Publications

In this work we present an algorithm to construct an infinitely differentiable smooth surface from an input consisting of a (rectilinear) triangulation of a surface of arbitrary shape. The original surface can have non-trivial genus and multiscale features, and our algorithm has computational complexity which is linear in the number of input triangles. We use a smoothing kernel to define a function $\phi$ whose level set defines the surface of interest. Charts are subsequently generated as maps from the original user-specified triangles to $\mathbb {R}^3$. The degree of smoothness is controlled locally by the kernel to be commensurate with the fineness of the input triangulation. The expression for~$\Phi$ can be transformed into a boundary integral, whose evaluation can be accelerated using a fast multipole method. We demonstrate the effectiveness and cost of the algorithm with polyhedral and quadratic skeleton surfaces obtained from CAD and meshing software.

It is difficult to quantify structure–property relationships and to identify structural features of complex materials. The characterization of amorphous materials is especially challenging because their lack of long-range order makes it difficult to define structural metrics. In this work, we apply deep learning algorithms to accurately classify amorphous materials and characterize their structural features. Specifically, we show that convolutional neural networks and message passing neural networks can classify two-dimensional liquids and liquid-cooled glasses from molecular dynamics simulations with greater than 0.98 AUC, with no a priori assumptions about local particle relationships, even when the liquids and glasses are prepared at the same inherent structure energy. Furthermore, we demonstrate that message passing neural networks surpass convolutional neural networks in this context in both accuracy and interpretability. We extract a clear interpretation of how message passing neural networks evaluate liquid and glass structures by using a self-attention mechanism. Using this interpretation, we derive three novel structural metrics that accurately characterize glass formation. The methods presented here provide a procedure to identify important structural features in materials that could be missed by standard techniques and give unique insight into how these neural networks process data."

Detecting community-wide statistical relationships from targeted amplicon-based and metagenomic profiling of microbes in their natural environment is an important step toward understanding the organization and function of these communities. We present a robust and computationally tractable latent graphical model inference scheme that allows simultaneous identification of parsimonious statistical relationships among microbial species and unobserved factors that influence the prevalence and variability of the abundance measurements. Our method comes with theoretical performance guarantees and is available within the SParse InversE Covariance estimation for Ecological ASsociation Inference (SPIEC-EASI) framework (SpiecEasi R-package). Using simulations, as well as a comprehensive collection of amplicon-based gut microbiome datasets, we illustrate the methods ability to jointly identify compositional biases, latent factors that correlate with observed technical covariates, and robust statistical microbial associations that replicate across different gut microbial data sets.

Inspired by the possibility that generative models based on quantum circuits can provide a useful inductive bias for sequence modeling tasks, we propose an efficient training algorithm for a subset of classically simulable quantum circuit models. The gradient-free algorithm, presented as a sequence of exactly solvable effective models, is a modification of the density matrix renormalization group procedure adapted for learning a probability distribution. The conclusion that circuit-based models offer a useful inductive bias for classical datasets is supported by experimental results on the parity learning problem.

A common problem in cosmology is to integrate the product of two or more spherical Bessel functions (sBFs) with different configuration-space arguments against the power spectrum or its square, weighted by powers of wavenumber. Naively computing them scales as $N_{\rm g}^{p+1}$ with $p$ the number of configuration space arguments and $N_{\rm g}$ the grid size, and they cannot be done with Fast Fourier Transforms (FFTs). Here we show that by rewriting the sBFs as sums of products of sine and cosine and then using the product to sum identities, these integrals can then be performed using 1-D FFTs with $N_{\rm g} \log N_{\rm g}$ scaling. This "rotation" method has the potential to accelerate significantly a number of calculations in cosmology, such as perturbation theory predictions of loop integrals, higher order correlation functions, and analytic templates for correlation function covariance matrices. We implement this approach numerically both in a free-standing, publicly-available \textsc{Python} code and within the larger, publicly-available package \texttt{mcfit}. The rotation method evaluated with direct integrations already offers a factor of 6-10$\times$ speed-up over the naive approach in our test cases. Using FFTs, which the rotation method enables, then further improves this to a speed-up of $\sim$$1000-3000\times$ over the naive approach. The rotation method should be useful in light of upcoming large datasets such as DESI or LSST. In analysing these datasets recomputation of these integrals a substantial number of times, for instance to update perturbation theory predictions or covariance matrices as the input linear power spectrum is changed, will be one piece in a Monte Carlo Markov Chain cosmological parameter search: thus the overall savings from our method should be significant.

