Marrying Optimal Transport and ODEs for Unified Continuous-Time 4D Reconstruction and Tracking

Existing unified 4D reconstruction and point tracking approaches typically rely on heuristic interpolations or just predict at integer timestamps, lacking kinematic coherence and failing to model dynamics at any arbitrary timestamp. In this paper, we propose Uni4R, a framework that unifies these tasks by learning continuous velocity fields through the synergy of Optimal Transport (OT) and Ordinary Differential Equation (ODE). Importantly, this continuous velocity field acts as a kinematic prior that mutually benefits both 4D reconstruction and point tracking. Specifically, we propose the Flow Matching Guided Decoder (FMGD). A global velocity branch first extracts anchor features that capture the global dynamic state of the sequence. Then, FMGD leverages Flow Matching (FM) theory to formulate a probability path defined by OT on the anchor feature manifold, instantiating it as FM-guided velocity features for velocity prediction. This establishes a robust kinematic inductive bias. Meanwhile, a point reconstruction branch provides geometric features. The local velocity prediction module then joint above features and time embeddings, to decode velocities at arbitrary timestamps. To overcome the absence of high-quality ground-truth velocities in fractional frames, we propose an integral-consistency training strategy. This strategy uses an ODE solver to integrate velocities to recover target pointmaps, enabling the model to be supervised end-to-end directly from integer timestamps. Experimental results demonstrate that Uni4R achieves SOTA performance in both 4D reconstruction and point tracking, and achieves SOTA in our new kinematics-aware benchmark at continuous time.

Causal Graphs, Markov Properties and Do-calculus for Stochastic Differential Equations

Stochastic differential equations (SDEs) are widely used to model continuous-time dynamical systems, but graphical causal models for them are not yet well-understood. We consider systems of causal SDEs that are equipped with an explicit causal semantics. We pose solvability conditions for systems of causal SDEs such that they have well-defined observational and interventional distributions - even after marginalisation - and provide a general class of Lipschitz semimartingale SDEs that satisfies these conditions. As core results we establish the $σ$-separation Markov property and the do-calculus in terms of the system's causal graph for probabilistic independence and interventions on the level of sample paths. For a class of additive-noise SDEs we prove a stronger $d$-separation Markov property, even if the system is cyclic. As a corollary of the do-calculus, we obtain an explicit causal interpretation of the graph: that the absence of a directed path implies the absence of a causal effect. We further introduce time-split systems, which consider the causal relations between the processes when evaluated on disjoint intervals or time-points, and use them to reason about subsampled time-series, continuous-time Granger non-causality and local independence. Finally, we discuss how constraint-based causal discovery algorithms (PC, FCI, CCD, CCI) apply directly to SDEs within our framework when conditional independence between sample paths can be consistently tested.

On the Necessity of a Liquid Substrate for Mesh Intelligence

A mesh of sovereign agents has no center: no shared clock, no shared model, and no coordinator to gather data or retrain. Its competence rests on each agent folding the projections its peers emit into a single internal state, online, from observations that arrive at irregular, unscheduled times, on a substrate whose weights it cannot retrain. Any one of these constraints is tractable on its own; folding optimally under all three at once is not. We ask what such a substrate must be, and prove two necessary conditions from one model of a self-evolving latent observed at irregular, exogenous times. Because the latent changes, its optimal estimator is time-varying: an adaptive timescale is necessary, and every fixed-gain filter is strictly suboptimal. And because arrivals are clock-free, the optimal estimate depends on the elapsed gap between them, which no gap-blind network recovers at any width or depth. This second condition is capacity-independent: scale cannot substitute for the missing dependence. The two conditions intersect in the continuous-time liquid class. An LSTM satisfies the first, a fixed continuous-time filter the second, and a multi-timescale liquid network both. Synthetic experiments confirm each: the network attains the timescale, and the separation is computed exactly. The characterization is necessary, not sufficient, and binds fixed-weight substrates: a network free to retrain reaches the class by other means. Proved per agent, the necessity binds every agent of a mesh, a structural condition on mesh intelligence.

