arXiv subjects

Jonathan Weare

Publications and source records attributed to Jonathan Weare.

At least 19 recordsLinked to original sources

arXiv2026

Restrained Diffusion Bridges: Targeted Denoiser Training and Refinement

Diffusion models are often inaccurate in critical, data-sparse regions of the target distribution. Sampling from restrained distributions, biased distributions used in classical enhanced sampling to concentrate data near chosen configurations, can control where training data are collected but biases the learned distribution. We establish that this bias cancels exactly when forward noising from restrained data is conditioned on the restrained configuration at an appropriately chosen intermediate time. This allows a pretrained model fine-tuned on restrained data to achieve high accuracy in any chosen region of the target distribution, regardless of its probability, using standard denoising or flow matching objectives without importance weights. When no target data are available, Gibbs sampling enables inference from a model trained exclusively on restrained data.

cs.LG↗

arXiv2026

Machine learning kinetics from molecular dynamics data

Most molecular transitions occur on time scales far beyond direct molecular dynamics simulations. The committor, the probability that a configuration reaches a product state before a reactant state, is a central kinetic statistic, providing a mechanism-independent reaction coordinate and a foundation for transition path theory and the calculation of rates. This review surveys modern approaches for estimating the committor and related kinetic statistics from molecular simulations, with an emphasis on self-supervised methods that learn solutions of their defining dynamical equations rather than relying on labeled shooting data. We develop a common operator viewpoint connecting generator-based partial differential equations, variational principles, Markov state models, dynamical Galerkin approximation, and neural networks. Empirical and theoretical evidence points to the efficiency of these methods. We provide theoretical and practical guidance for realizing their full potential in applications, including strategies for treating non-Markovian effects and for sampling. We conclude by identifying opportunities for further research, including connections to reinforcement learning and generative modeling.

physics.comp-ph↗

arXiv2026

LiveCoMS Roundtable: Discussing Markov state models after two decades of development v1.0

Markov state models (MSMs) play a central role in the analysis of molecular dynamics (MD) simulations for biomolecules, and MSM methodologies have been under continual development since their introduction to the field in the early 2000s. MSMs take MD trajectory data as input and produce a more compact, i.e., coarse-grained, and interpretable representation of the system in terms of conformational states and quantitative observables such as the equilibrium populations of states and the timescales of interconverting between states. Numerous strategies and considerable mathematical formalism have been developed to improve coarse-graining in both time and space to enable the most reliable representation of the underlying biomolecular dynamics. Nevertheless, there remains considerable discussion in the literature regarding best practices and MSM limitations, as well as proposals to abandon or extend the first-order Markovian framework in common use. This Roundtable article gathers together diverse opinions from $> 10$ authors active in MSM development, and by design, each author or author group offers a separate contribution. No consensus view is provided. Instead, the article is meant to be a guide for readers wishing to be exposed to a wide set of expert perspectives.

q-bio.BM↗

arXiv2026

Sampling with collective variables: Relative error analysis for umbrella sampling under general conditions

We consider the problem of estimating the expectation of a function (observable) with respect to a high-dimensional probability distribution, under the assumption that a low-dimensional collective variable map captures all the directions of slow mixing for the distribution. More precisely, under the assumption that the fibers of the collective variable map satisfy a log-Sobolev inequality, we establish non-asymptotic relative-error guarantees for this problem using the eigenvector method for umbrella sampling (EMUS). Our analysis applies to both log-Lipschitz observables and indicator of convex sets, and gives complexity bounds that are polynomial in regularity parameters and inverse relative accuracy, and exponential in the dimension of the collective variable space. As a relative error guarantee, our bounds apply to rare event estimation. We consider Gaussian biasing windows defined on a low-dimensional collective variable space and analyze a modified sparse matrix version of EMUS. The bounds quantify the effect of the map through mixing of the conditional distribution and fluctuation of the observable along the fibers.

