Publications

Markov state models (MSMs) are widely employed to analyze the kinetics of complex systems. But despite their effectiveness in many applications, MSMs are prone to systematic or statistical errors, often exacerbated by suboptimal hyperparameter choice. In this article, we attempt to understand how these choices affect the error of estimates of mean first-passage times and committors, key quantities in chemical rate theory. We first evaluate the performance of the recently introduced “stopped-process estimator” [Strahan, J. Long-time-scale predictions from short-trajectory data: A benchmark analysis of the trp-cage miniprotein. J. Chem. Theory Comput. 2021, 17, 2948–2963. 10.1021/acs.jctc.0c00933.] that attempts to reduce error caused by choosing a too-large lag time. We then study the effect of statistical errors on Markov state model construction using the condition number, which measures an MSM’s sensitivity to perturbation. This analysis helps give an insight into which factors cause an MSM to be more or less sensitive to statistical error. Our work highlights the importance of choosing a good sampling measure, the measure from which the initial points are drawn, and has implications for recent work applying a variational principle for evaluating the committor.

We present a family of integral equation-based solvers for the heat equation, reaction-diffusion systems, the unsteady Stokes equation and the incompressible Navier-Stokes equations in two space dimensions. Our emphasis is on the development of methods that can efficiently follow complex solution features in space-time by refinement and coarsening at each time step on an adaptive quadtree. For simplicity, we focus on problems posed in a square domain with periodic boundary conditions. The performance and robustness of the methods are illustrated with several numerical examples.

The randomized SVD is a method to compute an inexpensive, yet accurate, low-rank approximation of a matrix. The algorithm assumes access to the matrix through matrix-vector products (matvecs). Therefore, when we would like to apply the randomized SVD to a matrix function, f(A), one needs to approximate matvecs with f(A) using some other algorithm, which is typically treated as a black-box. Chen and Hallman (SIMAX 2023) argued that, in the common setting where matvecs with f(A) are approximated using Krylov subspace methods (KSMs), a more efficient low-rank approximation is possible if we open this black-box. They present an alternative approach that significantly outperforms the naive combination of KSMs with the randomized SVD, although the method lacked theoretical justification. In this work, we take a closer look at the method, and provide strong and intuitive error bounds that justify its excellent performance for low-rank approximation of matrix functions.

We present a spectrally accurate fast algorithm for evaluating the solution to the scalar wave equation in free space driven by a large collection of point sources in a bounded domain. With $M$ sources temporally discretized by $N_t$ time steps of size $\Delta t$, a naive potential evaluation at $M$ targets on the same time grid requires $\mathcal O(M^2 N_t)$ work. Our scheme requires $\mathcal{O}\left((M + N^3\log N)N_t\right)$ work, where $N$ scales as $\mathcal O(1/\Delta t)$, i.e., the maximum signal frequency. This is achieved by using the recently-proposed windowed Fourier projection (WFP) method to split the potential into a local part, evaluated directly, plus a smooth history part approximated by an $N^3$-point equispaced discretization of the Fourier transform, where each Fourier coefficient obeys a simple recursion relation. The growing oscillations in the spectral representation (which would be present with a naive use of the Fourier transform) are controlled by spatially truncating the hyperbolic Green's function itself. Thus, the method avoids the need for absorbing boundary conditions. We demonstrate the performance of our algorithm with up to a million sources and targets at 6-digit accuracy. We believe it can serve as a key component in addressing time-domain wave equation scattering problems.

Recent advances in mechanistic interpretability have revealed that large language models (LLMs) develop internal representations corresponding not only to concrete entities but also distinct, human-understandable abstract concepts and behaviour. Moreover, these hidden features can be directly manipulated to steer model behaviour. However, it remains an open question whether this phenomenon is unique to models trained on inherently structured data (ie. language, images) or if it is a general property of foundation models. In this work, we investigate the internal representations of a large physics-focused foundation model. Inspired by recent work identifying single directions in activation space for complex behaviours in LLMs, we extract activation vectors from the model during forward passes over simulation datasets for different physical regimes. We then compute "delta" representations between the two regimes. These delta tensors act as concept directions in activation space, encoding specific physical features. By injecting these concept directions back into the model during inference, we can steer its predictions, demonstrating causal control over physical behaviours, such as inducing or removing some particular physical feature from a simulation. These results suggest that scientific foundation models learn generalised representations of physical principles. They do not merely rely on superficial correlations and patterns in the simulations. Our findings open new avenues for understanding and controlling scientific foundation models and has implications for AI-enabled scientific discovery

Foundation models have transformed machine learning for language and vision, but achieving comparable impact in physical simulation remains a challenge. Data heterogeneity and unstable long-term dynamics inhibit learning from sufficiently diverse dynamics, while varying resolutions and dimensionalities challenge efficient training on modern hardware. Through empirical and theoretical analysis, we incorporate new approaches to mitigate these obstacles, including a harmonic-analysis-based stabilization method, load-balanced distributed 2D and 3D training strategies, and compute-adaptive tokenization. Using these tools, we develop Walrus, a transformer-based foundation model developed primarily for fluid-like continuum dynamics. Walrus is pretrained on nineteen diverse scenarios spanning astrophysics, geoscience, rheology, plasma physics, acoustics, and classical fluids. Experiments show that Walrus outperforms prior foundation models on both short and long term prediction horizons on downstream tasks and across the breadth of pretraining data, while ablation studies confirm the value of our contributions to forecast stability, training throughput, and transfer performance over conventional approaches. Code and weights are released for community use.

