We introduce the Riemannian Proximal Sampler, a method for sampling from densities defined on Riemannian manifolds. The performance of this sampler critically depends on two key oracles: the Manifold Brownian Increments (MBI) oracle and the Riemannian Heat-kernel (RHK) oracle. We establish high-accuracy sampling guarantees for the Riemannian Proximal Sampler, showing that generating samples with ε-accuracy requires O(log(1/ε)) iterations in Kullback-Leibler divergence assuming access to exact oracles and O(log2(1/ε))iterations in the total variation metric assuming access to sufficiently accurate inexact oracles.

The physical sciences are replete with dynamical systems that require the resolution of a wide range of length and time scales. This presents significant computational challenges since direct numerical simulation requires discretization at the finest relevant scales, leading to a high-dimensional state space. In this work, we propose an approach to learn stochastic multiscale models in the form of stochastic differential equations directly from observational data. Drawing inspiration from physics-based multiscale modeling approaches, we resolve the macroscale state on a coarse mesh while introducing a microscale latent state to explicitly model unresolved dynamics. We learn the parameters of the multiscale model using a simulator-free amortized variational inference method with a Product of Experts likelihood that enforces scale separation. We present detailed numerical studies to demonstrate that our learned multiscale models achieve superior predictive accuracy compared to under-resolved direct numerical simulation and closure-type models at equivalent resolution, as well as reduced-order modeling approaches.

The diffusion inversion problem seeks to recover the latent generative trajectory of a diffusion model given a real image. Faithful inversion is critical for ensuring consistency in diffusion-based image editing. Prior works formulate this task as a fixed-point problem and solve it using numerical methods. However, achieving both accuracy and efficiency remains challenging, especially for few-step models and novel samples.

Reinforcement learning (RL) has achieved significant success across a wide range of domains, however, most existing methods are formulated in discrete time. In this work, we introduce a novel RL method for continuous-time control, where stochastic differential equations govern state-action dynamics. Departing from traditional value function-based approaches, our key contribution is the characterization of continuous-time Q-functions via a martingale condition and the linking of diffusion policy scores to the action gradient of a learned continuous Q-function by the dynamic programming principle.

The Volterra signature extends the classical path signature by incorporating general matrix-valued kernel into its iterated integral structure, yielding a flexible notion of memory for time series. Its components can be viewed as successive Picard iterates of linear controlled Volterra equations, making their exact computation of additional mathematical interest. However, the kernel introduces substantial algorithmic challenges. We provide a resolution by first decomposing the Chen-type convolution relation established in [13] into analytic and arithmetic parts, and then introducing several efficient algorithms: a general approximative scheme with quadratic complexity O(J2) in the number of time steps J, an FFT-based acceleration with complexity O(J logJ) for convolution kernels on uniform grids, and an exact recursion with complexity O(JR2) for kernels admitting a state-space representation of dimension R; retaining standard signature complexity in the path dimension and truncation level N. We further show that the number of factors in matrix-valued kernels of the form K(t,s) = P p kp(t s)Ap do not increase the asymptotic complexity in J and N. Finally, we derive a finite-difference predictor-corrector scheme for the associated Volterra signature kernel. All algorithms are implemented in the publicly available JAX-based package tensordev.

In this paper, we examine the long-run behavior of regularized, no-regret learning in1 finite games. A well-known result in the field states that the empirical frequencies2 of no-regret play converge to the game's set of coarse correlated equilibria; however,3 our understanding of how the players' actual strategies evolve over time is much4 more limited - and, in many cases, non-existent. This issue is exacerbated by5 a series of recent results showing that only strict Nash equilibria are stable and6 attracting under regularized learning, thus making the relation between learning7 and pointwise solution concepts particularly elusive. In lieu of this, we take a more8 general approach and instead seek to characterize the setwise rationality properties9 of the players' day-to-day play. To that end, we focus on one of the most stringent10 criteria of setwise strategic stability, namely that any unilateral deviation from the11 set in question incurs a cost to the deviator - a property known as closedness under12 better replies (club).

We introduce a data-driven learning framework that assimilates two powerful ideas: ideal large eddy simulation (LES) from turbulence closure modeling and neural stochastic differential equations (SDE) for stochastic modeling. The ideal LES models the LES flow by treating each full-order trajectory as a random realization of the underlying dynamics, as such, the effect of small-scales is marginalized to obtain the deterministic evolution of the LES state. However, ideal LES is analytically intractable. In our work, we use a latent neural SDE to model the evolution of the stochastic process and an encoder-decoder pair for transforming between the latent space and the desired ideal flow field. This stands in sharp contrast to other types of neural parameterization of closure models where each trajectory is treated as a deterministic realization of the dynamics. We show the effectiveness of our approach (niLES - neural ideal LES) on two challenging chaotic dynamical systems: Kolmogorov flow at a Reynolds number of 20,000 and flow past a cylinder at Reynolds number 500. Compared to competing methods, our method can handle non-uniform geometries using unstructured meshes seamlessly. In particular, niLES leads to trajectories with more accurate statistics and enhances stability, particularly for long-horizon rollouts.

We study kernel quadrature rules with convex weights. Our approach combines the spectral properties of the kernel with recombination results about point measures. This results in effective algorithms that construct convex quadrature rules using only access to i.i.d.