math.ST↗

arXiv2026

Dimension-independent eigenvalue approximation for sparse matrices

Many applications in science and engineering require solving sparse Hermitian eigenvalue problems that are so large that even algorithms whose runtime scales linearly with the matrix dimension are unacceptably slow. Responding to this challenge, there is a growing body of work on sparsified iterative methods and submatrix selection methods that often perform well in practice. However, theoretical understanding of these methods is limited. In this paper, we take a step toward better understanding sublinear-time eigenvalue computation by analyzing the sparsified power method. We prove that the method approximates a leading eigenpair to a prescribed accuracy in a number of iterations comparable to the standard power method. The required number of retained nonzeros depends explicitly on the target accuracy, the initial overlap with the leading eigenvector, the compressibility of the leading eigenvector, and the relative spectral gap. When these quantities and the column sparsity are independent of dimension, the runtime is independent of dimension. In contrast, we show that a comparable worst-case guarantee cannot hold for a prototypical sparsified submatrix selection method. Submatrix selection is the approach of choice in many quantum chemistry applications. However, we show that, even with substantial initial eigenvector overlap, a single iteration of the method can produce a vector that is orthogonal to the leading eigenvector, after which the method cannot recover. Nevertheless, experiments on matrices from the SuiteSparse matrix collection and quantum chemistry problems illustrate the practical effectiveness of submatrix selection methods in a variety of domains despite these worst-case limitations.

math.NA↗

arXiv2026

Nuclear Quantum Effects as a Denoising Problem

Nuclear quantum effects are rigorously captured by imaginary-time path integrals, which map the quantum Boltzmann distribution onto a ring polymer of classical replicas. Yet the nuclear masses, the coupling to the environment, and the boundary conditions of the path remain hard-wired in the simulation or the trained model, even though this quantum context enters the path measure only through a quadratic action known in closed form. Here we show that a denoiser trained on classical Boltzmann statistics alone, composed at sampling time with an analytic Gaussian component carrying the entire quantum context, yields the quantum Boltzmann distribution of the nuclei. Such a composition exists and is exact whenever the training noise does not exceed the intrinsic quantum uncertainty of the target ensemble, and it is invariant across all quantum contexts admitted by this bound. We show exact transfer across temperature, isotopic mass, dissipation strength, and the boundary conditions of the path in theory and in numerical experiments, without retraining. The last yields the end-to-end displacement and momentum distributions of a tagged nucleus from open imaginary-time paths. The same invariance extends in principle to the permuted boundary conditions of bosonic exchange, with the identical denoiser. In this view, the noise of generative modeling and the quantum fluctuations of the nuclei are two faces of the same quadratic structure.

physics.chem-ph↗

arXiv2026

Delocalization of bias in unadjusted Hamiltonian Monte Carlo and underdamped Langevin

Unadjusted samplers such as unadjusted Hamiltonian Monte Carlo and underdamped Langevin are well-known to be biased. Metropolis--Hastings adjustment has been conventionally incorporated into Hamiltonian Monte Carlo to eliminate the bias. However, this adjustment can significantly increase the iteration complexity due to the small step size required for reasonable Metropolis acceptance rates. In this work, we extend the \emph{delocalization of bias} phenomenon, previously established for the overdamped Langevin algorithm, to these two unadjusted algorithms. We show that to control the $W_2$ bias of any $K$-dimensional marginal of a high-dimensional distribution, $O(\sqrt{K})$ integration steps suffice up to $\log d$ terms, assuming either weak or sparse interactions among variables. The discrete-time integrators here introduce technical difficulties beyond those of the overdamped setting, which we address through a broadly applicable matrix-polynomial framework that characterizes their propagators. Our result for the underdamped Langevin algorithm is valid for all large friction parameters, implying that the Leimkuhler-Matthews integrator for the overdamped Langevin dynamics also exhibits delocalization of bias.

