Many problems in computational science and engineering become one-to-many after coarse graining, partial observation, or inverse reconstruction: a resolved state may not determine a unique subgrid forcing, a structural descriptor may not determine a unique effective response, and a low-resolution observation may correspond to many plausible high-resolution fields. In such settings, deterministic surrogates may learn a well-defined mathematical object while still missing application-relevant uncertainty. This tutorial develops a self-contained module centered on the conditional-mean barrier: the point at which a squared-loss predictor has reached the conditional mean and the remaining error is irreducible aleatoric variance. We give two diagnostics for locating this barrier, residual-feature orthogonality and the coefficient of determination against its explained-variance ceiling, and prove that adding latent randomness to a squared-loss predictor collapses it back to the conditional mean. Crossing the barrier therefore requires a loss that scores distributions rather than point predictions. We briefly organize common distributional objectives, including negative log-likelihood, moment and observable matching, variational objectives, adversarial divergences, and score matching, by the feature of the conditional law each targets. The emphasis is the boundary itself and a finite-data procedure for recognizing it, rather than a survey of methods beyond it. CPU-based demonstrations on a two-branch law and a two-scale Lorenz-96 closure problem show how the diagnostics distinguish deterministic underfitting from residual distributional variability.

Stochastic reduced-order models are widely used to represent the effective dynamics of complex systems, but estimating their drift and diffusion coefficients from data remains challenging. Standard approaches often rely on short-time trajectory increments, state-space partitioning, or repeated simulation of candidate models, which become unreliable or computationally expensive for high-dimensional systems, coarse temporal sampling, or unevenly sampled data. We introduce a data-driven calibration method based on a novel relationship between the coefficients of a stochastic reduced model and the conditional score of the finite-time transition density, defined as the gradient of the logarithm of the transition density with respect to the initial state. The resulting identity expresses derivatives of lagged correlation functions as stationary expectations over observed lagged pairs involving this conditional score and the unknown model coefficients. This formulation allows the drift and diffusion structure to be constrained directly from finite-lag statistics, without differentiating trajectories, partitioning state space, or repeatedly integrating candidate reduced models during calibration, yielding a least-squares fitting problem over stationary lagged pairs. We validate the approach on analytically tractable and data-driven nonequilibrium diffusions, demonstrating that the inferred models preserve the invariant statistics while accurately reproducing finite-lag dynamical correlations. The framework provides a scalable route for learning stochastic reduced-order models from data that reproduce prescribed statistical and dynamical properties.

Similar ideas can be found in recent developments inspired by attention, such as RWKV [Peng et al.,

Firstly, many machine learning models use optimization algorithms which require access to derivatives of the model.

We propose ResidualPlanner, a matrix mechanism for marginals with Gaussian noise that is both optimal and scalable.

GVQA (from VQA-CP) builds on stacked attention networks (SAN).13 However,SAN and,byextension, GVQA architectures donotevaluate for,andgeneralize poorly on,17 unseen object attributes (CLEVR-CoGenT) and linguistic structural pattern (CLOSURE) combinations. The language parser is not trained, constructs text (s) object graphs (Gs) using rules-based entity38 recognizer [L126].(W3)CLOSURE ClarityMinor clarifications -5a: corrected inthecamera-ready version.

Modelling across engineering, systems science, and formal methods remains limited by binary relations, implicit semantics, and diagram-centred notations that obscure multilevel structure and hinder mechanisation. Hypernetwork Theory (HT) addresses these gaps by treating the n-ary relation as the primary modelling construct. Each relation is realised as a typed hypersimplex - alpha (conjunctive, part-whole) or beta (disjunctive, taxonomic) - bound to a relation symbol R that fixes arity and ordered roles. Semantics are embedded directly in the construct, enabling hypernetworks to represent hierarchical and heterarchical systems without reconstruction or tool-specific interpretation. This paper presents the structural kernel of HT. It motivates typed n-ary relational modelling, formalises the notation and axioms (A1-A5) for vertices, simplices, hypersimplices, boundaries, and ordering, and develops a complete algebra of structural composition. Five operators - merge, meet, difference, prune, and split - are defined by deterministic conditions and decision tables that ensure semantics-preserving behaviour and reconcile the Open World Assumption with closure under rules. Their deterministic algorithms show that HT supports reproducible and mechanisable model construction, comparison, decomposition, and restructuring. The resulting framework elevates hypernetworks from symbolic collections to structured, executable system models, providing a rigorous and extensible foundation for mechanisable multilevel modelling.

This paper investigates how the dynamic articulatory model DYNARTmo accounts for articulatory tradeoffs between primary and secondary articulators, with a focus on lips-jaw and tongue-jaw coordination. While DYNARTmo does not implement full task-dynamic second-order biomechanics, it adopts first-order task-space gesture specifications comparable to those used in articulatory phonology and integrates a simplified mechanism for distributing articulatory effort across multiple articulators. We first outline the conceptual relationship between task dynamics and DYNARTmo, emphasizing the distinction between high-level task-space trajectories and their low-level articulatory execution. We then present simulation results for a set of CV syllables that illustrate how jaw displacement varies as a function of both place of articulation (labial, apical, dorsal) and vowel context (/a/, /i/, /u/). The model reproduces empirically attested patterns of articulatory synergy, including jaw-supported apical closures, lower-lip elevation in bilabial stops, tongue-jaw co-movement, and saturation effects in labial constrictions. These results demonstrate that even with computationally simplified assumptions, DYNARTmo can generate realistic spatio-temporal movement patterns that capture key aspects of articulatory tradeoff and synergy across a range of consonant-vowel combinations.