The paper introduces MVNN, a measure‑valued neural network designed to learn interaction forces in systems governed by McKean‑Vlasov dynamics directly from particle trajectory data. MVNN extends conventional neural networks to operate on probability measures by learning cylindrical features through an embedding network that maps distributions to scalable vector representations. The authors prove theoretical properties such as well‑posedness, propagation‑of‑chaos, universal approximation, and provide quantitative rates under low‑dimensional measure‑dependence, while demonstrating the method’s accuracy and strong out‑of‑distribution generalization on various first‑ and second‑order dynamical systems.
By Liyao Lyu, Xinyue Yu, Hayden Schaeffer
The paper introduces a new optimal transport framework that uses the Barron energy—an intrinsic neural network complexity measure—instead of the traditional kinetic energy. By modeling transport as deep residual networks that map a reference distribution to a target distribution, the authors define a new metric on probability measures and analyze its properties for generative modeling. They demonstrate that diffusion generative models are suboptimal in this Barron geometry and explore how adaptivity can improve alternative generative approaches.
By Evan Dogariu, Joan Bruna
arXiv:2606. 03820v1 Announce Type: cross Abstract: We develop a quantitative approximation framework for diffusion distillation, viewing few-step sampling as error propagation under compositions of learned flow maps.
By Weiguo Gao, Ming Li, Lei Shi, Hanfei Zhou
arXiv:2607. 02003v1 Announce Type: cross Abstract: Although neural networks are remarkably effective, their underlying optimization principles remain theoretically elusive, often characterized by non-convex landscapes and stochastic heuristics.
By Matej Benko, Pierre Bousquet, Iwona Chlebicka, B{\l}a\.zej Miasojedow
The paper presents a differentiable hybrid modelling framework that combines a JAX finite volume population balance solver with learnable neural network components. This approach learns constitutive laws and initial conditions directly from experimental data, improving the fidelity of transport models in chemical engineering. The framework’s differentiability also enables optimisation of experimental settings for desired process outcomes.
By Arthur Jessop, Mohammed Alsubeihi, Ben Moseley, Ashwin Kumar Rajagopalan
arXiv:2607. 15291v1 Announce Type: cross Abstract: We develop a physics-informed neural network (PINN) framework for nonlocal partial differential equations arising in dynamic density functional theory (DDFT).
By Dimitrios Gourzoulidis, Soumaya Elkantassi, Serafim Kalliadasis