The Temporal Markov Transition Field

Tabular foundation models (TFMs) provide a promising route to time-series classification, but their effectiveness depends on how sequential data are converted into tabular representations. Existing representations face two challenges: global aggregation can lose the order of temporal evolution, while features computed in independently fitted coordinate systems may not have consistent meanings across sequences. We therefore view representation design for TFMs as a problem in its own right: the representation should preserve local temporal transitions while maintaining a shared feature definition across samples. We propose SwitchPFN, which learns a shared projection and regime codebook from the training sequences, making local dynamic operators and transition features directly comparable across samples. Across the evaluated benchmarks, SwitchPFN achieves the highest mean accuracy among the evaluated methods, improving over the strongest baseline by 4.47% relatively. Ablation studies, parameter sensitivity analyses, and reduced-training-data experiments further examine the contributions of the representation, its main design choices, and its behavior when labeled data are limited.

Modeling the dynamics of non-stationary stochastic systems requires balancing the representational power of deep learning with the mathematical transparency of classical models. While classical Markov transition operators provide explicit, theoretically grounded rules for system evolution, their empirical estimation collapses due to severe data sparsity when applied to high-resolution, high-noise environments. We explore this statistical barrier using financial time series as a canonical, real-world testbed. To overcome the degeneracy of empirical counting, we introduce a framework that utilizes neural networks strictly as parameterization engines to generate explicit, time-varying Markov transition matrices. By constraining the neural network to output its predictions as a formal stochastic operator, we maintain complete structural interpretability. We demonstrate that these learned operators successfully capture complex regime shifts: the state-conditioned model achieves mean row heterogeneity $\barρ = 0.0073$ while the state-free ablation collapses to exactly zero, and operator row entropy correlates with realized variance at $r = -0.62$ ($p \approx 10^{-251}$), revealing that high-volatility regimes homogenize transition dynamics rather than diversify them. Furthermore, rather than enforcing the Chapman-Kolmogorov equations as a rigid structural requirement, we repurpose them as a localized diagnostic tool to pinpoint specific temporal windows where first-order memory assumptions break down. Ultimately, this framework demonstrates how neural networks can be constrained to make rigorous, classical operator analysis viable for complex real-world time series.

We present FM-ChangeNet, a pathwise-supervised framework for change detection that reformulates bi-temporal reasoning as continuous transport in feature space rather than static endpoint comparison. Given encoded pre and post-temporal representations, we construct intermediate latent states and learn a time-conditioned velocity field $\hat{v}_θ(z_t,t)$ along the transformation trajectory. This pathwise formulation constrains the predictor over a continuum of intermediate states, providing a denser and less ambiguous supervision signal than conventional endpoint-only segmentation and enabling the model to capture temporal evolution explicitly. The learned velocity field is not only a transport mechanism but also an interpretable representation of change: its magnitude serves as a spatially localized change cue that helps distinguish true structural variation from nuisance effects such as illumination shifts and spatial misalignment. We develop a hierarchical multi-scale architecture with cross-temporal alignment, time-conditioned coarse-to-fine flow decoding, and a unified objective that couples flow supervision, trajectory consistency, spatial regularization, and segmentation loss. Experiments on remote sensing benchmarks show that the proposed framework produces more structured and robust change representations while achieving state-of-the-art performance.