
sMAPEA
HYPRO
11.875=0.172
0.764.10.000 0006
0.36311..00
89.52410.555
Dual-TPP
13.05820.22
0.966
0.395:10.003
90.812 40.47
A-NHP
12.5122m.336
0.823
0.376:00000
92.81 22.8124 10.12
TAXI
NHP
13.377=0.184
0.922:
0.397+0.000
92.182
IFTPP
12.765=0.106
1.0044
0.383=+0.016
93.120 1..12000.520
TCDDM
11.885
1.121 +00002
0.385+t-0.000
90.703 0:336
CDi1
11.004
0.785
0.350
90.721 +:0.291
FIM-PP (ZS) (NLL)
13.820
1,190 +0.013
0,281 40.001
78.141 4:0.414
FIM-PP (zs) (S-MAPE)
19.425 10.127
1.959 10.015
0.299 #0.001
90.510 20.267
FIM-PP (NLL)
8.336 20.071
0.451
20.006
0.291 10.004
75.366 10.160
FIM-PP (f) (S-MAPE)
8.175 20.052
0.465 20.004
0.275 #0.001
73.529 10.138
HYPRO
21.547
1.527:0.008 0.005
0.0011
133.147:00
Dual-TPP
23.691 =0. 203
2.674:1..03
0.873+0.0110
139 20.:2211:00 00.348
A-NHP
21.68320.20
1.5141100015
0.60810.011
135.271 1.0.995
TAOBAO
NHP
24.068=0.331
2.769+0.008
0.855+0.013
137. 0.
IFTPP
23. 23.195=0.039
2.129
0.602+00.887
TCDDM
21.01220.52
2.598
0.610+0.022
CDif
21.221 10.176
1..16
0.535 10.016
10.366 10,366
FIM-PP (zS) (NLL)
31.880
2.024
1.955 40.011
170.278
4.0029
FIM-PP (zs) (S-MAPE)
32.249 10.041
1,822
40.004
3.538 +0.039
172.688
400194
FIM-PP (NLL)
27.974 10.162
1.325 20.010
14.954 20.253
145.821
11.120
FIM-PP (f) (S-MAPE)
27.940 20.075
1.358 20.009
14.844 20.055
153.499 #0.558
HYPRO
21.062#0.372
0.92 +0.0119
L.235+0.000
107.566 007.566+0.211 +0.218
Dual-TPP
21.229 394
0.936
1. 1.223:10.010
107 27.
A-NHP
22.019
0.978 023
1.225:0:007
1.710.167
STACKOVERFLOW
NHP
21.655=0.30
0.970
1.266+20.00
108. 008.867:1..08
IFTPP
22.330
0.970
1.251+0..00
405.674+t..3
TCDDM
22.042=0.198
1.205=+0.014
1.228++0.010
108. 108.1114 10.111
CDifE
20.191 20.455
0.916=200010
1.180+0.003
(02.367+0.260
FIM-PP (ZS) (NLL)
23.527
188 10.005
1,039 100003
92.919 1:0.56
FIM-PP (zs) (S-MAPE)
21.182 10.14
0.857 10.010
1,047 1:0.00
93.715 1.0.88
FIM-PP (0) (NLL)
19.938 20.093
0. 823 10.010
1.012 4:0004
87.503
4:0.00
FIM-PP (f) (S-MAPE)
19.846 20.152
0.832 20.010
1.004 20.003
87.110 +0.393
HYPRO
24.956=
1.765 0.009
0.442+00001
83.401 +:1.03
Dual-TPP
25.929 2.00
2,098 10.10
0.475++90000
82, 22.352 28.