stat.CO↗

arXiv2026

AI-boosted rare event sampling to characterize extreme weather

Weather extremes pose major societal risks, especially in a changing climate, but due to their rarity, they are difficult to study using limited observations or complex climate models. We introduce AI+RES, a framework coupling fast AI weather forecasts with a high-fidelity physics model using a rare-event algorithm to efficiently characterize extremes. This approach enables the study of the statistics and physics of very rare events, such as once per millennium heatwaves at two orders-of-magnitude lower computational cost. AI+RES can be applied broadly across climate science and other fields concerned with rare events.

physics.ao-ph↗

arXiv2026

Composing diffusion priors with explicit physical context via generative Gibbs sampling

Pretrained diffusion models provide powerful learned priors, but in scientific sampling the target distribution often depends on physical context that is not fully represented by one generative model. We introduce Generative Gibbs for Physics-Aware Sampling (GG-PA), a training-free framework that formulates the composition of learned partial priors and explicit physical context as inference over a joint target distribution in an augmented state space. We derive a Gibbs sampler for this joint target, show that it is asymptotically exact as the diffusion time approaches zero, and prove that in settings with quadratic interactions it remains exact at finite diffusion times. We further introduce replica exchange over diffusion time to accelerate mixing. Experiments on a double-well system, a $ϕ^4$ lattice model, and atomistic peptide systems show that GG-PA recovers context-induced distribution shifts and emergent collective behavior in interacting systems using partial priors without retraining. These results demonstrate GG-PA as a practical approach for combining pretrained generative priors with explicit physical context.

cs.LG↗

arXiv2026

Functional Estimation of the Marginal Likelihood

We propose a framework for computing, optimizing and integrating with respect to a smooth marginal likelihood in statistical models that involve high-dimensional parameters/latent variables and continuous low-dimensional hyperparameters. The method requires samples from the posterior distribution of the parameters for different values of the hyperparameters on a simulation grid and returns inference on the marginal likelihood defined everywhere on its domain, and on its functionals. We show how the method relates to many of the methods that have been used in this context, including sequential Monte Carlo, Gibbs sampling, Monte Carlo maximum likelihood, and umbrella sampling. We establish the consistency of the proposed estimators as the sampling effort increases, both when the simulation grid is kept fixed and when it becomes dense in the domain. We showcase the approach on Gaussian process regression and classification and crossed effect models.

stat.ME↗

arXiv2026

Quantum statistics from classical simulations via generative Gibbs sampling

Accurate simulation of nuclear quantum effects is essential for molecular modeling but expensive using path integral molecular dynamics (PIMD). We present GG-PI, a ring-polymer-based framework that combines generative modeling of the single-bead conditional density with Gibbs sampling to recover quantum statistics from classical simulation data. GG-PI uses inexpensive standard classical simulations or existing data for training and allows transfer across temperatures without retraining. On standard test systems, GG-PI significantly reduces wall clock time compared to PIMD. Our approach extends easily to a wide range of problems with similar Markov structure.

physics.chem-ph↗

arXiv2025

An exact multiple-time-step variational formulation for the committor and the transition rate

For a transition between two stable states, the committor is the probability that the dynamics leads to one stable state before the other. It can be estimated from trajectory data by minimizing an expression for the transition rate that depends on a lag time. We show that an existing such expression is minimized by the exact committor only when the lag time is a single time step, resulting in a biased estimate in practical applications. We introduce an alternative expression that is minimized by the exact committor at any lag time. The key idea is that, when trajectories enter the stable states, the times that they enter (stopping times) must be used for estimating the committor and transition rate instead of the lag time. Numerical tests on benchmark systems demonstrate that our committor and transition rate estimates are much less sensitive to the choice of lag time. We show how further accuracy for the transition rate can be achieved by combining results from two lag times. We also relate the transition rate expression to a variational approach for kinetic statistics based on the mean-squared residual and discuss further numerical considerations with the aid of a decomposition of the error into dynamic modes.