Phase separation of proteins plays a critical role in cellular organization. How phase-separated protein condensates underpin biological function and how condensates achieve specificity remain elusive. We investigated the phase separation of MUT-16, a scaffold protein in Mutator foci, and its role in recruiting the client protein MUT-8, a key component in RNA silencing in Caenorhabditis elegans. We employed a multi-scale approach that combined coarse-grained (residue-level CALVADOS2 and near-atomistic Martini3) and atomistic simulations. Simulations across different resolutions provide a consistent perspective on how MUT-16 condensates recruit MUT-8, enabling the fine-tuning of chemical details and balancing the computational cost. Both coarse-grained models (CALVADOS2 and Martini3) predicted the relative phase-separation propensities of MUT-16’s disordered regions, which we confirmed through in vitro experiments. Simulations also identified key sequence features and residues driving phase separation and revealed differences in residue interaction propensities between CALVADOS2 and Martini3. Furthermore, Martini3 and 350-μs atomistic simulations on Folding@Home of MUT-8’s N-terminal prion-like domain with MUT-16 M8BR cluster highlighted the importance of cation-interactions between Tyr residues of MUT-8 and Arg residues of MUT-16 M8BR. Lys residues were observed to be more prone to interact in Martini3. Atomistic simulations revealed that the guanidinium group of Arg also engages in interactions and hydrogen bonds with the backbone of Tyr, possibly contributing to the greater strength of Arg-Tyr interactions compared to Lys-Tyr, where these additional favorable contacts are absent. In agreement with our simulations, in vitro co-expression pull-down experiments demonstrated a progressive loss of MUT-8 recruitment after the mutation of Arg in MUT-16 M8BR to Lys or Ala, confirming the critical role of Arg in this interaction. These findings advance our understanding of MUT-16 phase separation and subsequent MUT-8 recruitment, key processes in assembling Mutator foci that drive RNA silencing in C. elegans.

Elliptic partial differential equations (PDEs) can model many physical phenomena, such as electrostatics, acoustics, wave propagation, and diffusion. In scientific machine learning settings, a high-throughput PDE solver may be required to generate a training dataset, run in the inner loop of an iterative algorithm, or interface directly with a deep neural network. To provide value to machine learning users, such a PDE solver must be compatible with standard automatic differentiation frameworks, scale efficiently when run on graphics processing units (GPUs), and maintain high accuracy for a large range of input parameters. We have designed the jaxhps package with these use-cases in mind by implementing a highly efficient and accurate solver for elliptic problems with native hardware acceleration and automatic differentiation support. This is achieved by expressing a highly-efficient solution method for elliptic PDEs in JAX (Bradbury et al., 2018). This software implements algorithms specifically designed for fast GPU execution of a family of elliptic PDE solvers, which are described in full in Melia et al. (2025).

Despite the rise of single particle cryo-electron microscopy (cryo-EM) as a premier method for resolving macromolecular structures at atomic resolution, methods to address molecular heterogeneity in vitrified samples have yet to reach maturity. With an increasing number of new methods to analyze the multitude of heterogeneous states captured in single particle images, a systematic approach to validation in this field is needed. With this motivation, we issued a challenge to the community to analyze two cryo-EM particle image sets of thyroglobulin that exhibit continuous conformational heterogeneity. The first dataset was experimental and the second was generated with a simulator, allowing control over the distribution of molecular structures and enabled direct comparison between participants’ submissions and the ground truth molecular structures and distributions. Participants were asked to submit 80 volumes representing the heterogeneous ensemble and estimate their respective populations in the image sets provided. Participation of the research community in the challenge was strong, with submissions from nearly all developers of heterogeneity methods, resulting in 41 submissions across both datasets. Submissions qualitatively exceeded expectations, with the molecular motions identified by methods resembling both each other and the ground truth motion. However, quantitatively assessing these similarities was a challenge in and of itself. In the process of assessing the submissions, we developed several validation metrics, most of which require reference to the underlying ground truth volumes. However, we have also explored the use of metrics that do not necessarily reference ground truth. This is particularly apt for experimental datasets where ground truth is inaccessible. These approaches allowed us to assess the similarity and accuracy in volume quality, molecular motions, and conformational distribution of di!erent submissions. These metrics and the e!orts of all participants help chart a path forward for the improvements of heterogeneity methods for cryo-EM and for future challenges to validate these new methods as they continue to be developed by the community.

The first-order continuum partial differential equation (PDE) model proposed by Bistritzer and MacDonald [Proc. Natl. Acad. Sci. U. S. A. 108, 12233–12237 (2011)] accurately describes the single-particle electronic properties of twisted bilayer graphene at small twist angles. In this paper, we obtain higher-order corrections to the Bistritzer–MacDonald (BM) model via a systematic multiple-scales expansion. We prove that the solution of the resulting higher-order PDE model accurately approximates the corresponding tight-binding wave function under a natural choice of parameters and given initial conditions that are spectrally localized to the monolayer Dirac points. Numerical simulations of tight-binding and continuum dynamics demonstrate the validity of the higher-order continuum model. Symmetries of the higher-order models are also discussed. This work extends the analysis from Watson et al., J. Math. Phys. 64, 031502 (2023), which rigorously established the validity of the (first-order) BM model.