Spatiotemporal flows of neural activity, such as traveling waves, have been observed throughout the brain since the earliest recordings; yet there is still little consensus on their functional role. Recent experiments and models have linked traveling waves to visual and physical motion, but these observations have been difficult to reconcile with standard accounts of topographically organized selectivity and feedforward receptive fields. Here, we introduce a theoretical framework that formalizes and generalizes the connection between 'motion' and flowing neural dynamics in the language of equivariant neural network theory. We consider 'motion' not only in physical or visual spaces, but also in more abstract representational spaces, and we argue that recurrent traveling-wave-like dynamics are not just useful but necessary for accurate and stable processing of any signal undergoing such motion. Formally, we show that for any non-trivial recurrent neural network to process a sequence undergoing a flow transformation (such as visual motion) in a structured equivariant manner, its hidden state dynamics must actively realize a homomorphic representation of the same flow through recurrent connectivity. In this "spatiotemporal perspective on dynamical computation", traveling waves and related flows are best understood as faithful dynamic representations of stimulus flows; and consequently the natural inclination of biological systems towards such dynamics may be viewed as an innate inductive bias towards efficiency and generalization in the spatiotemporally-structured dynamical world they inhabit.

Physical systems commonly exhibit spatially varying complexity, presenting a significant challenge for neural PDE solvers. While Graph Neural Networks can handle the irregular meshes required for complex geometries and boundary conditions, they still apply uniform computational effort across all nodes regardless of the underlying physics complexity. This leads to inefficient resource allocation where computationally simple regions receive the same treatment as complex phenomena. We address this challenge by introducing Adaptive Mesh Quantization: spatially adaptive quantization across mesh node, edge and cluster features, dynamically adjusting the bit-width used by a quantized model. We propose an adaptive bit-width allocation strategy driven by a lightweight auxiliary model that identifies high-loss regions in the input mesh. This enables dynamic resource distribution in the main model, where regions of higher difficulty are allocated increased bit-width, optimizing computational resource utilization. We demonstrate our framework's effectiveness by integrating it with two state-of-the-art models, MP-PDE and GraphViT, to evaluate performance across multiple tasks: 2D Darcy flow, large-scale unsteady fluid dynamics in 2D, steady-state Navier–Stokes simulations in 3D, and a 2D hyper-elasticity problem. Our framework demonstrates consistent Pareto improvements over uniformly quantized baselines, yielding up to 50\% improvements in performance at the same cost.

The 𝒪(N^2) complexity of attention over N tokens remains a computational bottleneck in transformer models. Vector-Quantized (VQ) attention reduces this to 𝒪(MN) by representing keys with M codewords, but applies uniform codebook capacity regardless of where attention mass concentrates: high-attention regions of key space may be coarsely approximated while low-attention regions waste representational capacity. We propose Adaptive Vector-Quantized (AVQ) Attention, which adaptively allocates codebook capacity based on attention importance. Starting from a small set of codewords, our method identifies the most important codes during the forward pass and refines them with pre-learned child codewords, achieving fine-grained quantization where it matters most while maintaining coarse quantization elsewhere. We develop an implementation using custom Triton kernels that enables the full adaptive refinement process, including importance scoring, child codeword insertion, and parent contribution replacement, to be carried out within the tiled computation paradigm of Flash Attention with minimal overhead. Our approach maintains 𝒪(MN) complexity while achieving improved accuracy-efficiency trade-offs compared to fixed-codebook VQ-attention.