cond-mat.stat-mech↗

arXiv2025

Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias

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 $d$ of the problem increases, the number of iterations required to ensure convergence within a desired error in the $W_2$ metric scales in proportion to $d$ or $\sqrt{d}$. In this paper, we argue that, despite this poor scaling of the $W_2$ error for the full set of variables, the behavior for a small number of variables can be significantly better: a number of iterations proportional to $K$, up to logarithmic terms in $d$, often suffices for the algorithm to converge to within a desired $W_2$ error for all $K$-marginals. We refer to this effect as delocalization 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 $W_{2,\ell^\infty}$ 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.

stat.ML↗

arXiv2025

Sampling parameters of ordinary differential equations with Langevin dynamics that satisfy constraints

Fitting models to data to obtain distributions of consistent parameter values is important for uncertainty quantification, model comparison, and prediction. Standard Markov chain Monte Carlo (MCMC) approaches for fitting ordinary differential equations (ODEs) to time-series data involve proposing trial parameter sets, numerically integrating the ODEs forward in time, and accepting or rejecting the trial parameter sets. When the model dynamics depend nonlinearly on the parameters, as is generally the case, trial parameter sets are often rejected, and MCMC approaches become prohibitively computationally costly to converge. Here, we build on methods for numerical continuation and trajectory optimization to introduce an approach in which we use Langevin dynamics in the joint space of variables and parameters to sample models that satisfy constraints on the dynamics. We demonstrate the method by sampling Hopf bifurcations and limit cycles of a model of a biochemical oscillator in a Bayesian framework for parameter estimation, and we attain performance that matches or exceeds the performance of leading MCMC approaches that require numerically integrating the ODEs forward in time. We describe numerical experiments that provide insight into the speedup. The method is general and can be used in any framework for parameter estimation and model selection.

stat.CO↗

arXiv2025

Improved energies and wave function accuracy with Weighted Variational Monte Carlo

Neural network parametrizations have increasingly been used to represent the ground and excited states in variational Monte Carlo (VMC) with promising results. However, traditional VMC methods only optimize the wave function in regions of peak probability. The wave function is uncontrolled in the tails of the probability distribution, which can limit the accuracy of the trained wavefunction approximation. To improve the approximation accuracy in the probability tails, this paper interprets VMC as a gradient flow in the space of wave functions, followed by a projection step. From this perspective, arbitrary probability distributions can be used in the projection step, allowing the user to prioritize accuracy in different regions of state space. Motivated by this theoretical perspective, the paper tests a new weighted VMC method on the antiferromagnetic Heisenberg model for a periodic spin chain. Compared to traditional VMC, weighted VMC reduces the error in the ground state energy by a factor of 2 and it reduces the errors in the local energies away from the mode by large factors of $10^2$--$10^4$.

physics.comp-ph↗

arXiv2025

Can AI weather models predict out-of-distribution gray swan tropical cyclones?

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 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.

physics.ao-ph↗

arXiv2025

Randomly sparsified Richardson iteration: A dimension-independent sparse linear solver

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 $10^{108} \times 10^{108}$. So far, a complete mathematical explanation for their 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 vector itself is too large to store.

math.NA↗

arXiv2025

The surprising efficiency of temporal difference learning for rare event prediction

We quantify the efficiency of temporal difference (TD) learning over the direct, or Monte Carlo (MC), estimator for policy evaluation in reinforcement learning, with an emphasis on estimation of quantities related to rare events. Policy evaluation is complicated in the rare event setting by the long timescale of the event and by the need for \emph{relative accuracy} in estimates of very small values. Specifically, we focus on least-squares TD (LSTD) prediction for finite state Markov chains, and show that LSTD can achieve relative accuracy far more efficiently than MC. We prove a central limit theorem for the LSTD estimator and upper bound the \emph{relative asymptotic variance} by simple quantities characterizing the connectivity of states relative to the transition probabilities between them. Using this bound, we show that, even when both the timescale of the rare event and the relative accuracy of the MC estimator are exponentially large in the number of states, LSTD maintains a fixed level of relative accuracy with a total number of observed transitions of the Markov chain that is only \emph{polynomially} large in the number of states.

cs.LG↗