Stochastic Optimal Control for Continuous-Time fMRI Representation Learning
Authors: Joonhyeong Park
Byoungwoo Park
Chang-Bae Bang
Jungwon Choi
Hyungjin Chung
Byung-Hoon Kim
Juho Lee
Venue: ICLR 2026 Poster (2026)
Keywords: self-supervised learning
neural differential equations
irregular time-series
fMRI
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We formulate fMRI representation learning as a stochastic optimal control problem over continuous-time latent dynamics, unifying SSL objectives (MAE, JEPA) into a scalable framework that yields robust and compact representations.

Abstract: Learning robust representations from functional magnetic resonance imaging (fMRI) is fundamentally challenged by the temporal irregularity and noise inherent in data from heterogeneous sources. Existing self-supervised learning (SSL) methods often discard critical temporal information by discretizing or averaging fMRI signals. To address this, we introduce a novel framework that reframes SSL as a Stochastic Optimal Control (SOC) problem. Our approach models brain activity as continuous-time latent dynamics, learning a robust representation of brain dynamics by optimizing a control policy that is agnostic to the temporal irregularity. This SOC framework naturally unifies masked autoencoding (MAE) and joint-embedding prediction (JEPA) to extract compact, control-derived representations. Furthermore, a simulation-free inference strategy ensures computational efficiency and scalability for large-scale fMRI datasets. Our model demonstrates state-of-the-art performance across diverse downstream applications, highlighting the potential of the SOC-based continuous-time representation learning framework.

Published as a conference paper at ICLR 2026
STOCHASTIC OPTIMAL CONTROL FOR CONTINUOUS- TIME FMRI REPRESENTATION LEARNING
Joonhyeong Park's Byoungwoo Park', Changbae Bang", Jungwon Choi', Hyungjin Chungl 1.3 Byunghoon Kim² Juho Lee KAIST Yonsei University", EverEx (clearclouds, bw park,, jungwon. choi, hyungjin. chg@gmaine com, {changbae bang, ecystaiJ@yonsel..co
kr
ABSTRACT
Learning robust representations from functional magnetic resonance imaging (IMRI) fundamentally challenged by the temporal irregularity and noise in- herent in data from heterogeneous sources. Existing self-supervised learning (SSL) methods often discard critical temporal information by discretizing or averaging fMRI signals. To address this, we introduce novel framework that reframes SSL as Stochastic Optimal Control (SOC) problem. Our approach models brain activity as continuous-tine latent dynamies, learning robust representation of brain dynamics by optimizing a control policy that agnostic to the temporal irreg- ularity. This SOC framework naturally unifies masked autocncoding (MAE) and loint-embeddiig prediction (JEPA) to extract compact, control-derived represen- tations. Furthermore, simulation-fies inference strategy ensures computational efficiency and scalability for large-scale fMRI datasets. Our model demonstrates wate-of-the of-the--rrrtt performance across diverse downstream applications, highlighting the potential of the SOC-based continuous-time representation learning framework.
INTRODUCTION
Linear Probing Performence vs. GPU Hours HCP-A.Age Regression
(BOLD) signals, captured by iMRI, reflect localized brain activity, providing an indirect mea- sure of underlying brain dynamics (Ogawa al., 1990; Heeger & Ress, 2002). These signals imply fundamental brain dynam- ics essential for understanding cognitive functions, behavior, and clinical characteristics (Park & Friston, 2013). While super- vised learning models tailored for specific tasks like diagnosis prediction have shown promise (Kawahara et al., 2017; Li et al., 2021; Kan et al., 2022), their dependence on labeled data limits flexibility and generalizability. This motivates the development of methods that leverage vast amounts of unlabeled fMRI data to learn rich, transferable brain representations (Abraham et al., 2017; Yamashita et al., 2020; Zhang & Metaxas, 2024). Build- ing on its success in fields such computer vision and natural language processing (Devlin et al., 2019; He et al., 2020; Chen et al., 2020; LeCun, 2022; He et al., 2022), the SSL framework has recently been applied to IMRI analysis (Caro al., 2024; Dong al., 2024; Yang et al., 2024).
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BOG
Pearson Correlation
Mala
BraicMaam
8.7
Brain Mariag
Bia Mi
0
8.6
Pretrsining GPU Hours
Figure 1: Our BDO outperforms other self-supervised approaches, demonstrating superior efficiency.
Despite their success at capturing global brain representations, existing SSL methods often compro- mise the fidelity of fMRI temporal dynamics by treating the time-series data as segmented patches or static connectivity graphs as described in Figure 2. Since fMRI signals are noisy and continuously evolve over time, it is desirable for models to incorporate temporal inductive biases that naturally reflect the true nature of fMRI data to fully exploit these dynamics. However, in SSL scenarios that must integrate heterogeneous fMRI datasets, temporal modeling becomes even more complex. key difficulty arises from variability in acquisition protocols, especially in repetition time (TR), the
Equual contribution
I Equal advising.

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(b)
(0)
Figure 2: Conceptual illustration of SSL approaches for fMRI data. (a) Image-based approach (Caro et al., 2024; Dong ct al., 2024), where time-series data for each region-of-interee (ROI) is divided into fixed-size windows. (b) Graph-based approach (Yang al., 2024), where static graphs constructed to represent functional connectivity between ROIs. Both approaches discard high-resolution temporal dynamics during data preprocessing. (c) Our approach, leveraging continuous-ting latent dynamics, directly captures the evolution of brain activity over time, making it robust to varying TRs across heterogeneous datasets and preserving fine-grained details.
interval between successive fMRI measurements. For example, the UK Biobank (UKB) uses TR of 0.735-second (Alfaro-Almagro al., 2018), whereas the Autism Brain Imaging Data Exchange (ABIDE) reports TRs ranging from 1.5 to 3.0-second (Di Martino et al., 2014). Integrating such datasets poses challenges in aligning heterogeneous time scales into a common time axis, resulting in the irregularity described in Figure 3. Therefore, such irregularity highlights the need for a unified framework that can model temporal dynamics across heterogeneous datasets with diverse TRs.
Continuous-time dynamical modeling has become natural solution for irregularly sampled data in fi- nance (Black & Scholes, 1973). climate (Menne et al., 2016), and healthcare (Goldberger et al., 2000). Neu- ral differential equations (Chen al., 2018 8; Rubanova et al., 2019; Li et al., 2020; Kidger et al., 2020; Zeng al., 2023) and continuous-discret state-space meth- ods (Schirmer et al., 2022; Ansari et al., 2023; Park al., 2024) learn continuous dynamics in compact latent space rather than on the noisy, high-dimensional observation space. While integrating these latent trajectories requires costly numerical solvers, recent simulation-free and amortized inference techniques remove this bottleneck, making continuous-time modeling practical for lcrge-scale, high-dimensional datasets (Park al., 2024) such as fMRI.
Dataset A
Dataset B
Dataset
Time
Figure 3: Diverse TRs induce multi-scale
irregularities on unified time axis.
Building these advances, we propose novel SSL framework, Brain Dynamics with Optimal control (BDO), designed to explicitly model continuous latent dynamics while accounting for the heterogeneous data acquisition protocols of multi-site fMRI studies. In BDO, continuous latent state evolution modeled as a stochastic differential equation (SDE) (Oksendal, 1992), which encodes multi-scale brain activity onto single continuous real-time axis. Crucially, we fundamentally reframe fMRI representation learning as a SOC problem (Fleming & Soner, 2006; Carmona, 2016). By viewing the control signal as the driving force that aligns latent dynamics with observed brain activity, we reinterpret the optimal control policy as a learned latent encoder. This formulation naturally yields unified objective that integrates (MAE) (He et al., 2022) and the joint embedding predictive architecture (JEPA) (LeCun, 2022; Assran al., 2023), to learn robust representations. Furthermore, to enhance computational efficiency, we employ simulation-free inference strategy based on locally linear approximations of the SDE. The learned control-base latent representations serve as compact and highly transferable representations for wide range of downstream applications. We summarize our contributions as follows:
complementary SSL schemes, the masked autoencoder
We explicitly model the temporal structure of fMRI time-series via continuous latent dynamical modeling, thereby ensuring robustness to heterogeneous data acquisition protocols.
Building on the SOC formulation, we incorporate MAE and JEPA into single SSL framework yielding compact, transferable control- -base features.
A simulation-fre free inference scheme eliminates costly numerical solvers, enabling scalable and efficient training on large, high -iih-nimmniiinal fMRI datasets as shown in Figure 1.
Our experiments empirically show that the learned control-based features outperform baselines across diverse downstream tasks while retaining high computational efficiency and scalability.
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2 RELATED WORK
Task-specific Models for fMRI. Supervised deep learning has driven fMRI analysis for diagnosis and trait inference. Convolutional approaches like BrainNeiCNN (Kawahara et al., 2017), graph- based models like BrainGNN (Li et al., 2021), and transformer-basee have achieved strong performance by capturing ROI interactions or functional modules. However, these task-specific models rely heavily on labeled data and fail to learn generalizable representations.
BrainNetTF (Kan et al., 2022)
To address this, SSL has emerged as a powerful paradigm designed to overcome the reliance on labeled data. By exploiting the abundance of unlabeled IMRI scans, SSL aims to distill task-agnostic representations that generalize across a wide spectrum of downstream fMRI analyses.
Self-Supervised Learning for IMRL. Building on the success in vision and language domains (He al., 2020; Grill et al., 2020; He et al., 2022; Assran al., 2023), existing SSL approaches for fMRI generally follow two main routes as illustrated in Figure 2: image-based and graph- based approaches.
Image-based based models, such BrainLM (Caro al., 2024) and Brain-JEPA (Dong et al., 2024), treat fMRI time-series as image-like fixed grids of spatiotemporal patches. However, this rigid discretiza- tion creates fundamental temporal mismatch when applying models to heterogeneous datasets with varying TRs. For instance, even when artificial downsampling is employed to approximate a common temporal resolution, inevitably fails perfectly synchronize the diverse time scales of multi-site data, forcing the model to learn inconsistent dynamics from patches representing disparate durations.
Conversely, graph-based approaches, including BrainMass (Yang et al., 2024), compress time-series into static functional connectivity graphs, inherently discarding fine-gr aineggriied temporal dynamics. While recent dynamic graph methods (Choi al., 2024) utilize sliding windows, they still average out intra-window fluctuations and struggle to align diverse sampling rates across datasets.
To address these issues, we adopt continuous- latent dynamical modeling- While the concurrent work BrainHarmonix (Dong et al., 2025) introduces multimodal framework unifying structure and function, its approach to fMRI attempts to mitigate TR heterogeneity via adaptive patch resizing, fundamentally remaining& discrete patch-based method. In contrast, by viewing each IMRI scan as noisy observations of a continuous SDE, our model naturally aligns data with different TRs on a shared continuous real-time axis, handles variable sequence lengths without resampling, and retains the fine-grained temporal dynamics that patch-based or static-graph approaches discard.
Continuous Dynamical Models. Continuous time dynamical models have been developed to capture irregular time-series dynamics and uncertainty. Deterministic approaches like Neural and Latent ODEs parameterize smooth trajectories via neural networks (Chen et al., 2018; Rubanova et al., 2019), while GRU-ODE-B incorporates Bayesian updates for uncertainty (De Brouwer et al., 2019). Stochastic Latent-SDr SDEs introduce variational inference schemes for SDE-driven latent trajectories (Li al., 2020; Hasan al., 2021: Zeng al., 2023). Classical continuous- discrrte state-space models generalize discrete transitions to SDE-governed updates (Jazwinski, 2007), inspiring neural CD-SSM variants that leverage locally linear approximations (Schirmer et al., 2022; Ansari et al., 2023).
Although these frameworks excel at modeling irregular time time-series, they rely on costly numerical solvers, hindering scalability for high-dimensional fMRI. Recently, simulation-free approaches have been proposed to bypass expensive integration (Bartosh et al., 2025; Park et al., 2024). Inspired by these simulation-fire paradigms, we introduce a scalable SSL framework with a closed-form objective, capturing fMRI dynamics and uncertainty without expensive numerical solvers."
METHOD
In this section, we introduce BDO, a novel SSL framework for modeling fMRI time-series data using continuous latent dynamics. We formulate representation learning as a SOC problem, allowing us to unify two complementary SSL schemes, MAE and JEPA, into a single training objective, in a scalable and simulation-free manner. Proofs and detailed derivations provided in Appendix B.
We refer readers to Section A for additional related work section including detailed hackgrounds on self-supervised learning and stochastic optimal control foundations.
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3.1 MODELING CONTINUOUS DYNAMICS IN LATENT SPACE FOR FMRI
Latent Dynamics. Here, we briefly introduce the core assumptions that guide our method. Directly working with raw fMRI time-series data, which call y == (yer.'... .Yt,. is challenging due to their high-dimensional nature
(where each data point
Yr, E R")) with time stamps T = (4,,-.. , 4x and inherent noise. Instead, we introduce a latent space, where the essential structure of the data is assumed to lie in lower-dimensional space of dimension d' < n. This approach enables more efficient modeling by reducing the dimensionality of the data (Fraccaro al., 2017; Rubanova et al., 2019; Li et al., 2020; Kidger et al., 2020; Zeng et al., 2023; Ansari et al., 2023; Park et al., 2024).
Based on this, we assume that the complex sig- nals we see in brain scans are produced by some underlying hidden dynamics that unfold contin- uously over time. We model these continuous latent states X, E Rd as being governed by an Itô diffusion process (Oksendal, 1992):
Continuous Latent Dynamics
Optimal Control Pollay
dx. ft, X, )dt + att)dW,,
(1)
Ye
Yi
Ye
Demognaphcs
Tolts
Diagrosts
where ftt, )) IRd Rd is the drift, (t) E R is the diffusion coefficient, and W. E Rd is a standard Wiener process. The stochastic process [oo.r] == (Xx) hejorr will serve as the prior understanding of the system before we look the actual fMRI data V Because fMRI data too complex, it is hard to make strong initial assumptions. So, a common starting point is to simplify the prior to dX,- = (W) r. This means that we initially assume the underlying changes are purely random, acknowledging the difficulty in modeling the intricate nature of fMRI signals beforehand.
Discrete Observatins
Figure 4: Conceptual illustration of our proposed Brain Dynamics with Optimal control (BDO).
To learn meaningful representations from fMRI time-series, our goal is to infer the latent dynamics that generate the observed complex signals V. A successful latent dynamics model should serve as the representation itself, providing a compressed and powerful summary of the brain's activity. Here, we define these dynamics as the posterior stochastic process, X;, which provides the best possible explanation for the fMRI data y that we have actually observed. The posterior states X; are also governed by an Itô diffusion process (Li et al., 2020; Park et al., 2024):
dx; = (1, x;) + (t)aa (t, X7; V)jdt + (t)dW,
(2)
where a" (t,,,) :)) Hd x Rank - dd is introduced as part of the drift term. Compared to the prior process in (1), the newly added term a* steers our initial understanding of the hidden dynamics. It achieves this by incorporating the information from the actual fMRI signals ) we collected, pushing the latent dynamics towards a path that better aligns with the observed fMRI data.
Problem setup. The core idea is that the prior process is aligned with the observed data y through the term a*, which plays an exclusive role in incorporating observed information into the latent states, thereby enabling the direct extraction of meaningful representation from a* Once an optimal profile of a² is determined for the given set of data y, this representation can then be employed for downstream tasks as illustrated in Figure 4. Indeed, a" is referred to as the optimal policy, which can be developed under stochastic optimal control (SOC) theory (Fleming & Soner, 2006; Carmona, 2016; Park et al., 2024). Drawing on the SOC theory, we propose our method specifically designed to model the fMRI time-series data, with a focus on SSL to extract meaningful representations.
