The review explores how control theory, optimal transport, probabilistic inference, non‑equilibrium thermodynamics, and machine learning are interconnected through the optimization of free‑energy‑like functionals under dynamical or statistical constraints. It presents a conceptual thread linking these five fields and illustrates the ideas with applications in reinforcement learning, variational inference, and generative modeling. The article is written for readers without prior familiarity, beginning with physics principles.
By Emmy Blumenthal, Nikolas Claussen, Benjamin Eysenbach, Catherine Ji, Gautam Reddy, Colin Scheibner, Benjamin Sorkin
arXiv:2606. 24999v1 Announce Type: new Abstract: High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging.
By Yanwei Jia, Du Ouyang, Huy\^en Pham, Xun Yu Zhou
The paper introduces a mesh‑free kernel method for continuum‑marginal optimal transport, aiming to recover the minimum‑energy velocity field that reproduces a continuous family of probability marginals. By embedding the weak continuity equation into a reproducing kernel Hilbert space, the authors obtain a sample‑only objective that eliminates spatial discretization. The velocity is represented via a linear‑in‑parameters dictionary or neural network and optimized with mini‑batch stochastic techniques, achieving accurate drift recovery and marginal consistency in synthetic experiments, and the framework also extends to the Nelson problem of stochastic optimal transport.
By Yumiharu Nakano
The paper investigates continuous‑time stochastic control problems with unknown drift and running reward functions, using an exploratory reinforcement learning framework that incorporates relaxed controls and entropy regularization. It develops policy‑iteration algorithms based on probabilistic representations of the optimal value function and its gradient, proving convergence and demonstrating performance through numerical examples. The study also extends to a special case with control‑dependent diffusion, requiring a Hessian representation.
By Jin Ma, Gaozhan Wang, Jianfeng Zhang, Xunyu Zhou
arXiv:2012.01194v3 Announce Type: replace-cross
Abstract: In this article, we introduce a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate th...
By Christian Beck, Sebastian Becker, Patrick Cheridito, Arnulf Jentzen, Ariel Neufeld
arXiv:2411. 01982v2 Announce Type: replace-cross Abstract: We study the problem of learning controlled stochastic differential equations (SDEs) \[ dX_t = b(t,X_t,u_t)\,dt + \sigma(t,X_t,u_t)\,dW_t, \] whose drift and diffusion depend nonlinearly on time, state, and control values.
By Luc Brogat-Motte, Riccardo Bonalli, Alessandro Rudi
The paper introduces QD-PINN, a physics-informed neural network framework for estimating optimal stopping boundaries in quickest detection problems involving high-dimensional diffusion processes. It establishes a theoretical equivalence between the free-boundary PDE and an infinite-dimensional stochastic optimization problem, justifying the use of finite-dimensional neural networks. The authors demonstrate the method’s robustness by quantifying uncertainty with deep ensembles and Monte Carlo dropout, showing low variability and close agreement with reference numerical solutions.
By Hang Ye, Yi Liu, Haipeng Xing
arXiv:2301. 06308v2 Announce Type: replace-cross Abstract: Sharpness-aware minimization (SAM) is a training method that seeks to find flat minima in deep learning, resulting in state-of-the-art performance across various domains.
By Hoki Kim, Jinseong Park, Yujin Choi, Jaewook Lee
The paper introduces a geometric framework for reinforcement learning that treats policies as mappings into the Wasserstein space of action probabilities. It establishes a Riemannian structure induced by stationary distributions, defines the tangent space of policies, and characterizes geodesics while addressing measurability concerns. The authors formulate a general RL optimization problem, construct a gradient flow via Otto's calculus, compute the gradient and Hessian of the energy, and demonstrate the approach with numerical examples for low‑dimensional problems and neural‑network‑parameterized policies for high‑dimensional settings.
By Mathias Dus (IRMA)
arXiv:2609.30274v1 Announce Type: new
Abstract: Machine Learning and more specifically Deep Learning involves solving large scale nonconvex optimization problems. Several algorithms have been propose...
By St\'ephane Galatolo, St\'ephane Chr\'etien
arXiv:2501.10162v3 Announce Type: replace-cross
Abstract: Optimal transportation of raw material from suppliers to customers is an issue arising in logistics that is addressed here with a continuous...
By Alexandre Caboussat, Anna Peruso
High-frequency control problems are often studied through time discretization, but refining the timestep $δ$ increases the discrete horizon $H=T/δ$ and can make standard finite-horizon regret guarante...