The work was conducted during the internship of Y uqi Chen and Y uchen Fang at Microsoft Research.

Temporal graphs are widely used to model dynamic systems with time-varying interactions. In real-world scenarios, the underlying mechanisms of generating future interactions in dynamic systems are typically governed by a set of recurring substructures within the graph, known as temporal motifs. Despite the success and prevalence of current temporal graph neural networks (TGNN), it remains uncertain which temporal motifs are recognized as the significant indications that trigger a certain prediction from the model, which is a critical challenge for advancing the explainability and trustworthiness of current TGNNs.

Reasoning system dynamics is one of the most important analytical approaches for many scientific studies. With the initial state of a system as input, the recent graph neural networks (GNNs)-based methods are capable of predicting the future state distant in time with high accuracy. Although these methods have diverse designs in modeling the coordinates and interacting forces of the system, we show that they actually share a common paradigm that learns the integration of the velocity over the interval between the initial and terminal coordinates. However, their integrand is constant w.r.t.

AI/ML models have rapidly gained prominence as innovations for solving previously unsolved problems and their unintended consequences from amplifying human biases. Advocates for responsible AI/ML have sought ways to draw on the richer causal models of system dynamics to better inform the development of responsible AI/ML. However, a major barrier to advancing this work is the difficulty of bringing together methods rooted in different underlying assumptions (i.e., Dana Meadow's "the unavoidable a priori"). This paper brings system dynamics and structural equation modeling together into a common mathematical framework that can be used to generate systems from distributions, develop methods, and compare results to inform the underlying epistemology of system dynamics for data science and AI/ML applications.

Learning governing equations from data is central to understanding the behavior of physical systems across diverse scientific disciplines, including physics, biology, and engineering. The Sindy algorithm has proven effective in leveraging sparsity to identify concise models of nonlinear dynamical systems. In this paper, we extend sparsity-driven approaches to real-time learning by integrating a cornerstone algorithm from control theory -- the Kalman filter (KF). The resulting Sindy Kalman Filter (SKF) unifies both frameworks by treating unknown system parameters as state variables, enabling real-time inference of complex, time-varying nonlinear models unattainable by either method alone. Furthermore, SKF enhances KF parameter identification strategies, particularly via look-ahead error, significantly simplifying the estimation of sparsity levels, variance parameters, and switching instants. We validate SKF on a chaotic Lorenz system with drifting or switching parameters and demonstrate its effectiveness in the real-time identification of a sparse nonlinear aircraft model built from real flight data.

The work was conducted during the internship of Y uqi Chen and Y uchen Fang at Microsoft Research.

We analyze offline designs of linear quadratic regulator (LQR) strategies with uncertain disturbances. First, we consider the scenario where the exogenous variable can be estimated in a controlled environment, and subsequently, consider a more practical and challenging scenario where it is unknown in a stochastic setting. Our approach builds on the fundamental learning-based framework of adaptive dynamic programming (ADP), combined with a Lyapunov-based analytical methodology to design the algorithms and derive sample-based approximations motivated from the Markov decision process (MDP)-based approaches. For the scenario involving non-measurable disturbances, we further establish stability and convergence guarantees for the learned control gains under sample-based approximations. The overall methodology emphasizes simplicity while providing rigorous guarantees. Finally, numerical experiments focus on the intricacies and validations for the design of offline continuous-time LQR with exogenous disturbances.

The discovery of equations from observational data is one of the fundamental pillars of the traditional scientific method. From the work of Johannes Kepler, who inferred the laws of planetary motion from meticulous astronomical observations [1] collected by Tycho Brahe [2], to Isaac Newton's theoretical formulations that consolidated classical mechanics, the process of identifying mathematical relationships underlying natural phenomena has historically been characterized by its manual nature, based essentially on systematic trial-and-error procedures. However, in recent decades, the advent of Big Data, characterized by the production of an immense volume of complex, mostly nonlinear, data, in several fields has driven a new search for physical laws. Faced with the need to analyze these data sets to understand their intrinsic structure and derive symbolic representations that capture the integral behavior of a system, the demand for advanced analytical methods has become growing and indispensable. With the emergence of modern computational techniques, this process has undergone a radical transformation, driving the widespread development and use of various regression techniques.

Graph Neural Networks (GNNs) are known to match the distinguishing power of the 1-Weisfeiler-Lehman (1-WL) test, and the resulting partitions coincide with the unfolding tree equivalence classes of graphs. Preserving this equivalence, GNNs can universally approximate any target function on graphs in probability up to any precision. However, these results are limited to attributed discrete-dynamic graphs represented as sequences of connected graph snapshots. Real-world systems, such as communication networks, financial transaction networks, and molecular interactions, evolve asynchronously and may split into disconnected components. In this paper, we extend the theory of attributed discrete-dynamic graphs to attributed continuous-time dynamic graphs with arbitrary connectivity. To this end, we introduce a continuous-time dynamic 1-WL test, prove its equivalence to continuous-time dynamic unfolding trees, and identify a class of continuous-time dynamic GNNs (CGNNs) based on discrete-dynamic GNN architectures that retain both distinguishing power and universal approximation guarantees. Our constructive proofs further yield practical design guidelines, emphasizing a compact and expressive CGNN architecture with piece-wise continuously differentiable temporal functions to process asynchronous, disconnected graphs.