A-NHP
24.11620.660 607
1.741 100039
0.454:10001
84.323
AMAZON
25.730=0.mm
1.843 053
0.491
89.135 1..02
IFTPP
26. 0.119
L.955 112
0.464110.066
89. 1..385
TCDDM
25.091 20.227
1.778
0.448++0.082
$2.105+1.500
CDifE
24.230
1.766
0.450+0.049
82. 12.124+2.000
FIM-PP (2S) (NLL)
21.736 10.113
1.141
40.010
0.449
120.894 +0.393
FIM-PP (zs) (S-MAPE)
21.418 10.142
1.198
10.008
0.445 100002
118.944 10.487
FIM-PP (0) (NLL)
18 428 10.124
1.091
110016
0.361 1.0001
87,264 10.323
FIM-PP (f) (S-MAPE)
18.566 20.072
1.153
10.000
0.352 1.0.002
82.555 1:0.276
HYPRO
31.743=0.00m
1.927 00.07
33,683 6..63343244
105.073 0733++0000 0.9.58
31.652 0.075
1.963
28,104++...0
106.72 +-0.774
A-NHP
30.337
1.823
26.31044
106.02 1:1011
RETWEET
30.817 080
1.71340.00
27.010+0.420
107.053+1.300
IFTPP
31.974=0.082
1.942.0.00 060
28.825:+0.221
106.07 016.014-:..55
TCDDM
32.006=0.074
1.789
29.124:1:.405
106. 106.738 73.
CDifE
31.237=u.ur
1.745
26.429+0.201
105.767 +0.771
FIM-PP (2S) (NLL)
31.027 20.031
2.355
40.032
27.085
97.590 40.152
FIM-PP (28) (S-MAPE)
30.986 10.104
2.396
40.012
28.305
#0.066
105.397 +0.496
FIM-PP (f) (NLL)
30.592 20.037
1.611
10.031
25.021
1.0.044
86,875 10.108
FIM-PP (f) (S-MAPE)
30.78 788 10.051
1.652
10.018
24.836
1.0044
85.27 35.222 100086
29

---

Published as a conference paper at ICLR 2026
Table 5: Prediction of events in test sequences of five real-world datasets. Error-bars indicate the standard deviation over 10 trials. Results for the baseline methods were extracted from Zeng et al. (2024). Best results bold.
Datasct
Method
OTD
RMSE.
RMSE&
sMAPEMr
HYPRO
5.952.10.128
0.500
0.32210.004
85. 0.
Dual-TPP
7.534
0.63611..00
0.34011..003
89. (0.7277:0 0..322
A-NHP
6.441
0.682+0.0010
0.317
89.070+0.15
TAXI
NHP
7.415=40.122
0.6-11+0.011
0.351+0.000
91.625++0.17
IFTPP
7.209=0.184
0.608
0.335++0.00
N0.512 169
TCDDM
5.877#0.008
0.648 +0.015
0.327 1:0.05
88, 05.00:11.00000 0.00
CDi:
5.966#s0.88
0.547 000.00
0.318
89. $9.535+:00 0.024
FIM-PP (ZS) (NLL)
6.773
0,655
0.246 +00001
74.912
FIM-PP (zs) (SMAPE)
6.763 20.018
0.654 10.005
0.248 #0.001
76.402 00.802
FIM-PP (NLL)
4.083 20.032
0.311 100007
0.250 10.002
71.108 100900
FIM-PP (f) (SMAPE)
4.103 #0.029
0.315 +0.005
0.248 10.001
73.187 #0.409
HYPRO
11.31720.111
0.817100007
0.57310.011
133.837:10
Dual-TPP
13.28 13.280=0.0m 092
1.87710.014
0.691 10.007
134.437:10.