Stochastic Optimal Control. SOC provides a robust theoretical foundation for the alignment process by combining optimization and probability theory to determine optimal control strategies for dynamical systems (Fleming & Soner, 2006; Carmona, 2016). To estimate the optimal latent trajectory in (2) which is aligned with the observed data V/ we introduce controlled SDE:
dx.
[ttt,x)) + (t)a (t, X(: dt + ott)dW
where a (1,,.) I x Bank - Rdd represents the parameterized (with a neural network) control policy we aim to optimize and X (o.7) represent the parameterized latent states controlled by a². In
Thh time stamps are ordered as 0 < f1 <... ty < T and the intervals between consecutive time stamps be arbitrary.
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our formulation, the control policy a" acts as an encoder by mapping the observed data y to the latent space. We parameterize this encoder a" as a transformer (Vaswani et al., 2017), recognized for its effectiveness in processing sequential data such time-series observations. The objective is to approximate the optimal control policy a" by a 0000" thereby aligning X = X (0.27. This optimization is related to the variational inference framework (Chen et al., 2018), and the solution is structured as follows (Theodorou, 2015; Kappen & Ruiz, 2016; Li et al., 2020; Park et al., 2024):
Proposition 3.1 (Evidence lower bound). Let us consider the following optimal control problem:
J() = Ex)=(() [.-0
2 1,, (t,x%
Σ Elogge SET
(4)
where ge (ylx) = Ny (x), I is parameterized (Gaussian) likelihood function with a non- linear decoder network D Fd - R The negation of the objective function [ .)) coincides with the evidence lower bound (ELBO) of the variational inference for the posterior dynamics (2):
ELBO b) 2 - ,3),
(5)
where the equality holds at the optimal control policy a = arg ming Ja.
With this parameterized encoder-decoder, (a ) Proposition 3.1 shows that the optimization problem with the cost function specified in (4) is equivalent to performing variational inference within an autoen- coder framework (Kingma& & Welling, 2014).
at
attz
att
Ot
Transformer Encoder
Decoder
FFN
FFN
FFN
FFN
Moreover, when related to reinforcement learning (RL) literature, it aligns with continuous-time RL with entropy regularization (Todorov, 2006), where
Attn
Attn
Attn
O'Yn
Ytz
Yta
Yt
the integral term 1
2 ||a (t. 3") enforces regular- ization to maintain proximity to the prior process (1) Figure 5: (Left) Encoder network architecture
and the negative log-likelihood, - loog (y ()) == (Right) Decoder network architecture.
Ily - (x)] acts as reward for posterior process to explain the observed fMRI time-series. 20
Once the optimal policy a* is obtained by solving the SOC problem, we can sample from the posterior states X." by simulating the optimally controlled SDE (3) via a numerical solver (Li al., 2020).
Universal Feature from Optimal Control Policy. After obtaining the optimal policy a", we can derive a universal feature A. This feature generated by aggregating the control signals from the optimal policy across relevant set of time points or intervals TT, using aggregation function f. which gives A = f(fat generalizable representation for various downstream tasks. The rationale behind its potential as universal feature is that A. derived from the optimal policy a", encapsulates the essential data-driven adjustments required to accurately model the observed fMRI time-series V, thereby capturing salient and transferable characteristics of the underlying neural dynamics. The process of aggregating the control signals heeT to extract the universal feature A is described in detail in Appendix D.4.
The aggregated feature A is designed to serve as a robust and
3.2 REPRESENTATION LEARNING WITH STOCHASTIC OPTIMAL CONTROL
Extracting robust and transferable features from fMRI time-series is crucial for downstream clinical and research applications. However, conventional approaches to representation learning often struggle to adequately capture the intricate underlying dynamics and subtle individual variations present in fMRI signals. This frequently results in features with limited generalizability across diverse tasks.
To overcome these challenges, we propose novel SOC framework which integrates two complemen- tary SSL schemes. First, we adopt a masked autoencoder (MAE) framework (He et al., 2022). In this framework, portions of the input IMRI time-series Y are masked, and the model is tasked with reconstructing the missing target Deaz from the unmasked context Deex- This process compels the encoder a to learn rich and meaningful latent representations that capture the underlying structure and temporal dependencies. Second, to mitigate the risk of representation collapse- phenomenon where learned features become trivial or con-discriminative-- introduce auxiliary variable ä
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Continuous Latent Dynamics
Encoder
(onta
Random Masking
EMA
SDESolver (1o: Eo); D, a, (to, "o,....,h tx))
Leaminng
Without Masking
rgs
Frozen
Decoder
Regulartzation
Encoder
(Emptrical
Reconstruction
Figure 6: Overview of representation learning of BDO, Randomly masked fMRI time-series Veex are encoded into latent states; encoded control signals {al JEET... steer the SDE to predict latent states at the masked time points Tear, which are then used to reconstruct the missing observations Vear A slowly updated EMA encoder provides latent targets, preventing representation collapse.
that serves as a stable, self- self-generated latent prediction target. This mechanism, developed to ensure feature stability, shares its core philosophy with the joint embedding predictive architecture (JEPA) framework (LeCun, 2022: Assran al., 2023), which also leverages latent prediction to shape repre- sentations. Grounded in SOC theory, our formulation successfully unifies these two complementary SSL schemes into a single framework, yielding robust and informative representations A.
Masked Autoencoders. To learn the control signals that align the latent dynamics with the observed data, we use an MAE, which has been successfully applied various domains (He et al., 2022). With an encoder-decoder pair (a", D&), the MAE approach can be adapted to our SOC framework. Specifically, we randomly mask ratio y of data in V. The objective is then to reconstruct missing target V'aar based the remaining context Vatx. We denote corresponding time points Tear and Tatx- In this formulation, the optimal control policy o' determined by solving the SOC problem:
[/- 1 (e dt
a" arg min E X
Σ log (v.
(6)
EETE
Here, the control a is generated by encoding the context observations Vex. However, this approach may be suboptimal for highly noisy data modalities like fMRI as the naïve negative log- likeelholooo function, log 96 (y|x) 2az lly (x)l directly decodes the latent states X into the ob-
served raw signals yr. noisy data like IMRI, this can lead to the encoder-decoder pair (a D&) overfitting noise, as the model attempts to fit the irrelevant fluctuations in the observed signals (As- sran et al., 2023; Dong et al., 2024). As a result, the model may not effectively capture semantically meaningful features of the underlying dynamics, potentially compromising the generalizability of A.
Joint Embedding Prediction. To mitigate overlitting to noise inherent in the pure signal reconstruc- tion, introduce auxiliary variable å, the latent predictive target for the control policy ou, and incorporate additional strueture into the observation likelihood function (-(x). Formally, we propose hierarchical generative process over the likelihood function go in (6) as follows:
ge (o)x))))) = ((
where Go (y:l.) == (NyyDD 2) represents the latent decoder mapping back to the observation space, and (() is the latent distribution of the auxiliary variable â,. To ensure the latent target ār both predictable from the given context Vorx and informative about the masked target ar we specifically define the latent distribution TT r--[X.) )) geometric mixture of two components:
To(Cu x x po X MEEx) (0 X )
(8)

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where p(ā, X cxxx) = =N(ārla" (ex)) Vexx). (V (t, X Vear), I)) is the empirical prior, and E [0, balances these two components.
is the auxiliary posterior, q0 (āz X Pta:)
The auxiliary posterior pa serves as the probabilistic estimate of the latent target dG, which is generated by the primary encoder a conditioned solely on the contextual information Vatx- Essentially, this component represents the model's 's probabilistic belief about the target based solely the observed context, before integrating any information from target-specific observations via the empirical prior.
Conversely, the empirical prior qa serves as the predictive target distribution. It is generated by a separate target encoder a", which has access to the actual masked observations Ytar The target encoder parameters 0 are not updated by backpropagation. Instead, they track the online primary encoder parameters / via EMA, yielding a slowly evolving and stable reference (Assran et al., 2023).
This design ensures that qa provides consistent, evolving predictive targets and the primary online encoder a is then trained to align its predictions with the stable targets. This process is a form of self-distillation that prevents the model from overfitting to noise and helps it learn robust features A. sharing its core philosophy with the established JEPA framework (LeCun, 2022; Assran et al., 2023).
Training Objective. The above formulations allow the model to flexibly combine information from both contextual understanding and direct evidence from target data, thereby aiming to produce robust and generalizable features. Combining both modeling approaches within single SOC framework, we define the following training objective properly scaled by variances of the likelihoods o²):
E x9 (3)
So
Σ TETEE (f-
E
[ | - (a
+1 +
af =11l"]] (9)
(9)
where T (1- (1-A)02 is a balancing factor. The parameters for the initial latent state and dynamics {ug: Eg. a²} and the decoder network {D are jointly optimized by minimizing the training objective (9). The detailed derivation of this training objective is provided in Appendix B.3.
of
3.3 EFFICIENT MODELING OF LATENT DYNAMICS
The numerical simulation of (3) via SDE solvers is computationally demanding. To tackle these computational challenges, we turn to simulation methods. These methods suggest that complex data modeling can be achieved using linear SDEs (Schirmer et al., 2022; Smith et al., 2023; Ansari et al., 2023; Deng et al., 2024; Park et al., 2024), thereby avoiding costly numerical solvers. In particular, by approximating the drift function of controlled SDEs in (3) with linear model, we obtain closed-form solutions for the latent states, removing the need for numerical integration. This approach drastically reduces computational demands, which is especially beneficial for high nigh-dimensional data such as fMRL. Moreover, when combined with the assumption of piecewise locally linear dynamics, it enables robust simulation- free inference.
Theorem 3.2 (Simulation-Frre Free Inference). Let us consider a sequence of semi remiiivsitive definite (SPD) matrices DIeT where each Dr, E dRdxd admits the eigen-decomposition Dr, VA, V with eigen-busis V E &Rdxd and eigen-values Av. E diag(Tk' (RK)) for all E {1,... .k} and approximation
9 of controls o oTg,, where each of E Rdd. Then, for an interval (t (41), consider the SDE:
dx. = (-D, + +al, (2)) dt + dW.,
(10)
where X8 ~ NN Eo). Then, for any time-stamps t, E T, the marginal distribution of the solution of (10) is Gaussian distribution i.e., x! ~N(M,, ).,) whose parameters computed as
e E firo
E -4)A,, Ar. (I-eltrttiAA))) an
Pr. = V
(1I)
1=
Ee, v e -2) (1 -()1 E1
1
2 د (1 (1:)) A.. (1) )
2
1=0
where x;= = V VXx dz, VT at. W. VTW, j = V no and E. = VTS.V
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Hence, given the matrices and controls (i HET, derive closed-form solution for the latent states X Moreover, this parameterization enables the use of the parallel sean algorithm (Blel- loch, 1990), allowing parallel computation of the moments {Me, Eg Hee. Consequently, we can evaluate the objective function (9) in (log k) time. Further details can be found Appendix C.2.
Remark 3.3. A key advantage of this parameterization, once the model is trained via the SOC formulation, is that it enables the direct inference of the optimal policy a² for downstream tasks. In other words, obtaining a" does not require simulation of X for downstream tasks. Instead, any necessary instances of X3 are themselves efficiently inferred using the trained encoder.
4 EXPERIMENTS
In this section, we present empirical results demonstrating that our novel SSL method, BDO, yields superior representations for fMRI analysis. BDO was pre-trained using the large-scale UK Biobank (UKB) dataset in self-supervised manner, leveraging resting- pesting-state fMRI recordings from 41,072 participants LlffaroAAlmaee et al., 2018). To evaluate its applicability, we conducted experiments across various downstream tasks, including trait prediction and psychiatric diagnosis classification. These experiments were performed on four datasets: Human Connectome Project in Aging (HCP- A; Bookheimer et al., 2019), Autism Brain Imaging Data Exchange (ABIDE; Di Martino et al., 2014), Attention Deficit Hyperactivity Disorder 200 (ADHD200; Brown et al., 2012), and Human Connectome Project for Early Psychosis (HCP-EP; Prunier & Shenton Martha; Breier, 2021; Jacobs et al., 2024). All fMRI data were preprocessed by dividing brain into 450 ROIS, using Schaefer-400 for cortical regions and Tian-Scale III for subcortical areas (Schaefer et al., 2017: Tian al., 2020).
We compared our performance against both task-specific (TS) models and self- supervised learning (SSL) models. Specif- ically, we compared BDO with TS mod- els: BraiNNetCCNN (Kawahara et al., 2017), BrainGNN (L.i et al., 2021), and BrainNetTF (also known as BNT) (Kan et al., 2022). as
Table 1: Internal prediction on UKB 20% held-out.
Methods
Age
Gender
MSE
ACC (%)
Fl (%)
BrainNetCNN
+:18
0.621
+012
=0.14
90.87
TS BrainGNN
0,914 +.024
0.430
+000
79.07
+1.08
79.03 +1.00
BruinNelTF
0.561 +:004
0.673
2003
91.19
=0.51
91.17
well as four SSL models: MoCo (He et al., 2020) and BYOL (Grill al., 2020) as gen- eral SSL methods; BrainLM (Caro et al., 2024), and BrainMass (Yang et al., 2024) as
MoCo (90M) (90M)
0,933 2.022
0.413 2.10
80.11
20.73
80.11 20.73
BYOL (90M)
#:0
s0.13
72.97 20.13
LP Brain (85M)
0,737
+0019
0.547
+011
84.51
-0.4
84.51 +0.64
BrainMass (90M)
1.104..015
0,358
006
80.33
.0.86
80.34 20.56
BDO (86M)
0.600 x.004
0.635
=000
88.25
=0.78
88.21 20.79
IMRI-specific approaches. For a fair com- parison, all results are averaged over three runs with different data splits. The best- performing results are highlighted in bold?.
MoCo (90M)
0.751 +.025
0.545
#0013
85.95
=1.01
85.90
(YOI.(90M)
+:00
003
83.17
FT BrainLM (85M)
0,648 1.024
0,620
2.118
89.31
s1.13
89.29 a1.12
BrainMass (9UM)
0.727 1:15
0.573
00
87.03
20.86
86.99 20.87
BDO (86M)
0.481 =0010
0.722 =.007
92.59 20.68
92.57 #0.69
4.1 INTERNAL AND EXTERNAL EVALUATION.
We assessed robustness and transfcrability using both linear probing (LP) and full fine-tuning (FT) protocols, with detailed procedures described in Appendix D.4. Notably, even under the simpler linear probing evaluation, BDO matched or exceeded the performance of task-specific models on both internal and external benchmarks, demonstrating strong generalizability across datasets.
Internal tasks: Age and Gender Prediction. To evaluate in-domain generalization, we performed age and gender prediction on a 20% held-out subset of the UKB data. As shown in Table 1, BDO achieved state-of-the-are results, surpassing both task-specific and SSL models on all metrics.
External tasks: Trait and Diagnosis Prediction. For external validation, tested BDO on trait and psychiatric diagnosis prediction across four public datasets: HCP-A, ABIDE, ADHD200, and HCP. EP. Tasks included predicting demographics and cognitive scores in HCP-A shown in Table 2, and classifying autism, ADHD, and schizophrenia in the other datasets as presented in Table 3. BDO outperformed all baselines in accuracy and Fl score, demonstrating strong clinical potential. Notably, we also confirmed that these powerful representations, pre- pre-trained exclusively on resting-state fMRI, generalize effectively to task-based fMRI paradigms. See Appendix D.6 for the detailed analysis.
