Related papers: Structure-Aware Variational Learning of a Class of…
In machine learning energy potentials for atomic systems, forces are commonly obtained as the negative derivative of the energy function with respect to atomic positions. To quantify aleatoric uncertainty in the predicted energies, a widely…
High-dimensional data must be highly structured to be learnable. Although the compositional and hierarchical nature of data is often put forward to explain learnability, quantitative measurements establishing these properties are scarce.…
Learning physical dynamics from data is a fundamental challenge in machine learning and scientific modeling. Real-world observational data are inherently incomplete and irregularly sampled, posing significant challenges for existing…
Real-world data generation often involves certain geometries (e.g., graphs) that induce instance-level interdependence. This characteristic makes the generalization of learning models more difficult due to the intricate interdependent…
We develop an approach to learn an interpretable semi-parametric model of a latent continuous-time stochastic dynamical system, assuming noisy high-dimensional outputs sampled at uneven times. The dynamics are described by a nonlinear…
Based on a system-reservoir model, where the reservoir is driven by an external stationary, Gaussian noise with arbitrary decaying correlation function, we study the escape rate from a metastable state in the energy diffusion regime. For…
Energy-based models, a.k.a. energy networks, perform inference by optimizing an energy function, typically parametrized by a neural network. This allows one to capture potentially complex relationships between inputs and outputs. To learn…
Modeling dynamical systems and unraveling their underlying causal relationships is central to many domains in the natural sciences. Various physical systems, such as those arising in cell biology, are inherently high-dimensional and…
Using equilibrium fluctuations to understand the response of a physical system to an externally imposed perturbation is the basis for linear response theory, which is widely used to interpret experiments and shed light on microscopic…
Sampling from unnormalized densities is analogous to the generative modeling problem, but the target distribution is defined by a known energy function instead of data samples. Because evaluating the energy function is often costly, a…
Diffusion models have recently emerged as powerful stochastic frameworks for high-dimensional inference and generation. However, existing applications to partial differential equations (PDEs) predominantly rely on physics-informed training…
Modern single-particle-tracking techniques produce extensive time-series of diffusive motion in a wide variety of systems, from single-molecule motion in living-cells to movement ecology. The quest is to decipher the physical mechanisms…
We propose a method for learning dynamical systems from high-dimensional empirical data that combines variational autoencoders and (spatio-)temporal attention within a framework designed to enforce certain scientifically-motivated…
While there is currently a lot of enthusiasm about "big data", useful data is usually "small" and expensive to acquire. In this paper, we present a new paradigm of learning partial differential equations from {\em small} data. In…
This book presents the core principles that have guided the development of diffusion models, tracing their origins and showing how diverse formulations arise from shared mathematical ideas. Diffusion modeling starts by defining a forward…
We propose a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central…
A new method is proposed to numerically extract the diffusivity of a (typically nonlinear) diffusion equation from underlying stochastic particle systems. The proposed strategy requires the system to be in local equilibrium and have…
In this paper, we propose Lagrangian Gaussian Processes (LGPs) for probabilistic and data-efficient learning of dynamics via discrete forced Euler-Lagrange equations. Importantly, the geometric structure of the Lagrange-d'Alembert…
Classical diffusion models typically rely on isotropic Gaussian noise, treating all regions uniformly and overlooking structural information important for high-quality generation. We introduce an edge-preserving diffusion process that…
Dynamic nonlinear systems exhibit distortions arising from coupled static and dynamic effects. Their intertwined nature poses major challenges for data-driven modeling. This paper presents a theoretical framework grounded in structured…