Generative models can produce nonsensical text, unrealistic images, and unstable materials faster than simulation or human review can absorb; without per-sample confidence, trust erodes. Existing fixes run $k$ ensembles or stochastic trajectories at $k\times$ compute, measuring variability between models, not model confidence. We propose Flow Matching with Confidence (FMwC). FMwC injects input-dependent multiplicative noise at selected layers, propagates its variance through the network in closed form, and integrates it along the ODE trajectory, yielding a per-sample confidence score at standard sampling cost. The score supports multiple uses: filtering improves image quality and thermodynamic stability of crystals; editing rewinds trajectories to the points where the model commits and redirects them; and adaptive stepping concentrates ODE compute where the flow is ambiguous. We find that the confidence score correlates with the magnitude of the divergence of the learned velocity field, which gives us a window to understand the generative process, opening up surgical forms of guidance that target the moments that matter, new sampling algorithms and interpretability of generative models.

Generative models are increasingly used to capture correlations in many-body systems, but the representations they learn remain largely opaque to physical interpretation. Here, we establish an intuitive criterion that quantifies the capacity of a variational autoencoder (VAE) to faithfully reconstruct the joint probability distribution of a many body system. In a nutshell, a bound on the VAE capacity is obtained by comparing the rate of the latent channel to the bipartite mutual information of the data. Using this bound, we show that the conditionally independent decoder of any successful VAE is structurally identical to a finite-size mean-field factorization. Hence, a successful reconstruction is direct evidence for a latent mean-field theory and the microscopic parameters of that theory can be read off the trained decoder. We validate these conclusions on a hierarchy of solvable models with scalar (Curie-Weiss), vector (Hopfield) and tensor (Maier-Saupe) order parameters, recovering the full Hopfield pattern matrix from equilibrium samples alone. We find that, when applied to Salamander retinal recordings, a two-latent VAE reproduces the population statistics with only two effective collective variables allowing us to recover the `stored patterns' of the neural population and write a generalized Hopfield model which correctly models the experimental data.

Low-Rank Adaptation (LoRA) has become the standard for fine-tuning large pre-trained models at reduced computational cost. However, its low-rank point-estimate updates limit expressiveness, leave a persistent gap relative to full fine-tuning accuracy, and provide no built-in uncertainty quantification, limiting its applicability in settings where reliability matters as much as accuracy. We introduce BaLoRA, a Bayesian extension of LoRA with a novel input-adaptive Bayesian parameterization of LoRA matrices that adds minimal parameters and compute. Surprisingly, not only does the Bayesian extension yield well-calibrated uncertainty estimates, but the adaptive noise injection underlying our approach also significantly improves prediction accuracy, narrowing the gap with full fine-tuning across both natural language reasoning and vision tasks. When applied to band gap prediction in metal-organic frameworks, BaLoRA produces zero-shot test-time uncertainty estimates that correlate more strongly with model error than a trained ensemble of LoRA models, and improve monotonically with compute without sacrificing accuracy.

We propose kernel-gradient drifting, a one-step generative modeling framework that replaces the fixed Euclidean displacement direction in drifting models with directions induced by the kernel itself. Standard drifting is attractive because it enables fast, high-quality generation without distilling a large pretrained diffusion model, but its theory is currently understood mainly for Gaussian kernels, where the drift coincides with smoothed score matching and is identifiable. Our gradient-based reformulation exposes this score-based structure for general kernels: the resulting drift is the score difference between kernel-smoothed data and model distributions, yielding identifiability for characteristic kernels and a smoothed-KL descent interpretation of the drifting dynamics. Since kernel gradients are intrinsic tangent vectors, the same construction extends naturally to Riemannian manifolds and to discrete data via the Fisher-Rao geometry of the probability simplex. Across spherical geospatial data, promoter DNA and molecule generation, kernel-gradient drifting enables state-of-the-art one-step generation beyond the Euclidean setting without distillation.

Stochastic-gradient MCMC methods enable scalable Bayesian posterior sampling but often suffer from sensitivity to minibatch size and gradient noise. To address this, we propose Stochastic Gradient Lattice Random Walk (SGLRW), an extension of the Lattice Random Walk discretization. Unlike conventional Stochastic Gradient Langevin Dynamics (SGLD), SGLRW introduces stochastic noise only through the off-diagonal elements of the update covariance; this yields greater robustness to minibatch size while retaining asymptotic correctness. Furthermore, as comparison we analyze a natural analogue of SGLD utilizing gradient clipping. Experimental validation on Bayesian regression and classification demonstrates that SGLRW remains stable in regimes where SGLD fails, including in the presence of heavy-tailed gradient noise, and matches or improves predictive performance.