Experimental details are provided in Appendix D.
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Table 2: Demographics and trait prediction on HCP-A.
Age
Gender
Flanker
Methods
MSE
P
ACC (%) f
Fl (%)
MSE
P
BrainNeiCNN
0.472 +:054
0.727 +.040
72.36 +1.66
71.42 +4.03
1.001 +.097
0,310 +0083
TS
BrainGNN
0,570 a.050
0.657 a.031
66.81 #2.54
65.22 #2.14
1.137 2.049
0.229 =.051
BrainNerTF
0,389 2.038
0.780 a.036
75.00 #2.28
74.06 +2.78
0.959 #.058
0.357 #.071
MoCo (90M)
0.817 4.037
0.591 2.007
64.12 #1.18
64.06 +1.25
1,572 #:158
0,283 #.022
BYOL (90M)
0.609 4.038
0.619 2.030
64.81 #1.32
64.58 +3.49
0,960 #.072
0,304 +.071
LP
BrainLM (85M)
0.756 4.057
0.636 a.027
65.28 #3.00
64.99 #2.96
1.181 0:081
0,375 1.0
BrainMass (90M)
0.743 4.117
0.630 2.077
66,20 20.65
66.17 20.56
1.082 4.013
0,313 a.013
BDO (86M)
0.404 2.010
0.768
72.00 =2.95
71.30
0.856
0,450
MoCo (90M)
0.532 4.023
0.697 2.016
65.28 #1.96
64.63 #1.98
0,976 0.06
0,370 #.058
BYOL (90M)
0.531 +.037
0.694 #.021
69.68 #2.15
68.82 +2.75
1.317 #.031
0,216 +.059
FT
BrainLM (85M)
0,340 1.019
0.818 4.012
72.78 #2.12
72.36 #2.22
0.859 000
0,461 +0055
BrainMass (9UM)
0.471 4.030
0.728 4.020
67.82 21.99
66.90 =2.17
0,996 055
0.339 #.0
BDO (86M)
0.273 2.100
0,851 2.006
79.40 =4.07
78.98 #4.38
0.847 2.037
0.464 z.072
Table 3: Psychiatric diagnosis prediction on clinical IMRI datasets.
ABIDE
ADIID200
HCCPEP
Methods
ACC (9t f
F1 (%)
ACC (%) f
F1 f
(C)
F1 (%) f
BruinNerCNNN
64.39 12.17
64.23 =2.27
55.49 +4.39
53.62 15.15
70.29 26.90
58.07 19.52
BrainGNN
56.82
56.73 +3.43
$2.78 +3.27
$1.59 +2,89
73.14
65.46
BrainNeiTF
#3.66
66.30 #3.67
$4.29 +3.02
#6.52
#10.3
MoCo (90M)
60.76
60.74 #3.16
57.64 =1.77
57.72 +1.76
71.43 #2.33
70.86 #2.03
BYOL (90M)
58.48 +2.39
58.44 #2.43
60.15 +2.13
60.21 +2.30
70.48 #1.35
70. 15 +2.23
LP
Brain M (85M)
59.92 43.88
59.77 #3.77
60.44 +2.37
60.27 42.27
72.38 +3.56
73.04 +3.10
BrainMass (90M)
62.27 11.93
62.18 #2.07
61.15 #2.16
60.39 42.62
73.33 #3.56
73.32 23.20
BDO (86M)
66.67 #1.13
66.58 #1.02
61.40 #1.97
61.49 +2.92
76.19 #4.86
74.63 #4.96
MoCo (90M)
65.45 #2.26
65.16 #2.30
61.15 =1.28
59.39 +2.13
75.24 +1.35
74.78 #2.37
BYOL (90M)
60.76 +1.30
60.61 #1.41
58.15 #0.35
57.16 40.53
72.38 #1.35
73.08 +1.06
FT
BruinLM (85M)
61.36 14.28
61.28 24.19
61.65 #2.68
59.27 43.22
74.29 24.04
74.12 13.92
BrainMass (90M)
67.27
66.66 #3.46
63.91 +1.23
62.55 +0.74
76.19
76.25
BDO (86M)
69.32 +2.24
68.32 #1.78
64.16 +1.15
64.27 +1.08
82.86 #2866
82.87 #2.65
4.2 SCALABILITY AND EFFICIENCY.
Model Sea whblity (Age)
Data Sc Scalabiliy (Age)
en
Scalability. We evaluated BDO's s scalability by varying both model size and pre-training data volume as depicted in Figure 7. For model scaling, we pre-trained three BDO variants with increasing parameter sizes: 5M, 21M, and 86M, and tracked HCP-A age-regression performance tra- jectories over pre-training epochs well disease pre- diction accuracy across various datasets. Additionally, to examine the effect of pre- pre-training data volume, we trained BDO (86M) on progressively larger subsets of the UKB dataset. In both experiments, larger models and greater data sizes clearly yielded higher performance. Detailed performance metrics and spatial scalability analysis are available in Appendix 1.7.1 and D.7.2, respectively.
0.7
0,7
0.6
206
0.5
10ep
100mp
200ep
100p
1.0kp
200es
Model Scalability (Diagnozis)
Data Sca ability (Diagnesis)
spaiite
ADHO2000
HCHEP
ABIDE
ADHD200
HOHEP
Figure 7: (Left) Model scalability (Right) Data scalability HCP-A age regression and diagnosis prediction.
Efficiency. Figure shows that BDO significantly outperforms other SSL methods in both resource and parameter efficiency. Remarkably, regression performance comparable to other SSL models (He et al., 2020; Grill et al., 2020; Caro al., 2024; Yang et al., 2024). See Appendix D.8 for the detailed metrics and experimental setup.
the smallest BDO variant (5M) achieves HCP-A age

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Recomsbrutton
intrrtttiionn Lees
4.3 ABLATION STUDY.
11
48
Mask ratio y. Figure 8 illustrates the effect of the mask ratio y for MAE in Section 3.2. The performance increases with 7; reaching an op- timal peak 75%. Beyond this point, perfor- mance degrades, likely due to excessive informa- tion loss hindering reconstruction. Additionally, a detailed analysis of the importance of the MAE objective itself can be found in Appendix D.9.
11
Mask Nallo
1.0
236
30C
0.0
23K
10k
3.63
Figure 8: (Left) Training curve (Right) HCP-A age regression Pearson correlation p as the mask ratio and balancing factor T are varied.
Balancing factor 7. Our ablation on the balancing factor in (9) reveals that incorporating the JEPA regularizer (r > 0) enhances performance over the MAE-only setting (r = 0), as shown in Figure 8. This demonstrates that the empirical prior contributes to learning more robust and meaningful representations. See Appendix D.10 for a more comprehensive MAE-JEPA ablation.
Number of timesteps. During the pre-training, BDO learns from sequence segments randomly sampled from full fMRI recordings. The length of these segments, referred to as the number of timesteps, directly impacts downstream per- formance, as shown in Table 4. While using more timesteps consistently boosts performance,
Table 4: LP performance of BDO (86M) with varying number of timesteps (NoTs) HCP-A.
NoTs Age (MSE) J. Age (0) Gender (ACC) Gender (FI)
0.587
0.645 +005
63.67 +2.00
61.98 +0.92
160
0.404 1.00
0.768 a.008
72.00 12.95
71.30
0.805
72.22 21.13
71.34
likely by capturing richer brain dynamics, it also raises the computational cost of pre-training stage.
4.4 INTERPRETABILITY OF THE UNIVERSAL FEATURE A
Embedding Space of Universal Feature A. To evaluate the clinical relevance of the universal feature A. we projected A into two-dimensional space using PCA and UMAP, as shown in Figure 9. PCA revealed clear linear patterns closely aligned with age distributions, while UMAP preserved these meaningful separations. This indicates that & successfully encode biologically meaningful neural variations associated with aging. The ability to characterize aging is clinically critical, as deviations from typical aging trajectories may signal early vulnerability cognitive decline psychiatric disorders (Elliott et al., 2021). These results demonstrate that our framework learns meaningful representations reflecting aging-related neural dynamics, highlighting its clinical potential. Further qualitative analyses, covering feature attribution and latent dynamics, are detailed in Appendix D.11.
Biobank Held-out (UKB)
Human Connectome Project Aging (HCP-A)
70
45
70
PC1
UMAP1
Age
PC1
UIMA11
Age
Figure 9: Projected 2D features of the universal feature A using PCA and UMAP, coloured by age.
(Left) Embedding space of UKB held-out split. (Right) Embedding space of HCP-A dataset.
CONCLUSION AND LIMITATIONS
We introduced BDO, novel SSL framework leveraging SOC to model continuous latent brain dynamics from fMRI, yielding robust and transferable universal feature A. BDO achieves state-of- the-art performance across diverse demographic and clinical downstream tasks, demonstrating strong generalization to external cohorts and notable computational efficiency.
While our simulation- free inference based on locally linear approximations has proven effective for scalability, it may introduce variational gap and potential error accumulation over long-term temporal analyses. Further challenges include the inherent complexity of the BDO framework and achieving detailed neurobiological interpretation of learned continuous dynamics beyond the aggregated universal feature A. Future work may focus on developing more flexible inference approximations and enhanced interpretability methods for such sophisticated models of brain activity.
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ACKNOWLEDGMENTS
This research was supported by NRF grant funded by the Korea government (MSIT) (RS-2021- NRO56917) and the Bio & Medical Technology Development Program of the NRF funded by the MSIT (RS-2025-02263045),
This work also supported by Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No. RS-2019-11190075. Artificial Intelligence Graduate School Program (KAIST); No. RS-2022-1120713, Meta-lcarning Applicable to Real-world Problems; No. RS-2024-005599279 Global AI Frontier Lab), the National Institute of Health (NIH) research project (No. 2025-ER0806-00 and grant of the Korean ARPH-H Project through the Korea Health Industry Development Institute (KHIDI), funded by the Ministry of Health Welfare, Republic of Korea (RS-2025-25455095)
REPRODUCIBILITY STATEMENT.
Appendix D contains all implementation details required for reproducibility, including a full break- down of our datasets, preprocessing pipelines, network architectures, and training protocols with hyperparameters. Our source code is available in the supplementary material.
ETHICS STATEMENT.
Our work introduces a novel SSL framework for fMRI analysis, intended strictly as a tool to advance neuroscience research in areas such as demographic trait analysis and psychiatric diagnosis. We acknowledge the significant ethical responsibilities associated with such technology, particularly the risks of perpetuating societal biases from training data and the potential for misapplication in clinical settings. Crucially, this model is not validated diagnostic tool and should not be used for such purposes without rigorous clinical trials and expert oversight. This study relies on publicly available, anonymized data; any future applications on sensitive clinical data would require strict privacy protocols and institutional review. A large language model (LLM) was used solely to improve the manuscript's grammar and clarity, and not for generating scientific ideas results.
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Drozdstoy Stoyanov, Katrin Aryutova. Sevdalina Kandilarova, Rositsa Paunova, Zlatoslav Arabadzhiev, Anna Oodeva-Radneva, Stefan Kostianev, and Stefan Borgwardt. Diagnostic task specific activations in functional mri and aberrant connectivity of insula with middle frontal gyrus can inform the differential diagnosis of psychosis. Diagnostics, 11(1):95, 2021.
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A
RELATED WORK
To facilitate a detailed understanding of the theoretical foundations, we present an additional related work section on self-supervised learning and stochastic optimal control for continuous-time dynamics.
Self-Supervise Learning. SSL has emerged as powerful paradigm for extracting rich, generaliz- able representations from large-scale unlabeled data. In the vision domain, prominent approaches fall into two main categories: latent representation alignment and reconstruction. Alignment-basee methods, such as MoCo (He et al., 2020) and BYOL (Grill et al., 2020), learn invariant features by maximizing agreement between different views of the same instance, often utilizing contrastive loss or momentum bootstrapping. Conversely, reconstruction -rresbrructio-bbbeeddd frameworks like MAE (He et al., 2022) force the model infer missing patches from partial observations, encouraging the capture of global structure. More recently, JEPA have demonstrated the effectiveness of predicting latent representations rather than raw pixels (Assran al., 2023; Bardes et al., 2024), thereby learning high-level semantics while avoiding low-level noise.
Extended Review of Self-Supervised Learning in fMRI. Image-based SSL approaches for fMRI, which patchify fMRI time-series into spatiotemporal tokens, represent each scan as fixed grid of spatiotemporal tokens. BrainLM pioneered the image-style paradigm and used MAE objective to reconstruct masked patches (Caro et al., 2024), and Brain-JEPA builds on this by replacing reconstruction with JEPA that learns predict the latent embedding of one view from another (Dong et al., 2024). Beyond the loss of fine-rrained temporal dynamics during patchification and the difficulty of aligning temporal resolution across datasets pointed out in Section 2, key limitation of this design its rigid dependence on sequence length in the SSL scenario. For example, model pre-trained on UKB data with 160 time points grouped into 10 non-overlapping patches of length 16 expects exactly the same 10-token layout during inference. However, datasets such as ABIDE contain scans with few 76 time points, which cannot be processed without upsampling. This length rigidity undermines the core premise of SSL, namely broad out-of-the-box transfer across heterogeneous unlabeled cohorts, and therefore poses a critical limitation for SSL fMRI.
In contrast, basph-baeed approaches compress the fMRI time-series into single static functional- connectivity (FC) graph, freeing them from constraints such as heterogeneous temporal resolution or scan-length variability; however, this coarse summarization inevitably discards fine-grained temporal dynamics BOLD signals. BrainMass, for example, generates augmented static graphs by randomly dropping time points to produce PSeudo-FC variants, and leverages latent representation alignment and masked-Ree ROII prediction (Yang et al., 2024). ST-JEMA tries to restore temporal information by partitioning the scan into multiple sliding windows and building dynamic graph sequence which it performs JEPA-style SSL objective (Choi et al., 2024). While this design captures slow changes in connectivity, it still averages out intra-window dynamics and therefore inherits the same limitations as image-based patchification such as loss of high-frequent temporal details and the practical difficulty of matching window size to the diverse TRs across different datasets as pointed out in Section
Stochastic Optimal Control for Continuous-Tine Dynamics. SOC provides a rigorous mathe- matical foundation for optimizing policies in stochastic systems governed by SDEs. In the context of generative modeling, Li al. (2020) bridged SOC and variational inference by deriving the ELBO to optimize non-linear latent dynamics. This control-theoretic perspective was subsequently adapted to approximate smoothing distributions various settings (Heng et al., 2020; Lu Wang, 2024) and to estimate Doob's fu-transform for online filtering (Chopin al., 2023). Building on these developments, Park et al. (2024) recently leveraged SOC to enable efficient amortized inference.
While previous works primarily utilized SOC for efficient inference filtering, we extend SOC to the domain of representation learning by fundamentally reframing the optimal control policy as a latent encoder. By framing the SSL objective control cost, our framework naturally integrates masked autoencoding (MAE) and joint-embedde prediction (JEPA) tasks into a single continuous time optimization problem, enabling robust feature extraction from irregular fMRI data.
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B PROOFS AND DERIVATIONS
B.1 PROOF OF PROPOSITION 3.1
Proof. Bayesian filtering and smoothing techniques Särkkä (2013) are foundational tools for estimat- ing latent states in probabilistic dynamical systems. The goal is to recover the posterior distribution of the latent trajectories given the observations:
p (( ())
go ( (xxx)...))(((...))