A-NHP
11.223
0.87310.022
0.55010.001
232.26610.583
TAOBAO
NHP
11.973+0.178
1.910++00081
0.712+0.011
131.693++..er
IFTPP
11.0522t.100
+0.049
0.601
126.320+t.690
TCDDM
11.609
1.690#0.023
0.675+0.000
129.0094 +0990099
CDif
10.147 10.140
0.730 10.019
0.51910.00
124. 334.339:10 1:0332
FIM-PP (zs) (NLL)
15.951
1,129
1.761 40.013
168, 299 #:0.249
FIM-PP (zs) (SMAPE)
15.955 #0.034
1,106 40.007
1.918 10.026
167 167.486 4:0.199
FIM-PP (NLL)
13.173 20.261
0.745 10.010
14.892 20.370
146.921 20.858
FIM-PP (f) (SMAPE)
13.572 10.166
0.759 10.005
17.384 10.416
150.562 #0.830
HYPRO
11.590
0.586++o.ro +0.09
1,227 40.018
109.01 (09.014+v..0 0.422
Dual-TPP
11.719
0.591 1:0026
1.29610.0110
106. 60780.378
A-NHP
11.595=20.19
0.57511.0000
1.18840.014
105. 015.79910.55
STACKOVERFLOW
NHP
11.807
0.59611.011
1.0611
108. 008.07440.66
IFTPP
13.12440.174
0.702+60.08
1.182+00039
108. 409
TCDDM
11.410=a0.19
0.630+to.015
1.20140.022
107. 893
CDifE
10.735=0.183
0.57 000012
1.153+0.011
100.586#0.
FIM-PP (ZS) (NLL)
11.520 10.057
0.657
1,030 100001
93.296 1.0506
FIM-PP (zs) (SMAPE)
11.334 10.055
0.631 1:0.004
1,020 10.000
91.864 1.0.43
FIM-PP (0) (NLL)
10.355 #0.051
0.527 4:0.004
0,990 4:0003
86.443 1:0.128
FIM-PP (f) (SMAPE)
10.341 20.013
0.525 20.007
0.984 #0.003
86. 1 133 #0.212
HYPRO
9.552 #0.122
1.397 +0.033
0,433++0.00s
82. $2.847 0.7.8
Dual-TPP
11.309 +0,093
L.742
0.476+0.010
86, 36.633++..8
A-NHP
9.430=2.131
1.11710.049
0.427:0.039
83.121 10.11....15
AMAZON
11.27322.108
1.431
0.501
90.59110.c6
IFTPP
10. 10.230=0.20
1. 1.66310.11 168
0.447
88. 20,90000..100
TCDDM
10.55770.30
1.409+0:203
0,460++..a22
82, 42.401+0.810
CDifE
9.478=0.00
1.326++0.022
0.421+
81.287 81.287
FIM-PP (2S) (NLL)
11.124
0.736 10.004
0.449 +0.004
119.129 40.746
FIM-PP (zs) (SMAPE)
11.029 10.033
0.735 10.005
0.445 10.003
117.793 1.0.65
FIM-PP (0) (NLL)
10.034 10.060
0.737 4:00006
0.341 10.004
78, 738 1:0.33
FIM-PP (f) (SMAPE)
10.004 10.019
0. 732 1.0.003
0.343 10.004
79.623 1:0.54
HYPRO
16. 14.44.00006
40.000
27.236++0.250
+
Dual-IPR
16.0500a.088
1.077
31.493++..100
101.322 1.117
A-NHP
16. 1e.1124+00000
1,058 0.009
29.247
105. (05.930+1
RETWEET
15.945=2.00m
1.113310.00
32.367+0.100
107. 107.022+11.0 1.077
IFTPP
16. 0..
1.313:10.011
30.853++0.110
(06.941+2.00
TCDDM
15.874220005 053
1.194
28.530+t..110
105.55 (05.570=0..m 0494
CDifE
15.8588#0.090
1.023 03.
26.078
106.620+1.000
FIM-PP (2S) (NLL)
15.747 +0.032
1.342 40.027
28.138
98.668
FIM-PP (28) (SMAPE)
15.780 $0.017
1.331
40.030
28.472
40.071
100.765 +0.803
FIM-PP (f) (NLL)
15.645 10.020
1.033
10.034
25.308
1:0.135
83.010 10.278
FIM-PP (f) (SMAPE)
15. 16.585 10.040
1.012 10.040
25.413
10.086
84.945 10.343
30

---

Published as a conference paper at ICLR 2026
Table 6: Summary of the parametrized base intensities and interaction kernels of Hawkes processes used in our synthetic data generation. The parameters of each configuration are sampled from uni- form distributions, covering wide range of processes.
