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We propose a method for efficiently incorporating constraints into a stochastic gradient Langevin framework for the training of deep neural networks. Constraints allow direct control of the parameter space of the model. Appropriately…

Machine Learning · Computer Science 2021-06-22 Benedict Leimkuhler , Timothée Pouchon , Tiffany Vlaar , Amos Storkey

Pervasive across diverse domains, stochastic systems exhibit fluctuations in processes ranging from molecular dynamics to climate phenomena. The Langevin equation has served as a common mathematical model for studying such systems, enabling…

Statistical Mechanics · Physics 2025-05-01 Youngkyoung Bae , Seungwoong Ha , Hawoong Jeong

We employ constraints to control the parameter space of deep neural networks throughout training. The use of customized, appropriately designed constraints can reduce the vanishing/exploding gradients problem, improve smoothness of…

Machine Learning · Computer Science 2021-06-22 Benedict Leimkuhler , Tiffany Vlaar , Timothée Pouchon , Amos Storkey

In this paper we investigate how gradient-based algorithms such as gradient descent, (multi-pass) stochastic gradient descent, its persistent variant, and the Langevin algorithm navigate non-convex loss-landscapes and which of them is able…

Disordered Systems and Neural Networks · Physics 2022-03-22 Francesca Mignacco , Pierfrancesco Urbani , Lenka Zdeborová

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…

Machine Learning · Statistics 2026-04-28 Ludovico T. Giorgini

Reconstructing complete 3D shapes from incomplete or noisy observations is a fundamentally ill-posed problem that requires balancing measurement consistency with shape plausibility. Existing methods for shape reconstruction can achieve…

Computer Vision and Pattern Recognition · Computer Science 2026-03-31 Linus Härenstam-Nielsen , Dmitrii Pozdeev , Thomas Dagès , Nikita Araslanov , Daniel Cremers

D. B. Br\"{u}ckner et al. [Phys. Rev. Lett. 125, 058103 (2020)] have described a novel method for inferring the dynamics of systems governed by an underdamped Langevin equation in the presence of measurement noise. While this is a…

Statistical Mechanics · Physics 2026-04-10 Yeeren I. Low

We introduce a constructive framework to learn effective Langevin equations from stationary time series. Unlike conventional approaches that require iterative calibration to match target statistics, our construction guarantees the observed…

Chaotic Dynamics · Physics 2026-02-16 Ludovico Theo Giorgini

A key challenge in machine learning is to explain how learning dynamics select among the many solutions that achieve identical loss values in overparameterized models - a phenomenon known as implicit bias. Controlling this bias provides a…

Machine Learning · Computer Science 2026-04-07 Nicola Aladrah , Emanuele Ballarin , Matteo Biagetti , Alessio Ansuini , Alberto d'Onofrio , Fabio Anselmi

Many living and complex systems exhibit second order emergent dynamics. Limited experimental access to the configurational degrees of freedom results in data that appears to be generated by a non-Markovian process. This poses a challenge in…

Quantitative Methods · Quantitative Biology 2020-07-29 Federica Ferretti , Victor Chardès , Thierry Mora , Aleksandra M. Walczak , Irene Giardina

Overdamped Langevin dynamics are reversible stochastic differential equations which are commonly used to sample probability measures in high-dimensional spaces, such as the ones appearing in computational statistical physics and Bayesian…

Numerical Analysis · Mathematics 2025-02-10 Tony Lelièvre , Grigorios A. Pavliotis , Geneviève Robin , Régis Santet , Gabriel Stoltz

We present a representation learning algorithm that learns a low-dimensional latent dynamical system from high-dimensional \textit{sequential} raw data, e.g., video. The framework builds upon recent advances in amortized inference methods…

Machine Learning · Computer Science 2020-01-29 Jung-Su Ha , Young-Jin Park , Hyeok-Joo Chae , Soon-Seo Park , Han-Lim Choi

The measured time series from complex systems are renowned for their intricate stochastic behavior, characterized by random fluctuations stemming from external influences and nonlinear interactions. These fluctuations take diverse forms,…

Statistical Mechanics · Physics 2025-03-19 Pyei Phyo Lin , Matthias Wächter , Joachim Peinke , M. Reza Rahimi Tabar

Model generalization of the underlying dynamics is critical for achieving data efficiency when learning for robot control. This paper proposes a novel approach for learning dynamics leveraging the symmetry in the underlying robotic system,…

Robotics · Computer Science 2022-10-17 Jee-eun Lee , Jaemin Lee , Tirthankar Bandyopadhyay , Luis Sentis

We consider numerical methods for thermodynamic sampling, i.e. computing sequences of points distributed according to the Gibbs-Boltzmann distribution, using Langevin dynamics and overdamped Langevin dynamics (Brownian dynamics). A wide…

Statistical Mechanics · Physics 2015-01-13 Benedict Leimkuhler , Charles Matthews , Gabriel Stoltz

The ability to discover physical laws and governing equations from data is one of humankind's greatest intellectual achievements. A quantitative understanding of dynamic constraints and balances in nature has facilitated rapid development…

Dynamical Systems · Mathematics 2016-04-27 Steven L. Brunton , Joshua L. Proctor , J. Nathan Kutz

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)…

Machine Learning · Computer Science 2025-11-26 Yujin Kim , Sarah Dean

Identifying the governing equations of a nonlinear dynamical system is key to both understanding the physical features of the system and constructing an accurate model of the dynamics that generalizes well beyond the available data. We…

Machine Learning · Computer Science 2022-08-16 Peter Y. Lu , Joan Ariño , Marin Soljačić

We study numerical methods for sampling probability measures in high dimension where the underlying model is only approximately identified with a gradient system. Extended stochastic dynamical methods are discussed which have application to…

Numerical Analysis · Mathematics 2016-03-08 Benedict Leimkuhler , Xiaocheng Shang

Extracting governing physics from data is a key challenge in many areas of science and technology. The existing techniques for equations discovery are dependent on both input and state measurements; however, in practice, we only have access…

Data Analysis, Statistics and Probability · Physics 2022-08-12 Tapas Tripura , Souvik Chakraborty