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ABSTRACT The unadjusted Langevin algorithm is commonly used to sample probability distributions in extremely high‐dimensional settings. However, existing analyses of the algorithm for strongly log‐concave distributions suggest that, as the dimension of the problem increases, the number of iterations required to ensure convergence within a desired error in the metric scales in proportion to or . In this paper, we argue that, despite this poor scaling of the error for the full set of variables, the behavior for asmall numberof variables can be significantly better: A number of iterations proportional to , up to logarithmic terms in , often suffices for the algorithm to converge to within a desired error for all ‐marginals. We refer to this effect asdelocalization of bias. We show that the delocalization effect does not hold universally and prove its validity for Gaussian distributions and strongly log‐concave distributions with certain sparse interactions. Our analysis relies on a novel metric to measure convergence. A key technical challenge we address is the lack of a one‐step contraction property in this metric. Finally, we use asymptotic arguments to explore potential generalizations of the delocalization effect beyond the Gaussian and sparse interactions setting. 

Abstract Recently, a class of algorithms combining classical fixed‐point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as . So far, a complete mathematical explanation for this success has proven elusive. The family of methods has not yet been extended to the important case of linear system solves. In this paper, we propose a new scheme based on repeated random sparsification that is capable of solving sparse linear systems in arbitrarily high dimensions. We provide a complete mathematical analysis of this new algorithm. Our analysis establishes a faster‐than‐Monte Carlo convergence rate and justifies use of the scheme even when the solution is too large to store as a dense vector. 

Predicting gray swan weather extremes, which are possible but so rare that they are absent from the training dataset, is a major concern for AI weather models and long-term climate emulators. An important open question is whether AI models can extrapolate from weaker weather events present in the training set to stronger, unseen weather extremes. To test this, we train independent versions of the AI weather model FourCastNet on the 1979–2015 ERA5 dataset with all data, or with Category 3–5 tropical cyclones (TCs) removed, either globally or only over the North Atlantic or Western Pacific basin. We then test these versions of FourCastNet on 2018–2023 Category 5 TCs (gray swans). All versions yield similar accuracy for global weather, but the one trained without Category 3–5 TCs cannot accurately forecast Category 5 TCs, indicating that these models cannot extrapolate from weaker storms. The versions trained without Category 3–5 TCs in one basin show some skill forecasting Category 5 TCs in that basin, suggesting that FourCastNet can generalize across tropical basins. This is encouraging and surprising because regional information is implicitly encoded in inputs. Given that current state-of-the-art AI weather and climate models have similar learning strategies, we expect our findings to apply to other models. Other types of weather extremes need to be similarly investigated. Our work demonstrates that novel learning strategies are needed for AI models to reliably provide early warning or estimated statistics for the rarest, most impactful TCs, and, possibly, other weather extremes. 

An issue for molecular dynamics simulations is that events of interest often involve timescales that are much longer than the simulation time step, which is set by the fastest timescales of the model. Because of this timescale separation, direct simulation of many events is prohibitively computationally costly. This issue can be overcome by aggregating information from many relatively short simulations that sample segments of trajectories involving events of interest. This is the strategy of Markov state models (MSMs) and related approaches, but such methods suffer from approximation error because the variables defining the states generally do not capture the dynamics fully. By contrast, once converged, the weighted ensemble (WE) method aggregates information from trajectory segments so as to yield unbiased estimates of both thermodynamic and kinetic statistics. Unfortunately, errors decay no faster than unbiased simulation in WE as originally formulated and commonly deployed. Here, we introduce a theoretical framework for describing WE that shows that the introduction of an approximate stationary distribution on top of the stratification, as in nonequilibrium umbrella sampling (NEUS), accelerates convergence. Building on ideas from MSMs and related methods, we generalize the NEUS approach in such a way that the approximation error can be reduced systematically. We show that the improved algorithm can decrease the simulation time required to achieve the desired precision by orders of magnitude. 

Identifying informative low-dimensional features that characterize dynamics in molecular simulations remains a challenge, often requiring extensive manual tuning and system-specific knowledge. Here, we introduce geom2vec, in which pretrained graph neural networks (GNNs) are used as universal geometric featurizers. By pretraining equivariant GNNs on a large dataset of molecular conformations with a self-supervised denoising objective, we obtain transferable structural representations that are useful for learning conformational dynamics without further fine-tuning. We show how the learned GNN representations can capture interpretable relationships between structural units (tokens) by combining them with expressive token mixers. Importantly, decoupling training the GNNs from training for downstream tasks enables analysis of larger molecular graphs (that can represent small proteins at all-atom resolution) with limited computational resources. In these ways, geom2vec eliminates the need for manual feature selection and increases the robustness of simulation analyses. 

Abstract Mercury’s orbit can destabilize, generally resulting in a collision with either Venus or the Sun. Chaotic evolution can causeg₁to decrease to the approximately constant value ofg₅and create a resonance. Previous work has approximated the variation ing₁as stochastic diffusion, which leads to a phenomological model that can reproduce the Mercury instability statistics of secular andN-body models on timescales longer than 10 Gyr. Here we show that the diffusive model significantly underpredicts the Mercury instability probability on timescales less than 5 Gyr, the remaining lifespan of the solar system. This is becauseg₁exhibits larger variations on short timescales than the diffusive model would suggest. To better model the variations on short timescales, we build a new subdiffusive phenomological model forg₁. Subdiffusion is similar to diffusion but exhibits larger displacements on short timescales and smaller displacements on long timescales. We choose model parameters based on the behavior of theg₁trajectories in theN-body simulations, leading to a tuned model that can reproduce Mercury instability statistics from 1–40 Gyr. This work motivates fundamental questions in solar system dynamics: why does subdiffusion better approximate the variation ing₁than standard diffusion? Why is there an upper bound ong₁, but not a lower bound that would prevent it from reachingg₅? 

Many chemical reactions and molecular processes occur on time scales that are significantly longer than those accessible by direct simulations. One successful approach to estimating dynamical statistics for such processes is to use many short time series of observations of the system to construct a Markov state model, which approximates the dynamics of the system as memoryless transitions between a set of discrete states. The dynamical Galerkin approximation (DGA) is a closely related framework for estimating dynamical statistics, such as committors and mean first passage times, by approximating solutions to their equations with a projection onto a basis. Because the projected dynamics are generally not memoryless, the Markov approximation can result in significant systematic errors. Inspired by quasi-Markov state models, which employ the generalized master equation to encode memory resulting from the projection, we reformulate DGA to account for memory and analyze its performance on two systems: a two-dimensional triple well and the AIB9 peptide. We demonstrate that our method is robust to the choice of basis and can decrease the time series length required to obtain accurate kinetics by an order of magnitude.