Dataset Configuration
Base Intensity p(t)
Interaction Kernel 7())
Parameter Distributions
Constant
Constant
Exponential Decay
<o 11030
Exponential Kernel interactions)
a(t) = co
71) = ae Pt
a 240.0055 L.0)
(95) = 0)
3 (000001, 10.0)
Constant Base & Exponential Kernel
Constant
Exponential Decay (1) de Pt
< (1(0001,
a(t) = co
a 24(00005, L.0)
3 Z((00001, 10.0)
Sinusoidal Base Exponential Kernel
Sinusoidal
Exponential Decay
0 24(0005 0.15)
((t) = Asin(u(t - 1)) + 0o
(1) de Pt
24(000 10.0)
w 4(0.11 15.0)
7 110.00
~ ((0.1,0.6)
8 ~ (4(00,
Gamma Base & Exponential Kernel
Gamma Shape Constant
Exponential Decay 7(1) = ac -ir
~ (10.1,
p(t)
A +
A ~ 4(10.0, 50.0)
p ~ ((100, 2.0)
U(L00, 10.1)
24(0.005, L.]
8. 2(00.001, 10.0)
Poisson Process
Constant p(t)
Zero Kernel (1) 0
CD 24(0011,
Constant Base & Rayleigh Kernel
Constant
Rayleigh
(00001, 1.3)
((()
(()) = op (t-tas)) exp( (t-ta)))
((00001, 1.0)
41 4(0.55
Gun (000.0..))
Table For each dataset configuration, we sample Hawkes processes with varying numbers (#) of marks, sequences and events per sequence.
# Marks
# Samples
# Sequences
# Events per Sequence
1000
2000
100
5
1000
2000
100
1000
2000
100
15
1000
2000
100
22
5000
2000
100
31


Reviews:
[{"review_id": "MH6VjI2UfE", "reviewer": "Reviewer_BVzB", "summary": "This paper introduces FIM-PP (Foundation Inference Model for Point Processes), a Transformer-based model for inferring conditional intensity functions of Marked Temporal Point Processes (MTPPs) through in-context learning. Instead of training a dedicated model for each event dataset, the authors pretrain FIM-PP on a large synthetic corpus of Hawkes processes, learning to infer conditional intensities directly from event histories and contextual sequences. Once pretrained, the model can be applied to new datasets, zero-shot or fine-tuned with minimal data.", "questions": "1. How does FIM-PP perform on non-Hawkes synthetic processes (e.g., power-law kernels or renewal processes)?\n\n2. What is the sensitivity of zero-shot performance to the number of context sequences provided?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. The idea is interesting.\n\n2. The paper pioneers the application of foundation inference models and in-context learning to temporal point processes—an original and timely contribution aligning with the broader movement toward general-purpose dynamical system learners.\n\n3. The writing is good.", "weaknesses": "1. The pretraining prior is entirely Hawkes-based; real-world processes often deviate from this form (e.g., non-linear, periodic, or exogenous influences). The authors do not quantify or discuss this distributional gap.\n\n2. While the concept of amortized inference for MTPPs is appealing, the paper provides little theoretical justification for why in-context learning generalizes to unseen, potentially non-Hawkes dynamics.\n\n3. Transformer-based FIMs may struggle with long event sequences and large contexts. The paper lacks a complexity or runtime analysis.", "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": "0FFt1ESprG", "reviewer": "Reviewer_VHQu", "summary": "The paper proposes FIM-PP, a marked temporal point process foundation model that extends the foundational inference model (FIM) pretraining paradigm. Specifically, FIM-PP simulates synthetic data from several predefined parametric intensity functions and leverages an autoregressive attention-based encoder-decoder framework, enabling in-context learning on novel datasets to estimate the parameters of a Hawkes process with exponential decay. Experimental results over long time horizons demonstrate that the proposed approach achieves competitive zero-shot and fine-tuned event prediction performance on four datasets, with mixed results for next-event prediction compared to dataset-specific baselines on two datasets.", "questions": "- Are all model parameters retrained during fine-tuning of FIM-PP?\n- Could you clarify why FIM-PP achieves mixed next-event prediction results compared to dataset-specific baselines? Additionally, how does the model's performance vary as the prediction time window increases beyond N=1? What is the largest value of N for which the model can still predict with reasonable performance?\n- Table 2: Could you provide complete results for all baselines across all datasets?\n- Are the representations learned by FIM-PP transferable to alternative dataset-specific models for initialization, in order to support dataset-specific modeling assumptions?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The paper is well written and easy to follow.