II 1E
(12)
z())
where Z()) is normalization constant:
II go (xx)
z()) = Ex~ (1))
(13)
LIET
Expanding this expectation, we have
log Z()) = log Ex~ (1)
(14)
pxX (0.21)
= log Ex~ (1)
p()1X (0.T)
(15)
p(X [0,77
(
pl X [o,
2 Ex~ (10)
logp()/XX0.7) (0.77) + log
(16)
pr
X
[0,r]
(XX (o.T)
=Ex~ (10)
Σ x() log
(17)
p(X:
LET
[0,7]
(i) Ex~ (10)
Σ (oly((X))
1
||ae|| di + L andW,
(18)
2
LIT
(Mi)
Exv (10)
Σ glyx||9)) X
1
T ||au|12
dt
(19)
2
LET
Ja,
(20)
where (1) results from Jensen's 's inequality, (ii) follows by applying Girsanov's theorem (Baldi, 2017,
Theorem 12.1), and in the final equality, (iii) holds because W, is martingale process.
B.2 PROOF THEOREM 3.2
Proof. Since each SPD matrix D. for / E T admits an eigen-decomposition D:; VA, VT, we can transform the original process X;, which expressed in the canonical basis, into a new process X V X that resides in the space spanned by the eigenbasis V. With this transformation, the dynamics in (10) can be rewritten, for any interval [ti, (++1) 1),
[--AA A, + at. dt + dW., dx
(21)
where X X di V at. W, V W, and initial condition X ~ No Σn) with Āo = V Ho and Eg = V Eo V. Since V is orthonormal, We retains the distribution W, d W for all / E [o. 7). allowing W. to be treated as standard Wiener process. Now, given that As, is diagonal, the linear SDE in equation (21) admits closed-form solution for any & [1i, (41):
(xg (. (e ds Cele-ti)A dW.
x (t))))
(22)
Since the initial condition Xg is Gaussian and the SDE is linear with Gaussian noise, the process X; remains Gaussian. Therefore, its first two moments the mean and covariance can be derived from
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the solution above. To derive the moments, firstly evaluate the deterministic integral involving ār,;
[r cle-bi)Au âr, ds= -- -A (I- (I-e((-)) de.
(23)
Taking the expectation of X and using the martingale property of the Wiener process W.. we obtain:
Pe = Exar (21) X = - pie - (t- A.,'' (I - ((-elttiiA:s) âs.. (24)
Next, compute the covariance of Xg:
e 2(t-d;)Ai; (x,, - pr + [ coo (xx, - Mes + [ cl c( Σ, ExOr (21)
(xi,
(25)
(X,, - me)) (X,, med) + +11 I" clo-ti)And
-2(1-(i)A, E.d,,Ey Exex (21)
(26)
i 2(1-1, A. E Exan (21)
(X1, Hn) (Xi, - -med + f 2(0- ds
(27)
2(t-ti)Ae, Et.
2(t-,i)A A.² (I- (((t-))))
(28)
where (i) follows from the martingale property of W, and (ii) follows from Itô isometry:
[ cl W.
J 2((0-t))As,d ds
E Ex-~ (21)
E x= (21)
(29)
Using the recursive forms for the mean and covariance, we can determine these moments at each discrete time step 4. For the mean ūe,, the recurrence relation is:
j/: -(1
(11-1 (tttt))rr pito -e - Ao) A (I (
âco
(30)
Pe, - E -1/)0s puc
(31)
- Ej -yylAs And (I- clti clti-toldrr) ) Oec
(e- N,-1 (I (I- ân
f4. = Ejsbl(yt1-t) -tyAAs prt
Ej AT. (()))t))))A)) ac.
(32)
Similarly, for the covariance Ee, the recurrence relation is:
E,, = -2(1--to) E1D
1 _
to),, A (T- ((t -t0)
(33)
2
f,, E/=o 2(15+1 11:11-11100 1,) E10
(34)
2112 = Σ) 2(1,+ (1,) Ago (I- 2(13 2000-0))
1
_ -2(1--1)AA, A (I
(tt
2(2
2
Σ, -2 - (1 Ei.
2 - (1,41 (1)A,, A.
(
(1-
(35)
1=0
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Now, since X; V X2, with ju V Hg and En ET V. express the mean and covariance in the original canonial basis as follows. For the mean fTeT, which is given by
Vd, V Eimbles ty Pioo
(I (-(ltt)--WAr de:
(36)
i-1
=V - - v 110
Σ
(+ Ani (1 (1-e (1-elln-tiian ) V an
c
1=0
(37)
= -1 -;) 1 -V
on A, (T-cltur-tuay ) Vion
(38)
= - -)) -
Σ
Σ (1+ 1,) V D 'v (1- (1- v vTan
1=0
(39)
=
Σ
- D.- (I- (1-elttr-ti)Dll an
(40)
-/4;
(41)
where we used Di, = VA,, VT and the orthonormality of V. Similarly, for the covariance S.ET we have
i-1 2
VåL V VT= =v (2E (-2
-1)A,, Ets
e -2 (H A,²1 (I - 2200)1=)))1))
V
(42)
= V -2 V
VTV.V
i-1
(5 Aij -1- (1-22(ut)-()a) ))
VT
2
1=0
(43)
1 i-
(t) Eg - V
Σ
2
A - -(ll))))()))))))))))))))
V
2
1=0
(44)
-1 Eg - V
Σ
2 -1, (I- (1-)
V
1=0
(45)
i-1
-22))-041111)))
Eo
- - (011-1))
D (1 (1- 2 20
(46)
Ett
(47)
Thus, both the mean Mr, and the covariance Σ, of X at each time step t; are correctly recovered, completing the proof.
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B.3 DERIVATION OF ELBO
We start the derivation by integrating the mixture distribution in (8) into the SOC problem (6) as follows:
log POir (O.T)) = log / (o(vellx))mo())((
(48)
-ogg| / (u(y)(āt))
1
( (po(ār(())* ((ar) dā,
(49)
log /
pa x qā (ās Dimr)
h(āg)dār log Z(X:)
(50)
x(x ((()s))
" / [log (yclae) + Ao X + (1 A) log (ar log h(ār)) h(ā,)dā log Z(X))
(51)
(4) [og (xe + (A log + (1 A) log (a p(ā, ( log Z(X²)
(52)
/ (yelae)) + (1 - log (as po (ās x()d + (1 AJC - log Z(X!)
[oog
(53)
(i0)
Co (yrlās) +(1 A) log d8 ((ar)))
(54)
MAB
JEPA
lly, D (āx))1 + (1 (1-1) |āo ad o0l??
-Earp((((X
(55)
20
20
where (1) follows from Jensen's 's inequality, and (ii) follows by setting proposal distribution h pe. (iti) follows from the definition of po, since the entropy of Gaussian with constant covariance:
f [a- 1)
1) /
X dā = (1 (1-1)
log po (ā, (a)x = (1- -)C 0.
(56)
Finally, (tv) follows from (1 A)C 2 and since the normalization constant Z(X)) is calculated as:
f
z(x;)
((oG(((())))(((((>0))0 1->
(57)
/ C1exp exp
X |ā -,.1
(1 A)
ài
8
8
(58)
20 p
20
= / C exp
1 (ār
T-
(axl
T
a
1
1-1
a
2
(59)
2
o
= C3 exp
T8-1
0
1 1- A
}
(60)
2
x²
p
where C₁
C2
(2m)d/2(o7)) (o3)
1-1
(of)
(oz)
1
1-
=
0 +
(1 Vimr) and =
+
I.
(61)
0
d2
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Consequently, get
1=
z(x)) = C3 exp
o
o
+
of
a
|2
1
a
of
p
A(1-A)
= C3 exp
:
|
a
(62)
2
05
It implies that loo ãi (X²) 2 O. Hence we can derive the desired inequality in (9):
1/"
- Pog (eax) S Exem(10)
2 1 dl
Σ
Bp x
So (y + (1 X) log (ā [Jirr)]
LET
(63)
= =Exe~(10)
[ 1/² 2
|aafl dt
Σ E a(ppp( (xf)
ly, - Dv(ā))) (1 2)
āi di
8/12
20
20
IET
(64)
= -L(0, v))
(65)
For stable learning, we train our model with rescaled training objective:
[(,,b) = E(00(11)) /'" of |1
||1]11a0 di
Σ DE Ti
E Barpp(d( (X!)
l - D ( +r
2
reconstruction
regularization
(66)
Here, T X determines the balance between reconstruction and regularization. See Section 4.3 for details on how controlling the regularization influences the performance of BDO.
C SIMULATION FREE INFERENCE
C.1 NON-MARKOV CONTROL FORMULATION
In this section we clarify whether the ELBO derived in Proposition 3.1 is valid under various control Donsker-Varadhan variational principle, which provides a variational
formulations. We adapt the Donsker Varadhan formula for a wide range of Markov processes, including Itó diffusion processes (Hartmann et al., 2017: Tzen & Raginsky, 2019), into our formulation as follows.
Lemma C.1 (([0,]] ([), and let E C((O,T]] Rd)
Varadhan variational principle). Let TP be Markov path measure be a bounded and measurable functional. Then
) ->
log Ep [c ((x min Q&E
EQ [E(Xp.m))] + (QL()
(67)
In particular, if we define the functional with the negative log- ligeliiod function with respect to y
{(X == log p (o.T))
(68)
and by plugging the controlled path measure To as a particular choice of the variational path measure Q on the right hand side of (69), then recover the desired ELBO from Lemma C.I with exactly the same computation as in (14-20):
log [p() (10))((OT)))
min
Bg
log log p()) + (KI(
(69)
Q<<P
log logp()(((0o7)) + 1 T
min QC<P
Eg
|| de 2
(70)
TeT
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where (1) follows from the Girsanov Theorem (Baldi, 2017). In other words, the ELBO in Propo- Donsker-Varadhan
sition 3.1 can be viewed as the Donsker- Varadhan variational formula restricted to the family of controlled path measures Pa. Note that Lemma C.I is stated for Markov processes P and Q. hence the controlled path measure Tpo a valid choice for Q in our formulation as long as the control a is an admissible and adapted process.
This should include (standard) Markov controls and under Markov control formulation, it is well known that the optimal control a" yields tight lower bound, via the relationship between Hamiltonian- Jacobi Bellman equation (Van Handel, 2007) and the ELBO through Hopf-cole transformation (Flem- ing & Soner, 2006). In this case, the value function induced by Markov optimal control solves the HJB equation and the corresponding optimal path measure attains the mimimum of (69), so the ELBO bound becomes tight.
However, the ELBO formulation itself is not limited to Markov control cases. Because our specific non Markov control formulation in (10) is still adapted with respect to the Brownian filtration, the ELBO derived in Proposition 3.1 remains valid under the non Markov (open loop) control parameterization, and the interpretation of the ELBO will be unchanged. A closely related discussion appears in (Park et al., 2024, Remark 3.9). where the Markov optimal control is approximated by a non Markov control parameterized by a high capacity neural network. They argue that if the network has sufficient capacity and is trained by gradient descent, the resulting local minimizer can approximate the optimal control under Markov formulation.
C.2 PARALLEL SCAN ALGORITHM
The computation of the first two moments the mean MaET and covariance ΣiEr- of the controlled distributions can be efficiently parallelized using the scan (all-prefix-sums) algorithm (Blelloch, 1990). Leveraging the associativity of the underlying operations, we reduce the computational complexity from (kk to (log k) time with respect to the number of time steps k. We have established the linear recurrence in Theorem 3.2 for the mean and covariance at each time step t;:
m A,m,,., B, Σ, = A,E,, + B,I
(71)
(72)
where we define A,(() = 1-4, A = A 141)A B = e 1 A (
A, e (t)A, and B, = - 2A, (t))) A,, (I (t.)... To apply the parallel scan algorithm to our recurrence, we define two separate sequences of tuples for the mean and covariance computations for all E [1, (1,....,,) k):
M, A ; B,ok.). S. = (A,
(73)
Now, we define binary associative operators @ for the sequences {M,) and (S.):
M, x M, A Ay o Bja,, + A B.ar).
(74)
S, c S, (A, A, A, c B, + B.).
(75)
where o denotes element-wise multiplication. We can verify that @ is an associative operator since it satisfies:
(M, @ M2) M,-- ( c A, Baas, + +Baa,) @ M,
(76)
( o (AroA.) ,
+ +BoO,)) + Buccn.)
(77)
= ((A, o A.) o Ao, (An o A.) Baar, + + o Brar, + Brac.))
(78)
M, 8 (M, & M.).
(79)
Thus, we get (M, @ M.) @ M, = M, @ (M, 8 M.), confirming associativity for M, Similarly,
(S, X S.) @ S., = (A, o A, A, o I B,I) & M,
(80)
(A, o A.) A. (A, o B,a, + +B.I) + B.I))
(81)
((A,, A.) A., (A, A.) B.I A, B,I + B.I)
(82)
=S. & (S, @ S.).
(83)
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Algorithm Parallel Scan for Mean and Covariance Algorithm 2 Parallelscan
I: Input. Given time stamps T {tz .2,..., txh. 1: Input.
Sequence of tuples
initial mean Wit and covariance Dtg. control policies
{Tı, T2...., Tx). associative
opera-
{okt (0tg,, :ieg: (Rtg): matrices (Ar A Atg....,AAA
tor
2: Initialize sequenees (M, ). and {S, )
Stage 1: Up-Sweep (Reduce).
3: for to K do in parallel
3: for d = 0 to [log, K] - do
4: Compute (t, = -4-1
4:
for each subtree of height d in parallel do 2d 2 - -1 for &
5: Compute = = A.(())
Let i 0.1....
6:
Compute ARAAA, A (I- (lltii
B
if < then
7:
T. TTTTe T., end if
7: Compute A; (1,AA,
8:
Compute
B.
end for
L (e-2a((4)A1; Ar
e (2.()))a.))
10:
11: Stage 2: Down-Sweep.
9: Set M, (A,, B,a+.)
12: Tx = I. where I is the identity element for @.
10: Set S; (A., B.).
13: for [log, K| downto do
11: end for
14:
for each subtree of height d in parallel do
Parallel
Scan
[=
15:
Let i 2d+1 + 2d+1 for k 0,
Paralles 1elllelsoe weeeleeennMMMM)..
13: Parallel
Scan
( K
16:
if < K then
Paralle1Scan((S1
17:
T. T T @ T;
for 1 do parallel
18:
T. = T-TT.-2
15: ,4, ( peo M
end if 20: end for
19:
16: Σ, = ) + ((
21:
end for
end
22: Return
Scanned
sequence
18: Return PIET. EIET
(T T T T T2 T,
where
Hence, (S, 2 S₁) S. = (S, X Su), confirming associativity for S,. Now, can apply the parallel scan described in Algorithm for both PIET and covariance ExE based on the recurrence (24, 28) and the defined associative operators 60. Employing the parallel scan algorithm offers significant computational benefits, especially for large-scale problems with numerous time steps k. The logarithmic time complexity ensures scalability, making it feasible to perform real-time computations or bandle high-dimensional data efficiently.
D EXPERIMENTAL DETAILS
D.1 DATA PREPROCESSING
Preprocessing Pipeline. The preprocessing pipeline involved several standard steps, including skull- stripping, slice-timing correction, motion correction, non-linear registration, and intensity normal- ization. All data were aligned to the Montreal Neurological Institute (MNI) standard space for consistency. A whole-brain mask applied exclude non-brain tissues, such as the skull, from further analysis. The fMRI data were parcellated into 450 regions of interest (ROIs), comprising 400 cortical parcels based the Schaefer-400 atlas Schaefer al. (2017) and 50 subcortical parcels defined by Tian's Scale III atlas Tian et al. (2020). The mean IMRI time-series for each ROI was extracted across all time points.
