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Although diffusion models now occupy a central place in generative modeling, introductory treatments commonly assume Euclidean data and seldom clarify their connection to discrete-state analogues. This article is a self-contained primer on…
We consider the problem of sampling from constrained distributions, which has posed significant challenges to both non-asymptotic analysis and algorithmic design. We propose a unified framework, which is inspired by the classical mirror…
In this paper, we propose a probabilistic physics-guided framework, termed Physics-guided Deep Markov Model (PgDMM). The framework targets the inference of the characteristics and latent structure of nonlinear dynamical systems from…
This paper introduces a hidden quantum Markov models (HQMMs) framework to the Affleck-Kennedy-Lieb-Tasaki (AKLT) state-a cornerstone example of a symmetry-protected topological (SPT) phase. The model's observation system is the physical…
Many consequential real-world systems, like wind fields and ocean currents, are dynamic and hard to model. Learning their governing dynamics remains a central challenge in scientific machine learning. Dynamic Mode Decomposition (DMD)…
We have been investigating the problem of the Anderson localization in a disordered one dimensional tight-binding model. The disorder is created by the interaction of mobile particles with other species, immobilized at random positions. We…
Spatial self-organization emerges in distributed systems exhibiting local interactions when nonlinearities and the appropriate propagation of signals are at work. These kinds of phenomena can be modeled with different frameworks, typically…
State-space models (SSMs) have emerged as a powerful foundation for long-range sequence modeling, with the HiPPO framework showing that continuous-time projection operators can be used to derive stable, memory-efficient dynamical systems…
This study presents a sampling-based method to guarantee robust stability of general control systems with uncertainty. The method allows the system dynamics and controllers to be represented by various data-driven models, such as Gaussian…
Over the past decades, a great body of theoretical and mathematical work has been devoted to random-matrix descriptions of open quantum systems. In these notes, based on lectures delivered at the Les Houches Summer School "Stochastic…
The ordinary contact process is used to model the spread of a disease in a population. In this model, each infected individual waits an exponentially distributed time with parameter 1 before becoming healthy. In this paper, we introduce and…
State Space Models (SSMs) and Hidden Markov Models (HMMs) are foundational frameworks for modeling sequential data with latent variables and are widely used in signal processing, control theory, and machine learning. Despite their shared…
The Poisson distribution is the default choice of likelihood for probabilistic models of count data. However, due to the equidispersion contraint of the Poisson, such models may have predictive uncertainty that is artificially inflated.…
The reliability of artificial intelligence (AI) systems in open-world settings depends heavily on their ability to flag out-of-distribution (OOD) inputs unseen during training. Recent advances in large-scale vision-language models (VLMs)…
Spectral statistics of quantum systems have been studied in detail using the nearest neighbour level spacings, which for generic chaotic systems follows random matrix theory predictions. In this work, the probability density of the closest…
In reinforcement learning (RL), world models serve as internal simulators, enabling agents to predict environment dynamics and future outcomes in order to make informed decisions. While previous approaches leveraging discrete latent spaces,…
Differential equations in general and neural ODEs in particular are an essential technique in continuous-time system identification. While many deterministic learning algorithms have been designed based on numerical integration via the…
Recent years have witnessed a booming interest in data-driven control of dynamical systems. However, the implicit data-driven output predictors are vulnerable to uncertainty such as process disturbance and measurement noise, causing…
Discrete dynamics arise naturally in systems with broken temporal translation symmetry and are typically described by first-order recurrence relations representing classical or quantum Markov chains. When memory effects induced by hidden…
This paper focuses on some simple models of limit order book dynamics which simulate market trading mechanisms. We start with a discrete time/space Markov process and then perform a re-scaling procedure leading to a deterministic dynamical…