Feedback-Coupled Memory Systems in Continuous Time

The Feedback-Coupled Memory Systems (FCMS) architecture formalizes closed-loop coordination through four abstract operators, two of which - the agent update operator $f_i$ and the environmental update operator $Ψ$ - are left axiomatically undefined in the original framework. To address this, $f_i$ is defined by Mechanism-Based Intelligence (MBI), where agents update locally through a decentralized price mechanism and economic principles, and $Ψ$ is defined by the Coupled Memory Graph Process (CMGP), a non-Markovian framework where the environment is treated as a physical substrate that records and responds to trajectory history coherently without external forcing. The resulting continuous-time FCMS instantiation achieves Lyapunov global dissipativity governed by the computable threshold $4β^2 < 2ημγ^2$. This generalizes both the discrete FCMS stability condition $4ηβ^2 < γ$ and CMGP's physical bifurcation threshold $α_c = 1/K$, confirming that memory dissipation must outpace feedback gain as a universal organizing principle. Numerical simulation with $N=2$ agents and mean-field validation at $N=10^6$ confirm the stability threshold and the self-reinforcing coordination cascade that emerges when it is violated.

Continuous-time Optimal Stopping through Deep Reinforcement Learning

Simulation based solvers for optimal stopping problems must discretize the stopping decision. Under classical dynamic programming, a coarse exercise grid with only a few stopping opportunities can materially undervalue the optimal expected reward, whereas on a very fine grid, approximation errors accumulate through the backward recursion. To remove this limitation, we develop a new reinforcement-learning inspired algorithm that enables us to learn the exercise rule at arbitrarily fine time resolution. Our CARLOS (Continuous-time Adaptive Reinforcement Learning for Optimal Stopping) algorithm utilizes an aggregate deep neural network (ADNN) to learn a joint space-time decision boundary. Starting from a coarse time grid, we progressively increase the frequency of stopping opportunities, while in parallel training the ADNN to refine its timing-value estimates. We moreover design an adaptive sampling strategy that gradually concentrates training effort near the stopping boundary. Benchmarked results show that CARLOS delivers higher prices than existing Bermudan solvers, approaching the American upper bound, and achieves high computational efficiency relative to non-RL comparators.

Non-Negative Matrix Factorization for Event Data

Continuous-time event data, in which entities emit instantaneous events over time, arises naturally across many domains such as neuroscience, seismology, and social networks. Non-negative matrix factorization (NMF) is a natural tool to uncover interpretable structure in such data, but it has so far only been applied after binning or smoothing the entity-level counting measures. This preprocessing step comes with the risk of erasing entity-level heterogeneities and fine-grained temporal features. In this paper, we introduce EventNMF, a continuous-time non-negative factorization model that operates directly on event times: each entity's events are modeled as a Poisson process whose intensity factorizes through a non-negative B-spline basis, and a simple estimation procedure recovers interpretable temporal templates shared across entities. The resulting method is mathematically principled, easy to implement, and computationally efficient. We further show that standard binned-count approaches arise as the special case of degree-zero splines, explore bias-variance tradeoffs and compare against existing methods on a synthetic latent factor model, and demonstrate the effectiveness of EventNMF on several real-world applications.

Policy Gradient for Continuous-Time Robust Markov Decision Processes

The framework of robust Markov decision processes (RMDPs) allows the design of reinforcement learning agents that satisfy performance guarantees under worst-case transition dynamics. Traditional RMDPs consider discrete-time dynamics and recently, sample-efficient policy gradient algorithms have been considered in this context. This paper investigates policy gradient algorithms within a continuous-time RMDP framework. Policy gradients and adversarial gradients are derived using pathwise and adjoint-based formulas for stochastic and ordinary differential equations. We propose double-loop optimisers to obtain linear convergence in the oracle-based setting and an $\tilde{\mathcal{O}}(\frac{1}{ε^2})$ sample complexity in the sample-based setting in an analysis which also derives novel tools for the framework of undiscounted total cost MDPs. Additionally, we propose mean-field optimisers as distributional optimisers with an $\tilde{\mathcal{O}}(\frac{1}{K})$ oracle-based convergence rate and an $\tilde{\mathcal{O}}(\frac{N^2}ε)$ sample complexity under $N$-particle approximation. The effectiveness of continuous-time policy gradient algorithms is confirmed for both optimisers on continuous-time RMDPs with neural ordinary differential equation dynamics.