Data Normalization. To ensure comparability across participants and reduce inter-subject variability. we applied two-step normalization process the fMRI data. First, participant- wise zero-mean centering was performed by subtracting the mean signal from each ROI within each subject. Second, a robust scaling procedure was applied, where the median signal was subtracted, and the resulting values were divided by the interquartile range (IQR), computed across all participants for each ROL. After normalization, each IMRI sample was represented as a matrix of size x N, where T corresponds to the number of timesteps and N corresponds to the number of ROIs (N =40)..
UK Biobank (UKB) The UKB a population based prospective study comprising 500,000 partici- pants in the United Kingdom, designed investigate the genetic and environmental determinants of
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disease Sudlow et al. (2015). This study utilized 41,072 rs-MMR scans from the publicly available, preprocessed UKB dataset Aamagro et al. (2018). The preprocessing pipeline included non- linear registration to MNI space using FSL's applywarp function, thereby ensuring standardized spatial alignment across participants Jenkinson et al. (2012).
Human Connectome Project in Aging (HCP-A) The HCP-A is a large-scale neuroimaging initiative focused on characterizing structural and functional connectivity changes associated with aging across a wide age range Bookheimer et al. (2019). This study accessed 724 rs-fMRI samples from healthy individuals between 36 and 89 years of age. Preprocessed rs-IMRI volumes provided by the HCP-A dataset were utilized for subsequent analyses.
Autism Brain Imaging Data Exchange (ABIDE) The ABIDE consortium aims to elucidate the neural mechanisms underlying autism spectrum disorder Di Martino et al. (2014). In the present work, 1,102 rs-fMRI samples were obtained from the Neuro Bureau Preprocessing Initiative Craddock et al. (2013a), which employs the Configurable Pipeline for the Analysis of Connectomes (C- PAC) Craddock et al. (b)1 The preprocessing steps included slice-timing correction, motion realignment, intensity normalization (with a 4D global mean set to 1000), and nuisance signal removal. Nuisance regression involved 24-parameter motion model, component-bai bbased noise correction (CompCor) Behzadi et al. (2007) with five principal components derived from white matter and cerebrospinal fluid signals, and linear/quadratic trend removal. Functional- totoonalltt-nntmmiica registration was performed via a houndary-based rigid-body approach, while anatomical to-standard registration utilized ANTs. Band-pass hand-pass filtering and global signal regression were not applied.
Attention Deficit Hyperactivity Disorder 200 (ADHD200) The ADHD200 dataset comprises 776 rs-IMRI and anatomical scans collected from individuals aged to 21, including 491 typically developing individuals and 285 participants diagnosed with ADHD Brown et al. (2012). A total of 669 rs-MMRI datasets were selected for this study, specifically the preprocessed versions provided by the Neuro Bureau Preprocessing Initiative (Athena Pipeline) Bellec et al. (2017).
Human Connectome Project for Early Psychosis (HCP-EP) The HCP-EP is a neuroimaging initiative focused on understanding early psychosis, defined as the first five years following symptom onset, in individuals aged 16-35. The cohort includes participants with affective psychosis, non- affective psychosis, and healthy controls Jacobs et al. (2024); Prunier & Shenton Martha; Breier (2021). For this study, 176 rs-fMRI scans were analyzed. Preprocessing was conducted using IMRIPrep Esteban et al. (2019), followed by denoising with Nilearn Nilearn contributors (2025). The denoising process employed a 24-parameter motion model (including translations, rotations, their derivatives, and quadratic terms) and CorpCor-derived components extracted from white matter and cerebrospinal fluid masks. Additionally, all confound variables were demeaned to ensure consistency.
D.2 PRE-TRAINING STAGE
Pre-training Data. For self-supervised pre-training, we utilized the large-scale UKB dataset, which comprises resting-state fMRI recordings and medical records from 41,072 participants (Alfaro- Almagro et al., 2018). We utilized 80% of the dataset for pre-training. while the remaining 20% held-out data was reserved for downstream evaluation. We used fixed random seed (42) to ensure reproducibility when partitioning the UKB dataset into pre-training and held-out subsets. All experiments, including the reproduction of foundation model baselines, were conducted using the same dataset split to maintain consistency.
Irregular Multivariate Time-Series Sampling, We introduce irregularity in the time-series data by subsampling both the observation timestamps Taks and the corresponding fMRI signals V Unlike conventional approaches that assume uniformly spaced time points (Caro et al., 2024; Dong et al., 2024), we select a uniformly sampled subset of timestamps from the full sequence, ensuring that only a fraction of the IMRI signal is observed. Specifically, from each full-length IMRI recording, we randomly sample 160 timesteps (T = 160), introducing variability in temporal resolution across different samples. This choice reflects the fundamental nature of brain dynamics, which evolve continuously rather than discretely, and encourages the model to infer missing states from incomplete sequences.
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Temporal Masking. To encourage robust representation learning and improve generalization, employ temporal masking. where a subset of the 160 sampled time points is randomly masked during training. We apply masking ratio of Y 0.75, meaning that 75% of the sampled timesteps are hidden while the model is trained to reconstruct them. In Figure 8, we vary across [0.4, 0.5, 0.6, 0.7, 0.75, 0.8, 0.9) to examine the effect of masking ratio in learning robust representa- tions. Actual reconstruction results are provided in the internal and external datasets as visualized in Figures 15 and 16.
Pre-training Algorithm. The pre-training of BDO follows the procedure outlined in Algorithm 3. Given an observed fMRI time-series Vots, we employ masked reconstruction strategy, where random proportion of the temporal signals is masked to encourage the model to learn meaningful representations. The pre-training objective leverages amortized inference to approximate latent dynamics. At each iteration, subset of observed time-series Vas. is used as context, while the masked portion Dar serves as the target for reconstruction. The encoder network Tg maps the context data to sequence of latent states Ta. which are then used to estimate drift terms and control policies, forming the basis for latent trajectory prediction. The decoder network De: reconstructs the missing target states, optimizing training objective L(8, vb) that aligns the predicted and true trajectories.
Pre-training Details. We trained BDO using a batch size of 128 and a total of 200 pre-training epochs. The learning rate was scheduled using a cosine decay scheduler (Loshchilov & Hutter, 2016) with a 10-epoch warm-up phase. During warm-up, the initial learning rate was set to 0.0001, which increased to peak learning rate of 0.001 before gradually decaying minimum learning rate of 0.0001. For optimization, we employed the Adam optimizer (Kingma & Ba, 2015). Across all BDO configurations, we used fixed number of basis ( 100 and consistently multiplied the observation times by a time-scale of 0.1 for all datasets. To update 0, Exponential Moving Average (EMA) momentum used and linearly increased from 0.996 to 1.0. It is worth noting that our models required minimal hyperparameter tuning, which demonstrates that the proposed approximation scheme operates stably and robustly.
D.3 MODEL ARCHITECTURE
To maintain the structural advantages of our formulation, we designed our encoder network architec- ture in straightforward manner. In this regard, the networks used for pre-training BDO listed below, where N=450 0 is the number of ROIs and d is the dimension of latent space Rd as described in Table for each model.
Encoder network qo:
Input (N) + Linear (d) + ReLU () > LayerNorm (d) > Linear (d)
R)LU() + LayerNorm(d) + 12 X [LayerNorm (d) + Attn
FF)
FFN:
Input (d) + LayerNorm (d) + Linear (4 d) + GeLU Linear (d)
Residual (Input )
Attn:
Input (Q, K, V) + (2rmalize + Linear (Q) + Linear (K) +
Linear (V) Attention (Q, K) + Softmax (d) - Dropout ()
Matmul (V) + LayerNorm (d) + Linear + Residual (0)
Decoder network D.: Input (d) + Linear (N) + () - Dropout () Linear (d)
Locally Linear Approximation. Recall that the locally linear SDE in (10) drift matrix of the form D., = VA, VT, where V E Rdxd is a eigen-basis and Ar, E Rdxd a diagonal eigen-value matrix whose entries vary with the time index f,. Here, we the matrix V as a single trainable matrix shared across all time-interval t E [0, enforced to remain orthogonal following (Lezcano Casado, 2019). The dependent eigen-value A,, is defined as " affine formulation with following components A., = El= w/ A!), where (A")) is set of trainable diagonal base matrices. The weight coefficients w., = {we ,,.,,,, neural network Wo. Wi softmax(Wo(or.))
time dependent
w (1) are produced by
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Table Pre-training hyper-parameteer
Train
Warm-up
Minimum
3d
(L.)
BDO (5M)
200
10
0.001
0.0001
0.0001
128
192
100
[0.996, 1]
BD0 (21M)
200
10
0.001
0,000]
0,0001
128
384
100
[0.996, 1]
BDO (86M)
10
0.0001
000011
768
Algorithm 3 Pre-tree Fre-training BDO
Algorithm 4 Fine tuning BDO for downstream tasks
1: Input. Time-series Von. = VesTa. masking ratio
1: Input. Time-scrics and label (Dins; Oots), pre-trained
Y, encoder network Ts, decoder network D
encoder network Tp.
2: for m = 1,..., M do
2: Sample Tixe ~ IeETT. qu (Ze()a
3: Get Var by masking 7% of temporal sig- 3: Compute optimal control policy areTa nals.
Bo ZeeTane
Sample TiETa IIIETT (exx))
4: Compute the universal feature A Tata
EeTToN as.
5: Compute (Di, ut, ai heTa.
6:
Estimate (f4, Te with parallel scan algo-
5: Predict Out = (A).
rithm.
6: if Linear probing then
7: Sample XlETT. ind BIET ((,, E.).
7: Freeze the pre-trained encoder network Ts..
Sample TIETu ~ ~II p(2)
8:
Compute L(0" (w) = Luu (Oess Oces) using (84).
9:
Compute E(0,d) vb) using (66).
9: Update with C., :w).
10:
Update (0, 10) with Vo.
10: else if Fine tuning then
11: Apply + EMA(
11: Unfreeze the pre-trained encoder network To
12:
end for
12: Compute 2(0',w) Lusk(Oara, Oata) using (84).
13: Update (0",w) with V a. Li0"
14:
end if
Table 6: Dataset Subject Demographics
Category
UKB
HCP-A
ABIDE
ADHD200
HCP-EP
of subjects
41,072
724 60.35 (15,74) 56.08 (406)
1,102 17.05 (8.04)
669 11.61 (2.97)
176
Age. mean (SD)
54,98 (7.53)
23.39 (3.95)
(n)
(21.480)
(163)
(242)
Parient, S: (n)
48.19 (531)
58. 15 (389)
68.1 18 (120))
Target Population
Healthy Population
Healthy Population
ASD
ADHD
Psychotic Disorder
Healthy Population
Healthy Population
Healthy Population
D.4 DOWNSTREAM EVALUATION STAGE
To assess the generalization and transferability of BDO, conducted experiments across multiple datasets and tasks, encompassing both demographic and psychiatric prediction. Datasets used in this evaluation have distinct temporal resolutions and varying numbers of timesteps, reflecting the irregularity of real-world fMRI data acquisition. Additional details are described in Table 6. Note that in the downstream evaluation, irregular sampling and temporal masking were disabled. The full sequence of IMRI signals, timestamps, and corresponding labels was used, denoted as (Doss Totss atts).
Internal Evaluation. For internal evaluation, we utilized 20% held-out subset of the UKB dataset, which was excluded from pre- areitraining. This evaluation focused on age regression and gender classification, leveraging both LP and FT to analyze how well the model retains and transfers knowledge acquired during pre-training.
External Evaluation. For external evaluation, we examined the ability of BDO to generalize to unseen datasets. Demographic and trait prediction was performed on the HCP-A dataset, where LP and FT were employed to assess model performance on age, gender, neuroticism, and flanker scores. Beyond demographic characteristics, we evaluated psychiatric diagnosis classification using clinical fMRI datasets. including ABIDE, ADHD200, and HCP-EP. These evaluations relied on LP, as it provides a controlled assessment of the learned representations and their applicability to clinical classification tasks.
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Random Splits. All the datasets are partitioned into training, validation, and test sets using 6:2:2 ratio to ensure fair and reprodueible evaluation. To maintain consistency, we perform partitioning with 3 consecutive random seeds, 0, 1, and 2.
For classification tasks, such as gender classification, stratified sampling is applied to preserve class distributions across the training, validation, and test sets.
For regression tasks, such as age regression, binning-basee stratified sampling is employed. In this approach, the continuous target variable is first discretized into bins before applying stratified sampling, ensuring a balanced distribution of the target variable and mitigating potential biases from uneven data partitioning. Additionally, to improve numerical stability and facilitate optimization, the target variable is normalized using Z-score normalization, where the mean is subtracted, and the result is divided by the standard deviation.
The distributions of the three random splits for age regression tasks with the UKB and HCP- datasets, and six classification tasks with UKB gender, HCP-A gender, ABIDE diagnosis, ADHD200 diagnosis, and HCP-EP diagnosis are described in Figures 12-14.
Extracting the Universal Feature A. To extract the universal feature A, we define f as mean-pooling over the sequence of control signals aseT, given by A == f(atet) = |7| ΣteT (tz, This formulation ensures that A serves as compact and transferable representation of the underlying spatio-temporal dynamics captured by the optimal control signals. To enhance biological interpretability, mean- pooling is chosen as it provides a global summary of the temporal evolution of the control sequence while suppressing high-frequency fluctuations that may arise due to local variations in 0t+ Although we believe that prean-pooling provides a robust and scalable approach for summarizing temporal dynamics, we acknowledge that more sophisticated aggregation methods, such as weighted pooling or recurrent architectures, could further enhance downstream performance. These approaches may offer additional advantages for analyzing temporal dynamics, such as facilitating interpretability through attention weight analysis or capturing long-range dependencies. We leave the exploration of these advanced aggregation strategies for future work.
Downstream Evaluation Algorithm. To evaluate the effectiveness of BDO on downstream tasks, follow the procedure outlined in Algorithm 4. Given an observed fMRI time-series Vats and its corresponding labels Oats. we extract the universal feature representation A using the pre-trained encoder Tox. This representation is subsequently used for classification or regression tasks through either LP FT.
In the LP setting, we freeze the pre-trained encoder Tg. and train only the task-specific head hw : =dd + RV (single linear layer). The objective function C("", ,w) measures the discrepancy between the predicted O.h. and ground-truth Oam. and is optimized with respect to
In the FT setting, the entire model, including Te., is optimized. Both the encoder and task-specific head hn are jointly updated to refine the feature extraction process for the target task.
Training Objective for Downstream tasks. The loss function for downstream tasks is defined based on the nature of the prediction problem: classification tasks use Binary Cross-Entroyy (BCE) loss, while regression tasks employ Mean Squared Error (MSE) loss.
Model Selection. To determine the optimal model for each downstream task, we performed a grid search over key hyperparameters such as learning rate and batch size. For each task, we evaluated multiple configurations using the validation set and selected the model that achieved the best performance based on the predefined metric. The set of hyperparameters is provided in Table 7.
1
Ei=1 Oobs. i log Ostsss + (1 - Oubs,s) log(1 - Octsss) if regression
if classification
Lusk (Oobs Oabs))
N
Σ (Oobs.t obs.t Óobs,))
(84)
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Table 7: Search space of end-to-end fine-tuning (FT) and linear probe (LP).
Configurations
FT
LP
Optimizer
AdamW (Loshchilov, 2017)
Adam (Kingma & Ba, 2015)
Training epochs
50
50
Batch size
[16, 32]
[16,32, 64]
LR scheduler
cosine decay
cosine decay
LR
[0.001]
[0.01, 0.005 [0.001. 0.005] [0] N.A.