There is a vast literature on representation learning based on principles such as coding efficiency, statistical independence, causality, controllability, or symmetry. In this paper we propose to learn representations from sequence data by factorizing the transformations of the latent variables into sparse components. Input data are first encoded as distributions of latent activations and subsequently transformed using a probability flow model, before being decoded to predict a future input state. The flow model is decomposed into a number of rotational (divergence-free) vector fields and a number of potential flow (curl-free) fields. Our sparsity prior encourages only a small number of these fields to be active at any instant and infers the speed with which the probability flows along these fields. Training this model is completely unsupervised using a standard variational objective and results in a new form of disentangled representations where the input is not only represented by a combination of independent factors, but also by a combination of independent transformation primitives given by the learned flow fields. When viewing the transformations as symmetries one may interpret this as learning approximately equivariant representations. Empirically we demonstrate that this model achieves state of the art in terms of both data likelihood and unsupervised approximate equivariance errors on datasets composed of sequence transformations.

Causal modeling of physical temporal phenomena must handle interventions that act along trajectories, nonstationary induced laws, path-dependent effects, and feedback mediated by dynamics, all challenging in standard causal models. We introduce Hamiltonian Causal Models (HCMs), a trajectory-level framework in which observed variables interact with local environments and interventions act as controls of Hamiltonian mechanisms. HCMs separate immutable equations of motion from intervenable mechanisms and define causal effects as discrepancies between interventional path laws. A key motivation for HCMs is their natural interface with non-equilibrium thermodynamics. Entropy production quantifies the irreversibility of a process and is a central causal observable: it is estimable from data and witnesses causal effects along the system's evolution that are invisible to endpoint and cumulative versions of the standard average treatment effect. As in physics, cause and effect are not primitives of the relation between two random variables but arise from the non-invertibility of the thermodynamic arrow. With this, our paper reconciles the language of statistical causal models and non-stationary thermodynamics, offering new tools to describe causality in a wide range of physical systems.

We introduce Purrception, a variational flow matching approach for vector-quantized image generation that provides explicit categorical supervision while maintaining continuous transport dynamics. Our method adapts Variational Flow Matching to vector-quantized latents by learning categorical posteriors over codebook indices while computing velocity fields in the continuous embedding space. This combines the geometric awareness of continuous methods with the discrete supervision of categorical approaches, enabling uncertainty quantification over plausible codes and temperature-controlled generation. We evaluate Purrception on ImageNet-1k $256 \times 256$ generation. Training converges faster than both continuous flow matching and discrete flow matching baselines while achieving competitive FID scores with state-of-the-art models. This demonstrates that Variational Flow Matching can effectively bridge continuous transport and discrete supervision for improved training efficiency in image generation.

In-context learning (ICL) enables Large Language Models (LLMs) to adapt to new tasks with only a small set of examples at inference time, thereby avoiding task-specific fine-tuning. However, in-context examples may contain privacy-sensitive information that should not be revealed through model outputs. Existing differential privacy (DP) approaches to ICL are either computationally expensive or rely on heuristics with limited effectiveness, including context oversampling, synthetic data generation, or unnecessary thresholding. We reformulate private ICL through the lens of a Product-of-Experts model. This gives a theoretically grounded framework, and the algorithm can be trivially parallelized. We evaluate our method across five datasets in text classification, math, and vision-language. We find that our method improves accuracy by more than 30 percentage points on average compared to prior DP-ICL methods, while maintaining strong privacy guarantees.