\n- The adaptation of FIM to marked temporal point processes using an autoregressive attention-based encoder-decoder framework that enables in-context learning is both interesting and novel.\n- Experimental results over long time horizons demonstrate that the proposed approach achieves competitive zero-shot and fine-tuned event prediction performance on four datasets.\n- The paper provides both the model and data, enabling reproducibility.", "weaknesses": "- The performance on novel datasets appears to depend heavily on the quality and diversity of the generated synthetic data. A focused discussion on how the synthetic data is generated, namely, which parametric intensity functions are assumed would be valuable. Additionally, it would strengthen the submission to include an experimental or theoretical analysis quantifying uncertainty at inference time, particularly in relation to the similarity between inference dataset and the synthetic data used during training, similar to approaches in Gaussian processes.\n\n- FIM-PP consistently estimates the parameters of a Hawkes process with an exponential decay kernel for any novel dataset. However, it is unclear how this approach would generalize to other triggering kernels, especially those with localized effects. The paper should also discuss and include benchmarks against methods such as [1,2], which assume more flexible triggering kernels for Hawkes processes.\n\n- The proposed approach appears computationally intensive compared to dataset-specific methods, especially given the mixed performance gains observed in zero-shot tasks. The paper should provide a detailed discussion of the computational complexity of the proposed method, including the number of parameters, memory requirements, training and inference time, and how the model scales with the number of marks. It is also unclear why the number of marks is capped at 22, particularly since many clinical MTPP datasets, such as MIMIC, contain more than 22 marks.\n\n- The results for next-event prediction, when compared to dataset-specific baselines on two datasets, are mixed and warrant further discussion.\n\n- The proposed approach does not explore transferring the learned FIM-PP representations to alternative dataset-specific models for initialization, which could enable the incorporation of dataset-specific modeling assumptions. Exploring this aspect would strengthen the submission.\n\n**References**\n- [1] Isik Yamac et al., \"Hawkes Process with Flexible Triggering Kernels\", MLHC  2023.\n- [2] Pan Zhimeng et al., \"Self-adaptable point processes with nonparametric time decays\", NeurIPS 2021.", "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": "C4VrDLcCZK", "reviewer": "Reviewer_mxko", "summary": "Until now, there have been no general-purpose foundation models for temporal point processes (TPPs). This paper introduces FIM-PP, the first foundation model for TPPs. FIM-PP can perform on real-world data as zero-shot model and can also be fine-tuned quickly. The paper's main contribution is this first attempt at a foundation model for TPPs. While many foundation models exist for vision and text, none existed for this task. Quantitatively, the model shows competitive performance in a zero-shot setting and outperforms the baselines when fine-tuned (Fig 3 and Table 1).", "questions": "1. Regarding the first weakness, the use of only five datasets may limit the ability to fully evaluate the foundation model’s generalization capability. Does the model perform well across diverse domains and dataset characteristics? If possible, it would be helpful to see the model’s performance evaluated on datasets with varying domains and dynamics.\n2. Concerning the statement, “In contrast to some other neural methods, … we design our model to parametrize TPPs per mark conditioned on the joined history of all marks,” is this modeling approach original to your work?\n3. With respect to fine-tuning the foundation model, how quickly can the reported performance be achieved?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- As the authors note, this work is the first to propose a foundation model for temporal point processes (TPPs), thereby introducing a new and challenging task that sets an important direction for future research.\n- The paper is well-organized and clearly written, making it easy for readers to follow the proposed methodology and understand the results.