Minimum LR
[0, 0.0001, 0.001]
Weight decay
[0. 0.01]
Layer-wise LR decay
(0.85, 0.90, 0.95]
D.5 ADDITIONAL BASELINES
D.5.1 COMPARISON WITH BRAIN-JEPA
In this section, we present a detailed comparison with Brain-JEPA (Dong et al., 2024), a prominent SSL baseline for fMRI analysis. To ensure the most rigorous comparison against the optimal version of the baseline, we utilized the official Brain-JEPA pre-trained weights (trained for 300 epochs) and compared Brain-JEPA against BDO (trained for 200 epochs) under both I.P and FT protocols. To isolate the effect of preprocessing, we adopted Brain-JEPA's original preprocessing pipeline for both models. Note that BDO and Brain-JEPA share exactly the same Transformer backbone.
Comparison on HCP-A. This dataset aligns well with the pre-training conditions of Brain-JERE Specifically, the effective TR after downsampling in HCP-A (0.72s x 3 R= 2.16s) is very close to the pre-training TR derived from UKB (0.735s x R 2.2055), and the sequence length remains sufficient. Even in this favorable setting, BDO outperforms Brain-JEPA across the majority of tasks on HCP-A, with the exception of the Flanker regression task in the LP setting as shown in Table 8.
Comparison on ABIDE, ADHD200, and HCP-EP. We also extended the comparison to ABIDE, ADHD200, and HCP-EP as summarized in Table 9. However, unlike BDO, the official pre-trained Brain-JEPA model is constrained by fixed input dimensions of 450 ROIS (Schacfer-400 and Tian- Scale III) and sequence length of 160 timesteps. Consequently, for datasets with sequences shorter than 160 timesteps, we explicitly applied data repetition satisfy this input constraint.
ABIDE and ADHD200 are multi- site datasets characterized by heterogeneous TRs in addition to short sequence lengths. For fixed-patch architecture like Brain-JEPA, this TR variability implies that input patches correspond to inconsistent physical durations across subjects, complicating the modeling of unified temporal dynamics. This structural mismatch, combined with the artifacts introduced by data repetition, likely limits performance.
This limitation is also evident in HCP-EP, where Brain-JEPA necessitates downsampling for TR alignment (0.7s 3 x 2.1s). Since this downsampling reduces the sequence length below the required threshold, we subsequently applied data repetition to meet the input constraints. In contrast, through continuous-time modeling, BDO natively adapts subject-specific TRs and variable lengths, processing data directly without such artificial adjustments. This difference likely contributes to the observed performance gap, underscoring the benefits of continuous modeling.
Table 8: Performance comparison on HCP-A.
Age
Gender
Flanker
Methods
MSE 4
P1
ACC (%) 1
F1 (%)
MSE 4
P1
Brain-JEPA (86M)
0.364 2.0
0.804 1.004
73.15 11.09
73.09
0.962 1.131
0.360 2.064
LP
BDO (86M)
0.298 +.022
0.839 +.010
74.48 +1.82
74.52 +3.81
0.966 +.03
0.343 +.059
Brain-JEPA (86M)
0.275 +.011
0.854 +.008
+1.21
82.38 +1.20
0.891 +:011
0.424 +:017
FT
BDO (86M)
0.262 +:009
0.862 +. 000
82.92 +1.92
82.88 +1.99
0.862 0.34
0.452 +.088
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Table 9: Performance comparison on ABIDE, ADHD200, and HCP-EP.
ABIDE
ADHD200
HCP-EP
Methods
ACC (%)
FI (%)
ACC (
FI (16) f
ACC (%) f
FI (%) f
Brain-JEPA (86M)
59.92 +3.74
58.85 #3.72
56.22 #1.06
56.34
72.38 +3.56
71.35 +3.89
BDO (86M)
64.09 #2.60
64.04 +2.60
58.40 #3.03
58.54 +2,99
73.33 #3.56
72.54 #3.41
Brain Brain-JEPA JEPA (86M)
64.55 #4.80
64,27 #4.82
62.15 #0.61
62.32
75.24 #1.35
74.47 #1.46
FT
BDO (86M)
67.90 #2.68
67.88 +2.67
63.15 +2.81
63.32
79.05 +2.69
78.65 #2.63
D.5.2 COMPARISON WITH SOLVER COLVER-BASED BASELINES
We further investigated the advantages of our simulation- scheme compared to solver-based approaches. To this end, we conducted experiments comparing BDO against Latent ODE (Rubanova et al., 2019) and GRU-ODE-Bayce (De Brouwer et al., 2019). Since these frameworks were not originally designed for large-scale SSL scenarios, we established them in a supervised setting on the HCP-A dataset. Specifically, we employed composite objective of reconstruction and task-specific losses, as described in their original papers.
empirical reference by training
Table 10: Comparison with solver-based continuous-time models on HCP-A.
Age
Gender
Methods
MSE
P
ACC (%) f
Fl (%)
Latent ODE GRU-ODE-Bard Bayes
0.556 +.049
0.671
#.042
64.22 #3.88
63. #3.32
0.587 #.04
0,652 #.039
63.30 #3.65
62.18 #3.24
BDO (86M) LP
0,404 +.010
0.768 +.008
72.00
+2.95
71.30 42.19
BDO (86M) FT
0.273 +.11
0.851 +00
79.40 +4.07
78.98 44.38
As presented in Table 10, BDO significantly outperforms both supervised solver-based baselines, even in the LP setting. This result highlights BDO's ability to leverage the benefits of large-scale pre-training learn robust and generalizable representations. In contrast, while employing numerical solvers theoretically allows for modeling complex dynamics, their high computational cost acts as severe bottleneck, preventing them from effectively scaling to massive datasets for pre-training.
By bypassing the bottlenecks of numerical integration, our simulation-free framework enables efficient large-scale SSL while achieving state-of-the-ar performance. Thus, while the SOC formalism provides the theoretical foundation for our objective, the simulation-free scheme serves as the practical engine that enables scaling to the high dimensionality and complexity of neuroimaging data.
D.5.3 COMPARISON WITH BRAINHARMONIX-F
We compare our framework with BrainHarmonix (Dong al., 2025), recently proposed multi- modal foundation model designed to unify brain structural morphology and functional dynamics. BrainHarmonix is originally designed unify brain structure and function by pre-training two separate unimodal encoders (BrainHarmonix-S for Te-weighted structural MRI and BrainHarmonix- for IMRI time-series) before fusing them through shared tokens. However, for a fair comparison, we specifically isolate the functional encoder,
and use it as baseline.
BrainHarmonix- was pre-trained on massive combination of the UK Biobank (UKB) and Adoles- cent Brain Cognitive Development (ABCD) datasets, using the Schaefer-400 atlas. Thus, also pre-trained our proposed model on the identical large-scale curation of datasets (UKB and ABCD) using the same parcellation scheme and preprocessing pipeline.
We evaluated the linear probing performance across multiple benchmark datasets. As shown Ta- ble 11 and Table 12, compared to previous baselines such Brain-JEPA. BrainHarmonix-F demon- strates overall performance improvements, which can likely be attributed to the inclusion of the large-scale ABCD dataset during pre-training and the of data augmentation techniques. Notably,
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the performance gain is particularly evident in disease diagnosis tasks, suggesting that the ability to handle heterogeneous TRs allows the model to be agnostic to temporal mismatches and arbitrary data repetition, thereby learning more robust clinical features. However, BDO's superior comparable performance across most tasks further indicates that modeling the continuous stochasticity brain dynamics offers distinct advantage over discrete tokenization.
Table 11: Linear probing performance comparison HCP-A.
Age
Gender
Flanker
Methods
MSE
P
ACC (%)
F1 (%) f
MSE
p
BrainHHrmonii--F (86M)
0.308
0.837 000
75.22 #1:21
75.02 21.24
0,892 4.09
0.424 1:018
BDO (86M)
0.275
0.850 =.008
75.74 +1.72
75.27 +1.87
0.864
0.447 1.10
Table 12: Linear probing performance comparison on ABIDE, ADHD200, and HCP-EP.
ABIDE
ADHD200
HCP-EP
Methods
ACC (%)
FI (%) 1
ACC
FI (%)
ACC (%) f
FI (%)
BrainHarmonix-F (86M)
64. 46 +4.48
64.44 +4.48
63.41 +2.32
63.39 #2.45
76. +5.87
75.88 #5.87
BDO (86M)
66.33
66.01 +4.77
62.37 43.15
62.39 #3.14
77.14 +2.33
76.92 +2.13
D.6 GENERALIZATION FROM RESTING-STATE TO TASK-BASED FMRI
To conduct challenging test of BDO's generalization capabilities, we evaluated whether its represen- tations, learned exclusively from unconstrained resting-state fMRI, could be effectively transferred to structured task-based fMRI, where brain dynamics driven by explicit external stimuli.
For this experiment, we used the BDO-S6M model, which was pre-trained solely on resting-stat UKB data. We then evaluated its performance on three distinct and cognitively demanding task paradigms from the HCP-A dataset. The evaluation was performed under the LP setting.
The results summarized in Table 13. While there is moderate performance decrease compared to the in-domain resting-state baseline (HCP-A-Rest) the model still achieves strong predictive performance across all three task paradigms
This successful transfer demonstrates that BDO learns fundamental, task-relevant neural dynamics that are not limited to the resting state. This underscores BDO's broad applicability as powerful feature extractor for diverse fMRI paradigms, even without any task-specific fine-tuning.
Table 13: Generalization from resting-state to task-based fMRI.
Dataset
Age (MSE)
Age (Pearson) f
Gender (Acc.) f
Gender (F1) t
ICCP-A-V VisMotor
0.526 1018
0.691 2015
68.53 43.77
67.39 23.36
NCC--A-FacNamm
0.459 1012
0.732 2009
66.20 43.44
65,29 43.72
HCP-A-CARIT
0.488 1025
0.713 20000
67.60 11.74
66.79 11.29
HCP-A-Rest
0.404
0.768
72.00
71.30 22.19
D.7 SCALABILITY ANALYSIS
D.7.1 MODEL AND DATA SCALABILITY
The numerical results used to generate the scalability analysis plot (Figure 7) are presented in Table 14. The table includes detailed linear probing performance for three BDO model variants (5M, 21M, and 86M), evaluated on HCP-A age regression and classification tasks across ABIDE, ADHD200, and HCP-EP datasets. Additionally, results from the data scalability experiment, conducted exclusively with the largest model (86M), reported at varying proportions (25%, 50%, and 75%) of the total
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dataset. The entry labeled BDO (86M) corresponds to the model trained with the full dataset (100%).
serving as the reference for both model and data scalability experiments.
Table 14: LP performance used for scalability analysis in Figure 7.
HCP-A
ABIDE
ADHD200
HCP-EP
Variants
Age (p f)
ACC (%)
ACC (%) 1
ACC (%) f
BDO (5M)
0.635 1031
62,42 22.68
59.65
73.33 27.50
BDO (21M)
0.729 2011
63.79 21.33
61.15 #1.97
71.43 24.04
BDO (25%)
0.686 10000
61.06 11.05
57.39 +3.90
72.38 15.95
BDO (50%)
0.702 2014
63.03 21.63
56.89 23.38
74.29 29.90
BDO (75%)
0.734 2011
65.45 42.70
58.15 $1.78
74.29 47.56
BDO (86M)
0.768 +008
66.67 +1.13
61.40 +1.97
76.19 +4.86
D.7.2 ROI PARCELLATION SCALABILITY
To investigate the impact of spatial resolution, we conducted an ablation study varying the granularity of the ROI parcellation. We trained BDO (86M) variants using the Schaefer atlas with 200, 400, 600, 800, and 1000 parcels (Schaefer et al., 2017). All models in this experiment were pre-trained for 100 epochs and evaluated on the HCP-A dataset under the LP protocol.
Table 15: LP performance for ROI scalability analysis on HCP-A.
Age
Gender
Atlas
MSE +
P
ACC (%) f
F1 (%) 1
Schaefer-200
0.616 +.061
0.641 #.042
67.82 #0.87
66.04 +2.79
Schaefer-400
0.563 +.6
0.668 0.0
68.06 #1.50
67.02 #1.31
Schaefer-600
0.557 +.038
0.678 #.028
69.91 #1.43
68.76
Schaefer-800
0.571 +.058
0.669 4.039
73.38 20.87
72.82 +1.20
Schaefer-1000 0.537 2.068
0.695 +.050
74.31 +3.00
73.75 +2.84
As summarized in Table 15, we observed a general trend where increasing spatial resolution leads to improved downstream performance. This aligns with prior findings suggesting that finer parcellations capture richer representations of neural activity (Sacerdote & Giraudo, 2012). Specifically, the model trained with 1000 ROIs achieved the best performance across all metrics, confirming that framework effectively scales to utilize more granular spatial information.
is important highlight the computational efficiency the BDO framework regarding spatial scal- ability. Many existing transformer-based IMRI models typically tokenize ROIs into a sequence (Caro al., 2024; Dong al., 2024; Yang et al., 2024). In such architectures, increasing the number of ROIs (N) linearly extends the input sequence length, which in turn leads to quadratic increase (O(N2) in the computational and memory costs of the self atlfnattention mechanism. This makes high-resolution modeling prohibitively expensive.
In contrast, BDO projects the N N-dimensional ROI space at each time step into fixed d-dimensional latent space via linear projection before temporal processing. Therefore, increasing N only affects the number of parameters in the initial projection and final decoder layers, while the sequence length (I) and the computational complexity of the Transformer encoder (O(T2) remain unchanged. This structural design ensures that BDO scales efficiently with higher spatial resolutions, enabling feasible training even with fine- iin-nndde parcellations like 1000 ROIs. Furthermore, this architectural efficiency a key factor contributing to the superior training speed of BDO compared to other SSL basclines, detailed in the following section Section D.8.
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Table 16: Comparison of pre-training efficiency and linear probing performance across SSL models.
Model (Parameters)
Age (Pearson) t
Gender (Acc.)
GPU Hours (x GPUs) +
MoCo (90M)
0.591
64.12
174 hrs
BYOL (90M)
0.619
64.81
165 hrs
BrainLM (85M)
0.636
65.28
496 hrs
BrainMass (90M)
0.630
66.20
244 hrs
BDO (86M)
0.768
72.00
15 hrs
D.8 COMPARISON OF SSL MODEL EFFICIENCY
This subsection presents the detailed experimental settings and the exact numerical results used to construct Figure 1, which shows that BDO surpasses other SSL models in both resource and parameter efficiency.
To evaluate the efficiency of SSL models, measured the pre-training time using 4 NVIDIA RTX 3090 GPUs, calculated in GPU hours as the total CUDA time recorded with the PyTorch library and multiplied by the number of GPUs. Each model was trained for 200 epochs using the largest batch size that fully utilized available GPU memory.
Table 16 presents the linear probing performance of each pre-trained model on age and gender predic- tion tasks on HCP-A dataset, alongside their respective pre-training times, Our results demonstrate that BDO achieves superior efficiency in pre-training, requiring significantly fewer GPU hours com- pared to other SSL methods while maintaining competitive or superior performance. This efficiency highlights the scalability of BDO, making it practical choice for large-scale applications.
D.9 DETAILED MASK RATIO ANALYSIS
In our framework, the MAE objective plays a critical role by explicitly requiring the reconstruction of masked temporal segments. This encourages the model to capture detailed temporal dependencies and fine-grat tne-gga dynamics inherent fMRI signals. To directly validate the importance the MAE objective, we extended our ablation study include a mask ratio (r = 0), which effectively removes the MAE reconstruction task. Starting from the no-masking condition, the HCP-A age regression performance in the I.P setting for BDO-86M results are summarized below.