In physical systems, whenever a continuous symmetry is spontaneously broken, the system possesses excitations called Goldstone modes, which allow coherent information propagation over long distances and times. In this work, we study deep neural networks whose internal layers are equivariant under a continuous symmetry and may therefore support analogous Goldstone-like degrees of freedom. We demonstrate, both analytically and empirically, that these degrees of freedom enable coherent signal propagation across depth and recurrent iterations, providing a mechanism for stable information flow without relying on architectural stabilizers such as residual connections or normalization. In feedforward networks, this results in improved trainability and representational diversity across layers. In recurrent settings, we demonstrate the same mechanism is valuable for long-term memory by propagating information over recurrent iterations, thereby improving performance of RNNs and GRUs on long-sequence modeling tasks.

We introduce Categorical Flow Maps, a flow-matching method for accelerated few-step generation of categorical data via self-distillation. Building on recent variational formulations of flow matching and the broader trend towards accelerated inference in diffusion and flow-based models, we define a flow map towards the simplex that transports probability mass toward a predicted endpoint, yielding a parametrisation that naturally constrains model predictions. Since our trajectories are continuous rather than discrete, Categorical Flow Maps can be trained with existing distillation techniques, as well as a new objective based on endpoint consistency. This continuous formulation also automatically unlocks test-time inference: we can directly reuse existing guidance and reweighting techniques in the categorical setting to steer sampling toward downstream objectives. Empirically, we achieve state-of-the-art few-step results on images, molecular graphs, and text, with strong performance even in single-step generation.

Originating from lectures delivered at the African Institute of Mathematical Sciences, this book presents a unifying perspective on traditional and modern methods in generative AI and stochastic thermodynamics. By relating the core topics in machine learning to the notion of (variational) free-energy, a bridge is built between methods such as latent variable models, variational auto-encoders, optimal control, optimal transport, normalizing flows and diffusion models and concepts such as entropy production and fluctuation theorems in stochastic thermodynamics. Structured into three main parts, the book commences by setting up the required mathematical and statistical physics preliminaries needed to make it broadly accessible. The largest part of the book then focuses on building intuition of major advances in generative AI by considering discrete time processes and their relationship to topics in stochastic thermodynamics. Finally, the authors take a short excursion to the continuous time domain for the more advanced learner.

We present Riemannian Gaussian Variational Flow Matching (RG-VFM), a geometric extension of Variational Flow Matching (VFM) for generative modeling on manifolds. Motivated by the benefits of VFM, we derive a variational flow matching objective for manifolds with closed-form geodesics based on Riemannian Gaussian distributions. Crucially, in Euclidean space, predicting endpoints (VFM), velocities (FM), or noise (diffusion) is largely equivalent due to affine interpolations. However, on curved manifolds this equivalence breaks down. For this reason, we formally analyze the relationship between our model and Riemannian Flow Matching (RFM), revealing that the RFM objective lacks a curvature-dependent penalty -- encoded via Jacobi fields -- that is naturally present in RG-VFM. Based on this relationship, we hypothesize that endpoint prediction provides a stronger learning signal by directly minimizing geodesic distances. Experiments on synthetic spherical and hyperbolic benchmarks, as well as real-world tasks in material and protein generation, demonstrate that RG-VFM more effectively captures manifold structure and improves downstream performance over Euclidean and velocity-based baselines.

We introduce \textsc{Mosaic}, a probabilistic weather forecasting model that addresses two sources of spectral degradation in ML-based weather prediction: training to predict the ensemble mean deterministically and compressive encoding creating an information bottleneck. \textsc{Mosaic} combines learned functional perturbations for ensemble forecasting with block-sparse attention, a hardware-aligned formulation that shares keys and values across spatially adjacent queries, enabling each block to dynamically attend to the most relevant regions. By capturing arbitrarily long-range dependencies at linear cost, \textsc{Mosaic} processes high-resolution weather data without compression. On IFS HRES data, \textsc{Mosaic} at 1.5° resolution matches or outperforms models trained on 0.25° data, with individual ensemble members exhibiting near-perfect spectral alignment across all resolved frequencies.

