Organizations: Sustainability and Data Sciences Laboratory (SDS Lab), Northeastern University, Boston, MA, USA · AI4CaS: AI for Climate and Sustainability, Institute for Experiential AI, Northeastern University, Boston, MA, USA · Pacific Northwest National Laboratory (PNNL), Richland, WA, USA
Abstract
We study the problem of assigning a scalar score to a short trajectory window that reflects its position on an ordered continuum of predictability regimes, spanning structured deterministic dynamics to unstructured stochastic noise. Existing methods address deterministic-versus-stochastic discrimination within a single system and do not produce scores with a consistent numerical interpretation across systems. We formalize this as ordinal estimation over a five-level predictability ladder and identify a structural source of cross-system ambiguity: ranking supervision alone leaves the score coordinate unfixed up to a monotone reparameterization, which we term the gauge freedom of ordinal scoring. We propose the Gauge-Fixed Ordinal Network (GON), a temporal convolutional model trained with an anchor-and-variance objective that pins level-wise score means to shared target coordinates. GON operates on 2-jet features that expose local trajectory geometry, preserved by smooth flows and disrupted by stochastic surrogate procedures. On five held-out dynamical systems, initializing from a pretrained GON checkpoint consistently outperforms training from scratch across all window budgets, with adaptation depth reflecting geometric proximity to the training family. Zero-shot scores retain ordinal structure at the stochastic boundary, where surrogate procedures most strongly disrupt nonlinear geometry, and pretrained initialization consistently beats scratch across all window budgets. Pairwise discrimination and globally coherent ordinal scoring are distinct properties requiring a stable score coordinate for cross-system transfer, with direct implications for predictability assessment, model selection, and early-warning diagnostics across natural and engineered dynamical systems.
The growing availability of trajectory datasets has fueled major advances in data-driven motion prediction. Yet, models trained on one dataset often fail to generalize beyond their training domain as a result of differences in scene layouts, agent behaviors, and sensing conditions. A framework that learns latent representations of datasets and quantifies their similarity using distributional metrics is presented. This large-scale study covers 24 major datasets, including the most widely used motion-prediction benchmarks, and shows that the resulting transferability scores strongly correlate with cross-dataset model performance. The results provide practical guidance for dataset selection, pretraining, and large-scale foundation models for motion prediction, paving the way toward more generalizable and robust predictive systems.
We introduce First-Order Trajectory Matching (FTM), a surrogate-modeling method that learns the first-order local transport of probability mass from trajectories of stochastic systems. By matching the symmetric first-order motion of trajectories, FTM learns the probability current velocity, whose flow preserves time marginals to match ensemble averages, while also capturing current-like trajectory quantities such as fluxes, circulations, and barrier-crossing currents. FTM learns the current velocity directly from trajectories, avoiding drift, diffusion, and score estimation. Our stability analysis separates discretization error from sampling variance and shows that the one-step simulation-free FTM loss is stable when temporal resolution and sample size are properly balanced. Across stochastic dynamical systems and PDE examples, we empirically demonstrate that FTM provides trajectory-aware ensemble predictions at low, deterministic-rollout cost.
We report a systematic failure mode in predictive representation learning. Across 2695 neural network configurations trained to predict linear-Gaussian dynamics, the optimal encoder tracks the environment rather than the system it is meant to model. The mean causal fidelity -- the fraction of encoder sensitivity allocated to system degrees of freedom -- is 0.49, and only 2.5% of configurations exceed 0.70. The failure intensifies with dimension: at N=100, the optimal encoder becomes causally blind (fidelity ~10^{-8}) while achieving 92% lower prediction error than the causal representation. We prove this is not an optimization artifact but a structural property of the predictive objective: when environment modes are slower or less noisy than system modes, every minimizer of the population risk encodes the former. The set of dynamics exhibiting this predictive-causal gap is open and of positive measure in parameter space. In a nonlinear Duffing-GRU sweep, unconstrained predictors learn environment-dominant representations in 55% of tasks (95% CI 41--68%) versus 24% under operational grounding (p=2.3e-3); the median out-of-distribution MSE inflation under environment shift is 1.82x versus 1.00x. Operational grounding -- restricting the loss to system observables -- partially suppresses the gap, but causal fidelity is never recovered without an explicit system-environment boundary. The results identify the predictive-causal gap as a structural limit of learning, with implications for self-supervised representation learning, world models, and the scaling paradigm.