We propose Cormorant, a rotationally covariant neural network architecture for learning the behavior and properties of complex many-body physical systems. We apply these networks to molecular systems with two goals: learning atomic potential energy surfaces for use in Molecular Dynamics simulations, and learning ground state properties of molecules calculated by Density Functional Theory. Some of the key features of our network are that (a) each neuron explicitly corresponds to a subset of atoms; (b) the activation of each neuron is covariant to rotations, ensuring that overall the network is fully rotationally invariant. Furthermore, the non-linearity in our network is based upon tensor products and the Clebsch-Gordan decomposition, allowing the network to operate entirely in Fourier space. Cormorant significantly outperforms competing algorithms in learning molecular Potential Energy Surfaces from conformational geometries in the MD-17 dataset, and is competitive with other methods at learning geometric, energetic, electronic, and thermodynamic properties of molecules on the GDB-9 dataset.

We present a new integral representation for the unsteady, incompressible Stokes or Navier-Stokes equations, based on a linear combination of heat and harmonic potentials. For velocity boundary conditions, this leads to a coupled system of integral equations: one for the normal component of velocity and one for the tangential components. Each individual equation is well-condtioned, and we show that using them in predictor-corrector fashion, combined with spectral deferred correction, leads to high-order accuracy solvers. The fundamental unknowns in the mixed potential representation are densities supported on the boundary of the domain. We refer to one as the vortex source, the other as the pressure source and the coupled system as the combined source integral equation.

We study the stability properties of explicit marching schemes for second-kind Volterra integral equations that arise when solving boundary value problems for the heat equation by means of potential theory. It is well known that explicit finite difference or finite element schemes for the heat equation are stable only if the time step $\Delta t$ is of the order $O(\Delta x^2)$, where $\Delta x$ is the finest spatial grid spacing. In contrast, for the Dirichlet and Neumann problems on the unit ball in all dimensions $d\ge 1$, we show that the simplest Volterra marching scheme, i.e., the forward Euler scheme, is unconditionally stable. Our proof is based on an explicit spectral radius bound of the marching matrix, leading to an estimate that an $L^2$-norm of the solution to the integral equation is bounded by $c_dT^{d/2}$ times the norm of the right hand side. For the Robin problem on the half space in any dimension, with constant Robin (heat transfer) coefficient $\kappa$, we exhibit a constant $C$ such that the forward Euler scheme is stable if $\Delta t < C/\kappa^2$, independent of any spatial discretization. This relies on new lower bounds on the spectrum of real symmetric Toeplitz matrices defined by convex sequences. Finally, we show that the forward Euler scheme is unconditionally stable for the Dirichlet problem on any smooth convex domain in any dimension, in $L^\infty$-norm.

Given the importance of understanding single-neuron activity, much development has been directed towards improving the performance and automation of spike sorting. These developments, however, introduce new challenges, such as file format incompatibility and reduced interoperability, that hinder benchmarking and preclude reproducible analysis. To address these limitations, we developed SpikeInterface, a Python framework designed to unify preexisting spike sorting technologies into a single codebase and to standardize extracellular data file operations. With a few lines of code and regardless of the underlying data format, researchers can: run, compare, and benchmark most modern spike sorting algorithms; pre-process, post-process, and visualize extracellular datasets; validate, curate, and export sorting outputs; and more. In this paper, we provide an overview of SpikeInterface and, with applications to both real and simulated extracellular datasets, demonstrate how it can improve the accessibility, reliability, and reproducibility of spike sorting in preparation for the widespread use of large-scale electrophysiology.

We present a family of integral equation-based solvers for the linear or semilinear heat equation in complicated moving (or stationary) geometries. This approach has significant advantages over more standard finite element or finite difference methods in terms of accuracy, stability and space-time adaptivity. In order to be practical, however, a number of technical capabilites are required: fast algorithms for the evaluation of heat potentials, high-order accurate quadratures for singular and weakly integrals over space-time domains, and robust automatic mesh refinement and coarsening capabilities. We describe all of these components and illustrate the performance of the approach with numerical examples in two space dimensions.