Finite-time driving of stochastic systems generates excess dissipation, causing the evolving probability distribution to lag behind the instantaneous equilibrium, and consequently degrading the convergence of nonequilibrium free energy estimators based on the Jarzynski equality. Escorted free energy simulations address the non-adiabatic lag by engineering control fields $\mathbf{u}$ that eliminate the lag, enforcing the trajectory-wise equality $\mathcal{W}_\mathbf{u} = Δ\mathcal{F}$, and yielding zero-variance estimators. However, constructing the escorting field in closed form remains a challenge, approached variously through flow-field methods, targeted free energy perturbation, or learned diffeomorphisms. In this work, we construct a complementary numerical framework based on gauge-type transforms instead of generalized coordinate transforms for perfect escorting based on the exact spectral decomposition of the time-dependent Fokker-Planck generator. The biorthogonal decomposition of the Liouville operator directly yields a counterdiabatic correction whose action on the instantaneous equilibrium distribution exactly cancels the non-adiabatic lag at arbitrary driving speed in formal analogy with shortcuts-to-adiabaticity techniques such as Berry's transitionless driving for quantum systems. Numerical verification for simulations of an overdamped particle in a time-varying double-well potential and harmonic traps confirms that the counterdiabatic condition is satisfied to machine precision, with the non-adiabatic lag suppressed by roughly twelve orders of magnitude in total variation distance and sixteen orders in KL divergence relative to the unescorted dynamics. As a diagnostic, we demonstrate vanishing dissipated work $\mathcal{W}_{\text{diss}}(t) \approx 0$ for the deterministically propagated Fokker-Planck density across all protocol speeds.

Self-attention in Transformers relies on globally normalized softmax weights, causing all tokens to compete for influence at every layer. When composed across depth, this interaction pattern induces strong synchronization dynamics that favor convergence toward a dominant mode, a behavior associated with representation collapse and attention sink phenomena. We introduce Krause Attention, a principled attention mechanism inspired by bounded-confidence consensus dynamics. Krause Attention replaces similarity-based global aggregation with distance-based, localized, and selectively sparse interactions, promoting structured local synchronization instead of global mixing. We relate this behavior to recent theory modeling Transformer dynamics as interacting particle systems, and show how bounded-confidence interactions naturally moderate attention concentration and alleviate attention sinks. Restricting interactions to local neighborhoods also reduces runtime complexity from quadratic to linear in sequence length. Experiments across vision (ViT on CIFAR/ImageNet), autoregressive generation (MNIST/CIFAR-10), and large language models (Llama/Qwen) demonstrate consistent gains with substantially reduced computation, highlighting bounded-confidence dynamics as a scalable and effective inductive bias for attention.

Machine Learning Interatomic Potentials (MLIPs) are becoming a central tool in simulation-based chemistry. However, like most deep learning models, MLIPs struggle to make accurate predictions on out-of-distribution data or when trained in a data-scarce regime, both common scenarios in simulation-based chemistry. Moreover, MLIPs do not provide uncertainty estimates by construction, which are fundamental to guide active learning pipelines and to ensure the accuracy of simulation results compared to quantum calculations. To address this shortcoming, we propose BLIPs: Bayesian Learned Interatomic Potentials. BLIP is a scalable, architecture-agnostic variational Bayesian framework for training or fine-tuning MLIPs, built on an adaptive version of Variational Dropout. BLIP delivers well-calibrated uncertainty estimates and minimal computational overhead for energy and forces prediction at inference time, while integrating seamlessly with (equivariant) message-passing architectures. Empirical results on simulation-based computational chemistry tasks demonstrate improved predictive accuracy with respect to standard MLIPs, and trustworthy uncertainty estimates, especially in data-scarse or heavy out-of-distribution regimes. Moreover, fine-tuning pretrained MLIPs with BLIP yields consistent performance gains and calibrated uncertainties.