Table 17: Extended ablation study the mask ratio 7-
Mask Ratio (7)
Age (MSE)
Age (Pearson)
0.0 (No Masking)
0.793 + 0.014
0.445 0,020
0.2
0,487 # 0.036
0.711 # 0.027
0.4
0.013
0.695 0.015
0.6
0.47600.01
0.727 = 0.011
0.75 (Optimal)
0.466 0.025
0.738 0.014
0.8
0.526 + 0.014
0.686 + 0.006
In the no-masking condition, we observed severely degraded downstream task performance, indicating that the model failed to learn meaningful, transferable representations. This result implies that without the reconstruction challenge introduced by masking, SSL pre-training becomes ineffective capturing the complex temporal structures necessary for high-quality representation learning.
D.10 DISSECTING THE CONTRIBUTIONS OF MAE AND JEPA OBJECTIVES
To dissect the individual contributions of the MAE and JEPA objectives, conducted ablation study by controlling their relative influence with the balancing factor 7. A setting of T = corresponds an MAE-only model, and we also evaluated JEPA-only model without the reconstruction term.
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Gynue
Middl Oodgiaa Gyru
Supwrr Dcztpaal Fyyux
Postrantna Gyne
Angular Oyrus Supemargial Gyru
Angular Oyeua
Gyta
Gyese
Supertr Oodgitzl Gyrus
Gytss byyus
Midde Dooipial
2.20
Right Lateral View
Left Latoral View
Right Lateral View
Left Lateral View
Cuneus
Calartie Conne
Superior Fronial Gyrus, Madal
Right Medial View
Left Medial View
Right Medial View
Left Medial View
Figure 10: Brain surface visualization of IG scores. (Left) Age regression on the HCP-A dataset.
(Right) Psychotic disorder diagnosis on the HCP-EP dataset.
The results in Table 18 reveal a clear synergy. The MAE-only model (r = 0) establishes a strong performance baseline, while the JEPA-only model performs poorly, indicating the latent prediction task alone is insufficient. Crucially, the unified model with an optimal balance (T = 0.03) surpasses the MAE-only baseline. This demonstrates that the JEPA objective acts as a beneficial regularizer that refines the learned features via MAE, highlighting the complementary nature of the two objectives.
Table 18: Ablation study on the MAE and JEPA components.
Age (MSE) +
Age (Pearson) 1
JEPA-only
0.719 = 0.040
0,521 = 0.036
0
0.480 # 0.010
0.717 0.006
7 0.03
0.466 = 0.025
0.738 = 0.014
0.1
0.663 = 0.027
0.572 0.028
D.11 EXTENDED NEUROBIOLOGICA AND QUALITATIVE ANALYSES
D.11.1 NEUROBIOLOGICAL INTERPRETATION VIA INTEGRATED GRADIENTS
Integrated Gradients (IG) is an attribution analysis method from the field of explainable AI (XAI) that quantifies the contribution of each input feature in our case, each brain ROI-to model prediction (Sundararajan et al., 2017). The resulting IG score reflects how much given ROI positively or negatively contributes to the model output, relative near zero indicate minimal influence. To highlight the most decisive features, we computed the absolute IG scores for each subject and normalized them across ROIs to enable comparison of relative importance. IG scores were computed from the trained models for both the HCP-A age regression and HCP-EP schizophrenia diagnosis tasks using the Captum um (Kokhlikyan et al., 2020) library. The top 10 ROIs with the highest absolute IG scores summarized in Table 19 and Table 20 for each task, alongside their corresponding Yeo-7 network and AAL atlas labels.
reference baseline input. Scores
The spatial distribution of these attribution scores is visualized in Figure 10. For the age regression in HCP-A, the analysis highlighted regions integral to motor, cognitive, and sensory functions known undergo gelated alterations, specifically the left precentral gyrus (Zhou al., 2020), left medial superior frontal gyrus (Lamballais et al., 2020), and bilateral angular and occipital gyri (Fjell al., 2009). In the HCP-EP diagnosis, the analysis emphasized areas crucial sensory perception, executive control, and self-awareness all domains notably impaired in psychotic disorders including the right postcentral gyrus (Ferro al., 2015), bilateral superior occipital gyri (Tohid al., 2015), right middle frontal gyrus (Stoyanov al., 2021), and left superior parietal gyrus (Guo et al., 2014).
1.11.2 QUALITATIVE ANALYSIS OF LATENT DYNAMICS
To explore the characteristics the learned continuous time latent dynamics, we performed a qualitative analysis of the step-wise latent displacement, defined as Az; 2t+1 2t, across time for
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Table 19: Top 10 ROIs with the highest IG scores in the HCP-A age prediction task.
Rank
Yeo-7 Network Label
IG Score
AAL Atlas Label
7Newworks I.H SomMot 26
0.0074
Precentral I.
2
0.0061
Soontal_SupMMeet
3
7Networks Baacikss__SSaaaa Wwe.tHSSSaeettttnn...mm
0.0060
SupraMarginal R
4
7Networks 7Networks.RH_Defineee RH_Default.Par.5
0.0059
Angular
5
7Networks RH_ Vis_
0.0057
Calcarine
6
7Networks RH_ WW..aHlSSVeeaaaaa VentAAtn Teenttttt.emmmeePPa...
0.0056
SupraMarginal.
7
7Networks LH Vis 24
0.0056
Occipital Sup L
8
TNetworks LH Vis 21
0.0055
Calcarine L.
9
0.0054
Angular_L.
10
CNetworks.LH... 1is_
0.0054
Occipital
Table 20: Top 10 ROIs with the highest IG scores the HCP-EP diagnosis prediction task.
Rank
Yeo-7 Network Label
IG Score
AAL Atlas Label
7Network__Ri BH_ RH_Ssss_SSSSSmML
0.0082
Postcentral_R
2
7Networks_ B__isss_________ RH_Vis._V V2s_29
0,0056
Cuneus_R
3
7Networks LH Vis 29
0.0049
Occipital Sup
4
7Networks LH. Vis 27
0,0047
Occipital Midd_L
5
7Networks_LL. L.t.oosss_LLSSSMMooLLLoL _SomMott33
0.0047
Postcentral_L.
6
7Networks. RH. (eCnAAttnPPFLLI
0.0046
Frontal Mid 2
7
7Networks RH. Vis.26
0.0045
Occipital Sup.R
8
0.0042
Parietal_Sup.
9
0.0042
Postcentral
10
7Networks RH DorsAttn Post 4
0.0041
Occipital Mid R
each participant. We specifically examined the distribution of these displacements between healthy controls and participants diagnosed with ADHD from the ADHD200 dataset.
As visualized in Figure 11, we noted that healthy controls tended exhibit larger Az magnitudes compared to participants diagnosed with ADHD. While the precise neurobiological meaning of the latent space requires further verification, this pattern could be interpreted as reflecting more ac- transitions in
Histograms of Vel Velecity Norms
Control
1.5
0.2
tive moment-to-mont the latent representation of healthy brain dynamics. This observation shares similarities with recent resting- state fMRI studies, which have re- ported reduced neural flcxibility and diminished state-transition fluidity in individuals with ADHD (Yin et al., 2022; de Lacy & Calhoun, 2018). Al- though establishing a direct link be- tween our latent trajectories and these neurophysiological concepts remains a subject for future research, we present computational perspective the dynamical differences associated with ADHD.
610
6m
Liver tam Nu 1/1 1an.oo
Figure 11: Qualitative analysis of learned latent dynamics via step-wise latent displacement in ADHD200, The plot compares the distribution of step-wise latent displacements (Az) between Healthy Controls and ADHD patients. The ob- served lower magnitude of A in the ADHD group indicates fluid state transitions Less. as an interesting preliminary finding that may offer a
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D.12 ATTENTION BASED TEMPORAL AGGREGATION STRATEGIES
In our main framework, we employed a simple mean-pooling strategy to aggregate the learned latent control signals o, into the universal feature A. This design choice primarily driven by the need for architectural simplicity and computational efficiency, as well as to ensure a fair comparison with existing baselines that predominantly utilize peoa--oolingg (Caro et al., 2024; Dong et al., 2024).
However, we acknowledge that simple averaging inherently abstracts away temporal granularity, which may limit the model's ability to capture precise time-dependent dynamics, such as transient neural events or rapid neurophysiological fluctuations. To empirically investigate the potential benefits of preserving richer temporal structures, we conducted an additional ablation study using Pooling by Multihead Attention (PMA) (Lee et al., 2019). Unlike static averaging, PMA allows the model to dynamically weigh temporal features based on their relevance.
We compared the standard mean-pooling approach with PMA-based aggregation on the HCP-A dataset under the LP protocol. As summarized in Table 21, the PMA-base aggregation yields performance improvements across both age regression and gender classification tasks. These results suggest that sophisticated aggregation mechanisms capable attending to specific temporal structures enhance the quality of the final representation. While mean-poo mean-pooling serves robust and efficient standard for our current framework, exploring advanced temporal aggregation methods represents a promising avenue for future research to further improve temporal fidelity and performance.
Table 21: Comparison of aggregation strategies HCP-A.
Age
Gender
MSE 4
pt
ACC (%) f
(%)
BDO (86M) - Mean Pooling
0.404 #.010
0.768 #.008
72.00 +2.95
71.30 +2.19
BDO (86M) PMA
0.385 +.021
0.772 2.012
73.84 +1.43
73.21 +1.30
D.13 SIMULATION STUDY FOR THE IMPACT OF TEMPORAL MISMATCH
To empirically validate the impact of temporal mismatch on representation learning, we conducted a simulation study on the HCP-A dataset. We compared our model against variants where the TR information provided to the model was intentionally distorted. This setup simulates scenarios where the modeled time scale deviates from the actual physical time scale of the data. Specifically, we tested two distortion scenarios: Compressed TR, where the input TR provided to the model was artificially decreased, and Dilated TR, where the input TR was artificially increased.
Table 22: Impact of TR distortion on HCP-A.
Age
Gender
Method
MSE J
ACC (%)
F1 (%) 1
BDO (86M) Compressed TR
0.532 4.022
0.678 #.017
67.59 #2.62
66.48 #2.84
BDO (86M) Dilated TR
0.558 2.018
0.660
=.014
67.82 +1.99
66.98 #1.83
BDO (86M) Standard
0.404 +.010
0.768 1.008
72.00 +2.95
71.30 #2.19
As shown in Table 22, both types of distortion led to consistent performance degradation across all metrics. These results suggest that deviating from the true physical time scale limits the model's ability to capture robust representations. This highlights the critical importance of the exact integration scheme employed in BDO, which enables the model to effectively learn from heterogeneous datasets without suffering from the temporal mismatches that plague discrete, fixed-gria approaches.
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UKB Held-out Age Distribution in Splits Across Random Seeds
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Figure 12: Age distribution across training, validation, and test splits for the UKB held-out age regression task under three different random seeds (0, 1, and 2). The dataset is partitioned using a 6:2:2 ratio, with binning-basee stratified sampling applied to maintain a balanced target variable distribution. To enhance numerical stability, Z-score normalization is applied to the age variable. Each row represents a different random seed, illustrating the consistency of the sampling procedure across splits.
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HCP-A Age Distribution in Splits Across 3 Random Seeds
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Figure 13: Age distribution across training, validation, and test splits for the HCP-A age regression task under three different random seeds (0, and 2). The dataset is partitioned using 6:2:2 ratio, with binning-based based stratified sampling applied to maintain a balanced target variable distribution. To enhance numerical stability, Z-score normalization is applied the age variable. Each row represents a different random seed, illustrating the consistency of the sampling procedure across splits.
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UKB Hald Gender D Darry Si
BH192
Oin HCPA Gester Den Splts
90l 15-10
Drg 52
Tokking Tod())
Tee INAI2N
Figure 14: Label distributions across six classification tasks (UKB held-out gender, HCP-A gender, ABIDE autism, ADHD200 ADHD, and HCP-EP psychotic disorder) for training, validation, and test splits. Each row corresponds to different task, with columns representing the proportion of samples per class across data splits. Stratified sampling ensures that label distributions remain consistent across splits, despite variations in sample composition. To illustrate this, we visualize the distributions using a single random seed (0). Gender classification tasks are divided into Female/Male categories, while discase classification tasks distinguish between Control and Patient groups (ASD Control for ABIDE, ADHD vs. Control for ADHD200, and Psychotic disorder vs. Control for HCP-EP).
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Terget ROI Signel Reconsstructon in Internal Dataset UKB Hell-out (g = 0.75)
Original Shampto
Contoel Ibemze
Recorniraciod itampto to
MAG Dhample
225
-2.
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Time
Time
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Original 15ample
Contant 1Semiiic
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MAK Sammpl
-25
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Time
Time
25
7.5
25
Figure 15: Reconstruction quality of BDO in the UKB held-out subset (internal dataset). Five samples are randomly drawn for visualization, with mask ratio of y = 0.75. Each column represents the original IMRI sample, context with masking patterns, reconstructed sample, and MAE (Mean Absolute Error) heatmaps. Although we set the mask ratio high 75%, the reconstruction quality remains robust, demonstrating that BDO efficiently captures the underlying brain dynamics and successfully reconstructs missing regions with high fidelity.
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Target: ROI Signal Reconstruction External Dataset HCP-A 0.75)
Time
Original Stampl
Strrpee
Reconemutted SSample
(Simate
Original Dutu thumplic
Cortocc (Sumple
Rocomoiractoal SSumpli
VAA 1tumalic
line
Time
Time
g
Original Data 1Sampla
Cartbot Stumple
Recoremacedd Sample
MAG Losa (Samata
Time
Timc
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Figure 16: Reconstruction quality of BDO in HCP-A (external dataset). Five samples are randomly drawn for visualization, with mask ratio of 0.75. Each column represents the original IMRI sample, context with masking patterns, reconstructed sample, and MAE (Mean Absolute Error) heatmaps. Although set the mask ratio as high as 75%, the reconstruction quality remains robust, demonstrating that BDO efficiently captures the underlying brain dynamics and successfully reconstructs missing regions with high fidelity.
