We take advantage of NF preconditioning and NF based Metropolis-Hastings updates for a faster convergence.

We analyze prediction error in stochastic dynamical systems with memory, focusing on generalized Langevin equations (GLEs) formulated as stochastic Volterra equations. We establish that, under a strongly convex potential, trajectory discrepancies decay at a rate determined by the decay of the memory kernel and are quantitatively bounded by the estimation error of the kernel in a weighted norm. Our analysis integrates synchronized noise coupling with a Volterra comparison theorem, encompassing both subexponential and exponential kernel classes. For first-order models, we derive moment and perturbation bounds using resolvent estimates in weighted spaces. For second-order models with confining potentials, we prove contraction and stability under kernel perturbations using a hypocoercive Lyapunov-type distance. This framework accommodates non-translation-invariant kernels and white-noise forcing, explicitly linking improved kernel estimation to enhanced trajectory prediction. Numerical examples validate these theoretical findings.

Coarse graining (CG) is an important task for efficient modeling and simulation of complex multi-scale systems, such as the conformational dynamics of biomolecules. This work presents a projection-based coarse-graining formalism for general underdamped Langevin dynamics. Following the Zwanzig projection approach, we derive a closed-form expression for the coarse grained dynamics. In addition, we show how the generator Extended Dynamic Mode Decomposition (gEDMD) method, which was developed in the context of Koopman operator methods, can be used to model the CG dynamics and evaluate its kinetic properties, such as transition timescales. Finally, we combine our approach with thermodynamic interpolation (TI), a generative approach to transform samples between thermodynamic conditions, to extend the scope of the approach across thermodynamic states without repeated numerical simulations. Using a two-dimensional model system, we demonstrate that the proposed method allows to accurately capture the thermodynamic and kinetic properties of the full-space model.

Current artificial intelligence systems show near-human-level capabilities when deployed in isolation. Systems of a few collaborating intelligent agents are being engineered to perform tasks collectively. This raises the question of whether robotic matter, where many learning and intelligent agents interact, shows emergence of collective behaviour. And if so, which kind of phenomena would such systems exhibit? Here, we study a paradigmatic model for robotic matter: a stochastic many-particle system in which each particle is endowed with a deep neural network that predicts its transitions based on the particles' environments. For a one-dimensional model, we show that robotic matter exhibits complex emergent phenomena, including transitions between long-lived learning regimes, the emergence of particle species, and frustration. We also find a density-dependent phase transition with signatures of criticality. Using active matter theory, we show that this phase transition is a consequence of self-organisation mediated by emergent inter-particle interactions. Our simple model captures key features of more complex forms of robotic systems.

Sampling is a fundamental and arguably very important task with numerous applications in Machine Learning.

We take advantage of NF preconditioning and NF based Metropolis-Hastings updates for a faster convergence.

In this work, we introduce LangevinFlow, a sequential Varia-tional Auto-Encoder where the time evolution of latent variables is governed by the underdamped Langevin equation. Our approach incorporates physical priors -- such as inertia, damping, a learned potential function, and stochastic forces -- to represent both autonomous and non-autonomous processes in neural systems. Crucially, the potential function is parameterized as a network of locally coupled oscillators, biasing the model toward oscillatory and flow-like behaviors observed in biological neural populations. Our model features a recurrent encoder, a one-layer Transformer decoder, and Langevin dynamics in the latent space. Empirically, our method outperforms state-of-the-art baselines on synthetic neural populations generated by a Lorenz attractor, closely matching ground-truth firing rates. On the Neural Latents Benchmark (NLB), the model achieves superior held-out neuron likelihoods (bits per spike) and forward prediction accuracy across four challenging datasets. It also matches or surpasses alternative methods in decoding behavioral metrics such as hand velocity. Overall, this work introduces a flexible, physics-inspired, high-performing framework for modeling complex neural population dynamics and their unobserved influences.

The seminal work of Jordan, Kinderlehrer, and Otto [33] developed what is now widely known as the JKO scheme, a foundational method for generating iterative algorithms to compute distributions and reshaping our understanding of sampling algorithms. The JKO scheme can be interpreted as a gradient flow of the free energy with respect to the Wasserstein metric, often referred to as the Wasserstein gradient flow. This interpretation has led to significant advancements in machine learning, including applications in reinforcement learning to solve policy-distribution optimization problems [55]. While the JKO scheme traditionally assumes that the underlying model is fully known, in this paper, we relax this assumption by allowing models with unknown parameters. We develop statistical approaches to estimate these parameters and adapt the JKO scheme to work with the estimated values. Specifically, Langevin equations--stochastic differential equations--play a key role in describing the evolution of physical systems, facilitating stochastic gradient descent in machine learning, and enabling Markov chain Monte Carlo (MCMC) simulations in numerical computing. For examples and detailed discussions, see [11, 8, 51, 22, 19, 39, 43]. Solutions to Langevin equations, known as Langevin diffusions, are stochastic processes whose distributions evolve according to the Fokker-Planck equations [27, 48].

Machine learning (ML) and artificial intelligence (AI) can provide powerful tools for the scientific community, as demonstrated by the recent Nobel Prize in Chemistry. Reversely, insights from traditional physics theories also contribute to a deeper understanding of the mechanism of learning. Ref. [1] contains a broad overview of the successful cross-fertilisation between ML and the physical sciences, covering a number of domains. One way to mitigate against possible scepticism with regard to using ML as a "black box" is by unveiling the dynamics of training (or learning) and explaining how the relevant information is engraved in the model during the training stage. To further develop this programme, we study here the dynamics of first-order stochastic gradient descent as applied to weight matrices, reporting and expanding on the work presented in Ref. [2]. When training ML models, weight matrices are commonly updated by one of the variants of the stochastic gradient descent algorithm. The dynamics can then be decomposed into a drift and a fluctuating term, and such a system can be described by a discrete Langevin equation. The dynamics of stochastic matrix updates is richer than the dynamics for vector or scalar quantities, as captured by Dyson Brownian motion and random matrix theory (RMT), with the appearance of universal features for the eigenvalues [3-9]. Earlier descriptions of the statistical properties of weight matrices in terms of RMT can be found in e.g.