Despite the rapid advancements in Artificial Intelligence (AI), Stochastic Differential Equations (SDEs) remain the gold-standard formalism for modeling systems under uncertainty. However, applying SDEs in practice is fraught with challenges: modeling risk is high, calibration is often brittle, and high-fidelity simulations are computationally expensive. This technical report introduces JointFM, a foundation model that inverts this paradigm. Instead of fitting SDEs to data, we sample an infinite stream of synthetic SDEs to train a generic model to predict future joint probability distributions directly. This approach establishes JointFM as the first foundation model for distributional predictions of coupled time series - requiring no task-specific calibration or finetuning. Despite operating in a purely zero-shot setting, JointFM reduces the energy loss by 21.1% relative to the strongest baseline when recovering oracle joint distributions generated by unseen synthetic SDEs.
Figures & tables
Figure 1: JointFM, the digital quant. Traditional quantitative modeling follows a three-stage select-calibrate-simulate pipeline (top): choose a system of stochastic processes, fit parameters to historical data, and then simulate future paths [ 22 , 12 ] . JointFM replaces this workflow by pretraining on a universe of synthetic SDE dynamics and directly predicting future joint probability distributions from context in a single forward pass (bottom). Image generated with Gemini.
Level
New Dynamic
SDE Term / Mechanism
0
Drift & deterministic trends
b(t,Xt)dt
1
Diffusion, nonlinear drift, sinusoidal forcing
Σ(t,Xt)dWt,g(Xt)dt,s(t)dt
2
Block correlations (intra-F, intra-Y)
Corr(dWF),Corr(dWY)
3
Cross-block correlations
ρFY
4
Full global correlation
ρglobal∈R(M+N)×(M+N)
5
Compound Poisson jumps [ 27 ]
J(t,Xt)dNt
Table 1: Complexity curriculum : each level introduces a new dynamic on top of all previous levels. The right column names the corresponding SDE term or mechanism: dWF and dWY denote the Brownian increments of the M exogenous features F and the N target series Y , respectively; ρFY is the cross-block correlation between features and targets; and ρglobal∈R(M+N)×(M+N) is the full joint correlation matrix. Results in this paper use levels 0–7, only well-behaved Markovian dynamics. Future work will explore non-Markovian memory and more complex regime structures.
Sector
Drift / Trend ( dt )
Diffusion / Volatility ( dWt )
Jumps ( dNt )
Memory / Regimes
Finance
Vasicek / Hull–White Mean-reversion in rates & spreads.
Correlated GBM State-dependent volatility with cross-asset dependence.
Merton / Bates Price gaps upon macro announcements.
Regime Switching Seasonal vs. non-seasonal buying regimes.
Table 2: Industry-Agnostic Generalization: Mapping SDE components to industry-specific dynamics.
Figure 2: Recovering joint distributions. Zero-shot synthetic distribution recovery across model families and baselines. Lower values indicate better distributional match on energy loss, marginal energy, and CRPS-sum. JointFM variants consistently outperform Historical Simulation and DCC-GARCH, with the performance gap becoming increasingly pronounced as the forecast horizon lengthens.
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Figure 3: Training data. This example stems from SDE sampler as it is configured for our recovery experiment but showing only four targets.
Time-series foundation models are pretrained sequence models designed to transfer across forecasting, classification, anomaly detection, and other temporal tasks.
Modern time series forecasting is evaluated almost entirely through passive observation of single historical trajectories, rendering claims about a model's robustness to non-stationarity fundamentally unfalsifiable. We propose a paradigm shift toward interventionist, exact-statistical benchmarking. By systematically titrating calibrated Gaussian observation noise into known chaotic and stochastic dynamical systems, we transform forecasting from a black-box sequence matching game into an exact distributional inference task. Because the underlying data-generating process and noise variance are mathematically explicit, evaluation can rely on exact negative log-likelihoods and calibrated distributional tests rather than heuristic approximations. To fully leverage this framework, we extend the Fern architecture into a probabilistic generative model that natively parameterizes the Symmetric Positive Definite (SPD) cone, outputting calibrated joint covariance structures without the computational bottleneck of generic Jacobian modeling. Under this rigorous evaluation, we find that state-of-the-art zero-shot foundation models behave consistently with the context-parroting mechanism, failing systematically under non-stationary regime shifts and elevated noise. In contrast, Fern explicitly captures the invariant measure and multivariate geometry of the underlying dynamics, maintaining structural fidelity and statistically sharp calibration precisely where massive sequence-matching models collapse.
Time series foundation models (TSFMs) are increasingly pre-trained on synthetically generated time series trajectories, where the data generating process is known. Current pre-training recipes are based on loss objectives which compare TSFM outputs to realized future values of each trajectory. We instead propose loss objectives which compare TSFM outputs to the conditional forecast distribution of each trajectory, a procedure we call synthetic data distillation (SDD). SDD corresponds to a Rao-Blackwellization of the training objective, in that it leaves the expectation of stochastic gradients unchanged while provably reducing the covariance of the stochastic gradient under the Loewner partial ordering. We empirically validate SDD on a TSFM model family of sizes from 4M to 2.5B parameters, and observe faster convergence of validation loss at every model size: on Gaussian Process data, SDD attains or improves upon the Status Quo loss whilst requiring 10%−40% less training iterations.
Probabilistic time series foundation models (TSFMs) provide coordinate-wise predictive distributions, but these marginals do not determine a joint distribution over multivariate future trajectories. We study training-free coupling of frozen TSFM marginals into multivariate forecast sample paths. Our primary evaluation fixes the empirical marginal sample multiset at every channel--horizon coordinate across methods, isolating the effect of coupling alone. Historical temporal and channel relations substantially improve their corresponding dependence diagnostics. The same pattern persists when the fixed-marginal constraint is removed and paths are sampled directly, and remains present under native multivariate backbone inference. These results support treating dependence reconstruction as a distinct post-processing problem for probabilistic TSFMs.
Jinmyeong Choi, Jinkwan Jang, Seul Lee +1
Graduate School of Data Science Seoul National University