41


Reviews:
[{"review_id": "KHCEBFO2ck", "reviewer": "Reviewer_9gcz", "summary": "The authors propose a stochastic optimal control framework as a way to view fMRI pre-training. The model they end up with is intuitively similar to a variational auto encoder that is trained with masked auto encoding and is regularized with a Joint Embedding Predictive Architecture (JEPA) objective function. The model outperforms baselines across a variety of tasks and datasets, and the model exhibits good scalability.", "questions": "1) On lines 84-85 what do the authors mean by \"..., which encodes multi-scale brain activity onto a single continuous real-time axis.\"?\n2) Line 121-122: Why do the authors assume the data is being governed by an Ito diffusion process? There is a rich literature of biophysical dynamical models for fMRI data, including the FitzHugh–Nagumo model [1], the Wilson-Cowan model [2], and the Kuramoto model [3]. The authors provide no biophysical or neuroscientific explanation for this design choice.\n3) In the same paragraph on Lines 131-132 the authors mention \"... we initially assume the underlying changes are purely random, acknowledging the difficulty in modeling the intricate nature of fMRI signals beforehand.\". Why do the authors not just use a learnable prior or infer an initial condition?\n\n\n\n[1] Ghosh, A., Rho, Y., McIntosh, A. R., Kötter, R., & Jirsa, V. K. (2008). Cortical network dynamics with time delays reveals functional connectivity in the resting brain. Cognitive neurodynamics, 2(2), 115-120. \\\n[2] Deco, G., Jirsa, V., McIntosh, A. R., Sporns, O., & Kötter, R. (2009). Key role of coupling, delay, and noise in resting brain fluctuations. Proceedings of the National Academy of Sciences, 106(25), 10302-10307. \\\n[3] Cabral, J., Hugues, E., Sporns, O., & Deco, G. (2011). Role of local network oscillations in resting-state functional connectivity. Neuroimage, 57(1), 130-139.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The authors perform a wide range of experiments, describe an interesting framework for fMRI pre-training, and thoroughly ablate their model.", "weaknesses": "Major weaknesses:\n1) No comparison to recent foundation models, and limited related work in the main text. The authors have moved most of their related work to the Appendix. Especially given how quickly the foundation model field is evolving, I believe it is important to have a more substantial related work section than is currently present in the paper. Moreover, a few key foundation models are missing from the evaluations, nor do they discuss in depth differences between their approach and BrainJEPA for example. Namely, BrainJEPA [1] and BNT [2]. I cannot recommend the paper for publication without fair comparisons to these works.\n2) The authors claim that their stochastic optimal control framework solves heterogeneity in repetition time (TR) for the fMRI data, but it is unclear why this is true given the authors' model choices. Specifically, they mention on Lines 42-45 that \"... existing SSL methods often fail to fully capture the inherent temporal dynamics of fMRI by treating the time-series data as segmented patches or static connectivity graphs as described in Figure 2.\". However, this is not necessarily true. Transformers can still capture dynamics even when they use temporal patching. For example POYO [3] and NDT [4] that show that transformers can still capture dynamics even with temporal patching in the computational neuroscience domain. Moreover, the authors say their framework solves heterogeneous TRs, but still use a temporal transformer (see Figure 5) to create embeddings. It is unclear to me how the authors fix heterogeneous TRs when the transformer is pre-trained on a dataset with a certain spacing between patches. The current transformer does not seem to solve this issue even when incorporated into a stochastic optimal control policy.\n\nMinor weaknesses/types/grammar:\n- The authors keep mentioning (e.g. on Lines 76-79) that fMRI data is high-dimensional, which is true, but then also use anatomical regions of interest (ROIs) to reduce the data to 450 spatial dimensions. It is unclear to me why the authors seem to focus so much on the high dimensionality of the data when they do not contribute a specific way to address this fact.\n- To the previous point, the authors do not ablate the ROI choice.\n- Theorem 2.2 can be moved to the Appendix because it is not a novel theorem but a direct consequence of the papers cited in the previous paragraph. Especially given the fact that the authors moved their related work to the Appendix, I believe these types of derivations can be swapped for a more substantial related work section. If anything some of the mathematical derivations, including but not limited to this theorem, take away from the main contributions this paper makes.\n- The \"Masked auto encoders\" paragraph on page 5 repeats many of the ideas/claims in paragraph 2.2, I think either of these paragraphs can be shortened.\n- Lines 152-153: \"The SOC proves ...\" -> SOC provides\n- Lines 242-243: \"..., we use the MAE ...\" -> we use an MAE\n\n[1] Dong, Z., Li, R., Wu, Y., Nguyen, T. T., Chong, J., Ji, F., ... & Zhou, J. H. (2024). Brain-jepa: Brain dynamics foundation model with gradient positioning and spatiotemporal masking. Advances in Neural Information Processing Systems, 37, 86048-86073. \\\n[2] Kan, X., Dai, W., Cui, H., Zhang, Z., Guo, Y., & Yang, C. (2022). Brain network transformer. Advances in Neural Information Processing Systems, 35, 25586-25599. \\\n[3] Azabou, M., Arora, V., Ganesh, V., Mao, X., Nachimuthu, S., Mendelson, M., ... & Dyer, E. (2023). A unified, scalable framework for neural population decoding. Advances in Neural Information Processing Systems, 36, 44937-44956. \\\n[4] Ye, J., & Pandarinath, C. (2021). Representation learning for neural population activity with neural data transformers. arXiv preprint arXiv:2108.01210.", "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": "lMMiLB4nkB", "reviewer": "Reviewer_SCN9", "summary": "This paper proposes Brain Dynamics with Optimal control (BDO), a novel self-supervised learning\n(SSL) framework for learning representations from fMRI time-series. The core problem this work\naddresses is the temporal irregularity and noise inherent in multi-site fMRI datasets, which is often\nhandled by existing methods by discarding or discretizing temporal information.\n\nBDO reframes fMRI representation learning as a Stochastic Optimal Control (SOC) problem. It models\nlatent brain activity as a continuous-time stochastic differential equation (SDE) and learns a control\npolicy (parameterized by a Transformer) that steers a prior SDE to a posterior SDE that best explains\nthe observed, noisy fMRI data. This control policy itself serves as the learned representation.\n\nThe framework provides a principled unification of Masked Autoencoding (MAE) and Joint-Embedding\nPredictive Architectures (JEPA). The SOC objective is shown to be equivalent to an ELBO (MAEstyle reconstruction), while a JEPA-style latent target (from an EMA-updated encoder) is introduced\nto prevent the model from overfitting to signal noise. To ensure scalability, the paper introduces a\nsimulation-free inference method based on a locally linear SDE assumption, which admits a closed-form\nsolution. In addition, the model is pre-trained on the large-scale UKB dataset and demonstrates stateof-\nthe-art performance on a wide range of downstream tasks, including trait and diagnosis prediction,\non both internal (UKB held-out) and external (HCP-A, ABIDE, etc.) datasets.", "questions": "1. Can you please clarify the apparent inconsistency in the control policy? Is the implemented\npolicy αθti(Y) (open-loop) or α(t,Xθt ,Y) (closed-loop)? If it is open-loop, how is this reconciled\nwith the SOC theory in Proposition 2.1, which is based on a state-dependent policy?\n\n2. The final feature A is a mean-pooling of the control policy, which discards all temporal information.\nWhy was this chosen? Have you experimented with using the learned latent states Xt\nor a more temporally-aware aggregation of {αt} (e.g., attention, final hidden state) as the final\nrepresentation?\n\n3. How is the drift matrix Dti = V ΛtiV ⊤ in the linear SDE (Eq. 10) parameterized? Does the\nTransformer encoder output V and all {Λti}? Is V shared across all time steps and subjects?\n\n4. In Appendix D.11, you mention a difference in per-sample zero-mean normalization might be\nresponsible for the failed Brain-JEPA reproduction. Could you elaborate on this? This seems\nlike a subtle difference to cause a full failure, and understanding it would strengthen the baseline\ncomparisons.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Novel and Elegant Formulation: Reframing SSL for fMRI as a Stochastic Optimal Control\nproblem is a highly novel and theoretically compelling contribution. This approach is a natural\nand powerful fit for modeling continuous-time, noisy, and irregularly-sampled data, which is a\nkey challenge in fMRI analysis.\n\n2. Principled Unification of MAE and JEPA: The paper does an excellent job of motivating\nand technically unifying two powerful SSL paradigms. Deriving the MAE-like objective from\nthe SOC formulation (Prop 2.1) is elegant. The subsequent introduction of a JEPA-style latent\ntarget (Eq. 7-9) to regularize this objective and avoid overfitting to noise is a well-reasoned and\ntechnically sound improvement.\n\n3. Impressive Computational Efficiency: A major contribution is the simulation-free inference\n(Theorem 2.2). By using a locally linear SDE approximation, the authors derive a closed-form\nGaussian posterior for the latent states, which can be computed in parallel with O(log k) complexity.\nThis makes a theoretically complex SDE/SOC framework computationally practical for\nmassive datasets like UKB. The empirical result of 15 GPU hours for pre-training (Table 10) is\nextremely strong compared to baselines.\n\n4. Strong and Comprehensive Empirical Validation: The experimental results are state-of-the-art and convincing. BDO outperforms a wide array of baselines (both task-specific and\nfMRI-specific SSL models) across numerous diverse tasks (age, gender, diagnosis) and datasets\n(UKB, HCP-A, ABIDE, ADHD200, HCP-EP). The generalization from resting-state pre-training\nto task-based fMRI (Appendix D.5) is also a strong result.", "weaknesses": "1. Inconsistency in Control Policy Formulation: There appears to be a critical inconsistency\nbetween the theoretical formulation and the practical implementation of the control policy α.\nThe SOC theory (Eq. 3, Prop 2.1) defines a closed-loop control policy α(t,Xθt ;Y), which depends\non the current latent state Xθt . This is standard for optimal control. However, the simulation-free implementation (Eq. 10) and the architecture diagram (Fig. 5) use what appears to be an\nopen-loop policy αθti(Y), which is pre-computed from the fMRI data Y and does not depend on\nthe current state Xθt . This is a significant simplification that deviates from the theory. While it\nenables the efficient closed-form solution, the paper needs to explicitly state this simplification\nand justify how the SOC theory (Prop 2.1) still holds for this open-loop policy.\n\n2. Simplistic Final Representation: The model’s universal feature A is derived by a simple\nmean-pooling of the control policy sequence {αt}. This design choice seems to contradict the\npaper’s primary motivation, as it discards the rich temporal structure that the continuous-time\nSDE model worked so hard to capture. It is surprising that this simple pooling outperforms\nother methods, and it feels like a missed opportunity.\n\n3. Clarity on Linear SDE Parameters: The locally linear SDE in Eq. 10 (dXθ\nt = [−DtiXθt +αθti ]dt + dWt) is key to the model’s efficiency. However, it is unclear how the drift matrix Dti\nis parameterized and learned. Theorem 2.2 states Dti = V ΛtiV ⊤, which implies it changes over\ntime. How are the shared eigenbasis V and the eigenvalues Λti generated by the encoder network?\nThis is a crucial, missing architectural detail.\n\n4. Baseline Reproducibility: The authors note in Appendix D.11 that they could not reproduce\nthe Brain-JEPA baseline with their own preprocessing, which is a minor red flag. While they\nprovide a fair comparison using the original Brain-JEPA preprocessing (where BDO still wins),\nit would be more convincing to understand *why* the reproduction failed.", "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": "Tn569HPA96", "reviewer": "Reviewer_cuG3", "summary": "The paper proposes a self-supervised learning framework called Brain Dynamics with Optimal Control (BDO) particularly developed to meet the challenges arising when dealing with fMRI data. It consists in explicitly modelling continuous-time latent brain dynamics using stochastic optimal control (SOC). Unlike existing methods that discretize or average fMRI signals and lose temporal information, BDO treats brain activity as a stochastic differential equation (SDE), capturing multi-scale temporal structure while being robust to heterogeneous sampling protocols. Within the SOC formulation, the framework unifies two major SSL paradigms, Masked Autoencoding (MAE) and Joint Embedding Predictive Architecture (JEPA), allowing the model to learn compact, control-based representations of brain dynamics. Furthermore, to ensure scalability, the authors introduce a simulation-free inference strategy based on locally linear SDE approximations. Experiments on large multi-site fMRI datasets show that BDO achieves state-of-the-art performance and strong transferability across diverse downstream tasks.", "questions": "Ca authors add positioning of their work w.r.t. following two nicked references, and potentially add this in Appendix A?\n\n- Hassan et al. *Identifying Latent Stochastic Differential Equations*.IEEE Signal Processing 2021\n\n- Bartosh et al.  *SDE Matching: Scalable and Simulation-Free Training of Latent Stochastic Differential Equations*. ICML2025", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "First, I find paper exceptionally well written. It posses all key qualities: clarity, smooth flow of ideas, good balance presenting both intuition and technical complexity, appendix complements adequately the main text. \n\nSecond, the paper tackles an important problem and proposes a method that is applicable beyond targeted fMRI applications. \n\nThird, the innovation to use SOC in SSL context is, up to my knowledge, clear and it also consists in combing MAE and JEPA.  While, focusing on linear drifts and exploiting Gaussian distributions to scale-up is relatively standard idea, it is executed well and is a very good first strategy for the proposed method.\n\nFinally, the experimental setup and baselines are adequate, and clearly shows the main benefits coming from BDO.", "weaknesses": "While I do not find any strong weakness of the paper, here is a few minor issues:\n\n**Minor**\n- While full names of acronyms such as JEPA and MAE are introduced in the abstract, it would be helpful to still write them in the main text when the concepts are introduced.\n\n- Typo in Eq. (3): in drift part should read $\\alpha^\\theta$", "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": "D3STFPGDTr", "reviewer": "Reviewer_8VFs", "summary": "This paper proposes BDO, a deep latent model for self-supervised representation learning from fMRI time series with heterogeneous temporal resolutions. The approach formulates representation learning as a stochastic optimal control (SOC) problem over continuous-time latent dynamics governed by SDEs, where the control policy acts as an encoder. The model combines masked autoencoding (MAE) and joint-embedding prediction (JEPA) in a single SOC framework and introduces a simulation-free inference scheme for computational efficiency. Trained on large-scale resting-state fMRI (UK Biobank) and evaluated on multiple public datasets (HCP-A, ABIDE, ADHD200, HCP-EP), BDO achieves higher downstream task performance (e.g., age and diagnosis prediction) compared to prior SSL baselines.", "questions": "- Did you examine or visualize the inferred latent trajectories to assess whether they capture meaningful brain dynamics (e.g., temporal smoothness, subject-specific signatures)?\n- How sensitive are the results to the dimensionality of the latent space?\n- How do your “simulation-free” results compare against a full latent-SDE baseline with numerical solvers?\n- Could the same results be achieved by a neural SDE-VAE or GRU-ODE-Bayes setup without the SOC formalism?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Tackles a highly relevant and timely topic: representation learning for continuous, irregularly sampled neuroimaging data.\n- The SOC formulation is mathematically elegant and could offer a principled way to model latent neural dynamics.\n- Integration of MAE and JEPA in a single control-theoretic objective is technically interesting.\n- Results on multiple benchmarks are empirically strong and suggest robustness across datasets.", "weaknesses": "1.\tLimited insight into learned dynamics.\nThe latent SDE component is central, but no qualitative or quantitative analysis of the learned latent trajectories or their neurobiological structure is provided. As a result, it is unclear whether the model learns meaningful neural dynamics or primarily acts as an advanced denoising / feature-compression scheme.\n\n2.\tClarity and accessibility.\nThe exposition is mathematically dense and often assumes familiarity with SOC and variational inference. The connection to reinforcement learning, while conceptually appealing, is not sufficiently explained or justified. Figures are mostly schematic and do not substantially aid understanding of the pipeline.\n\n3.\tPositioning and novelty.\nThe work seems closely related to recent latent SDE and neural ODE formulations for neuroimaging (e.g., ElGazzar & van Gerven, 2024), yet these are not cited or compared. It remains unclear whether the main novelty lies in the SOC framing or in the MAE-JEPA combination, and whether either leads to qualitatively different behavior.\n\n4.\tEvaluation design.\nThe chosen downstream tasks (e.g., age regression, gender classification) are convenient but not strongly motivated as demonstrations of dynamic modeling. More suitable evaluations—such as temporal forecasting, dynamic functional connectivity analysis, or representational alignment across TRs—would have made the contribution clearer.\n\n5.\tPresentation and writing.\nThe introduction and motivation sections are somewhat opaque; key intuitions are missing. While the results are broad, the paper feels like a mathematically strong model attached post-hoc to standard neuroimaging tasks rather than driven by neuroscientific questions.\n\nElGazzar, A., & Gerven, M. van. (2024). Generative Modeling of Neural Dynamics via Latent Stochastic Differential Equations (No. arXiv:2412.12112). arXiv. https://doi.org/10.48550/arXiv.2412.12112", "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)