\n- The foundation model demonstrates strong performance in estimating intensity functions - event time predictions (Table 1).", "weaknesses": "1. The claim of establishing a \"foundation model\" for Temporal Point Processes (TPPs) is not fully substantiated by the experimental evaluation. A core capability of any foundation model is its versatility across diverse data types and domains. The evaluation is limited to five datasets (Taxi, Taobao, StackOverflow, Amazon, Retweet), which primarily cover only two broad domains: Social/E-commerce and Transportation. To truly validate the model's generalized knowledge, the evaluation should be expanded to include critical TPP application domains such as Healthcare (MIMIC, EHR) and Finance (Financial Transactions), as noted in the TPP literature (e.g., [1], [2], [3]). The current limited domain coverage makes it difficult to definitively assess the model's claimed generalizability.\n\n2. The model exhibits a noticeable weakness in event type prediction, which is a critical measure for TPP models. Specifically, the results on the Taxi dataset show a significant performance gap in event type prediction. The fine-tuned model's performance (e.g., 0.69) is notably lower than that of established baselines (e.g., 0.91). While strong time prediction performance is valuable, the poor type prediction suggests the model struggles to accurately capture the inter-dependencies between different event marks, which is essential for complex marked TPPs.\n\n3. A key argument for using a foundation model is its rapid adaptability to new, specific tasks, as highlighted in the abstract (\"rapidly fine-tuned to target systems\"). The paper misses a quantitative analysis of this fine-tuning efficiency. Metrics such as the time or computational resources required for fine-tuning (e.g., convergence speed, epochs to reach peak performance) should be presented and compared against baselines to validate the model's advantage in rapid task adaptation.\n\n[1] Shchur, Oleksandr, Marin Biloš, and Stephan Günnemann. \"Intensity-free learning of temporal point processes.\" ICLR (2020).\n\n[2] Mei, Hongyuan, and Jason M. Eisner. \"The neural hawkes process: A neurally self-modulating multivariate point process.\" NIPS (2017).\n\n[3] Liu, Zefang, and Yinzhu Quan. \"Tpp-llm: Modeling temporal point processes by efficiently fine-tuning large language models.\" ICLR 2025 Workshop on Foundation Models in the Wild.", "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": "8bJqABxcxn", "reviewer": "Reviewer_UZoj", "summary": "*disclaimer* i did not use llm at all for reviewing this paper. \n\nThis paper proposes a foundational model approach for event prediction commonly modelled by marked TPPs.  The approach is modeling the conditional intensity given history context in the pretraining, specifically modeling the parameters of a modifed hawkes process intensity that inclusion excitation and inhibition effect.  Authors provides a zeroshot and fine tune version. The fine tune performs better on next event prediction on 4 real benchmarks. For multi event prediction task results are less prominent.", "questions": "1.\tHave the authors look at other transfer learning based approach for TPP models? Whats the benefit of modeling conditional intensity  over directly predicting time and type in the transfer learning setting? \n\n2.\tFor multi event prediction,  I think the current approach is still autoregressive, the FIM-PP models are less convinced to me that they will be better than CDiff / Dual -TPP models – since the latter ones are for long horinzon prediction. It will be beneficial to provide some theoretical insight?", "limitations": "", "rating": 2, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "A few strengths include 1) a new foundational model perspective on modeling conditional intensity.  The paper can be improved by mentioning/citing papers with other angles for foundational model for marked TPPs, ( I am sure there are a few ) and highlight how theirs differ from others) 2). Presentation is very clear, easy to follow 3). I think the thoroughness of experiments is reasonable – may include a few other datasets from benchmarks such as EasyTPP.", "weaknesses": "A few weaknesses include 1). The expressiveness of the modeling of conditional intensity form (can you think of something more expressive? 2.) I think the fine tune version is much prominent in downstream tasks, so it kind of diminish the claim that the transfer learning approach setting one does not have to train on individual datasets. The zeroshot version is okay. For fine tuning the authors should think about computation costs as well compare to baselines. 3). The author should consider add Transformer based baseline for a fair comparision since the proposed approach is transformer based.", "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)