Publications

We introduce an efficient numerical method for second-order linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. In oscillatory regions the solution is generated via a nonoscillatory phase function that obeys the nonlinear Riccati equation. We propose a defect correction iteration that gives an asymptotic series for such a phase function; this is numerically approximated on a Chebyshev grid with a small number of nodes. For analytic coefficients we prove that each iteration, up to a certain maximum number, reduces the residual by a factor of order of the local frequency. The algorithm adapts both the stepsize and the choice of method, switching to a conventional spectral collocation method away from oscillatory regions. In numerical experiments we find that our proposal outperforms other state-of-the-art oscillatory solvers, most significantly at low to intermediate frequencies and at low tolerances, where it may use up to \(10^6\) times fewer function evaluations. Even in high-frequency regimes, our implementation is on average 10 times faster than other specialized solvers.

We introduce a new class of computationally tractable scattering problems in unbounded domains, which we call decomposable problems. In these decomposable problems, the computational domain can be split into a finite collection of subdomains in which the scatterer has a "simple" structure. A subdomain is simple if the domain Green's function for this subdomain is either available analytically or can be computed numerically with arbitrary accuracy by a tractable method. These domain Green's functions are then used to reformulate the scattering problem as a system of boundary integral equations on the union of the subdomain boundaries. This reformulation gives a practical numerical method, as the resulting integral equations can then be solved, to any desired degree of accuracy, by using coordinate complexification over a finite interval, and standard discretization techniques.

Multiple scattering methods are widely used to reduce the computational complexity of acoustic or electromagnetic scattering problems when waves propagate through media containing many identical inclusions. Historically, this numerical technique has been limited to situations in which the inclusions (particles) can be covered by nonoverlapping disks in two dimensions or spheres in three dimensions. This allows for the use of separation of variables in cylindrical or spherical coordinates to represent the solution to the governing partial differential equation. Here, we provide a more flexible approach, applicable to a much larger class of geometries. We use a Green’s representation formula and the associated layer potentials to construct incoming and outgoing solutions on rectangular enclosures. The performance and flexibility of the resulting scattering operator formulation in two-dimensions is demonstrated via several numerical examples for multi-particle scattering in free space as well as in layered media. The mathematical formalism extends directly to the three dimensional case as well, and can easily be coupled with several commercial numerical PDE software packages.

This work focuses on the gradient flow dynamics of a neural network model that uses correlation loss to approximate a multi-index function on high-dimensional standard Gaussian data. Specifically, the multi-index function we consider is a sum of neurons $f^*(x) \!=\! \sum_{j=1}^k \! \sigma^*(v_j^T x)$ where $v_1, \dots, v_k$ are unit vectors, and $\sigma^*$ lacks the first and second Hermite polynomials in its Hermite expansion. It is known that, for the single-index case ($k\!=\!1$), overcoming the search phase requires polynomial time complexity. We first generalize this result to multi-index functions characterized by vectors in arbitrary directions. After the search phase, it is not clear whether the network neurons converge to the index vectors, or get stuck at a sub-optimal solution. When the index vectors are orthogonal, we give a complete characterization of the fixed points and prove that neurons converge to the nearest index vectors. Therefore, using $n \! \asymp \! k \log k$ neurons ensures finding the full set of index vectors with gradient flow with high probability over random initialization. When $ v_i^T v_j \!=\! \beta \! \geq \! 0$ for all $i \neq j$, we prove the existence of a sharp threshold $\beta_c \!=\! c/(c+k)$ at which the fixed point that computes the average of the index vectors transitions from a saddle point to a minimum. Numerical simulations show that using a correlation loss and a mild overparameterization suffices to learn all of the index vectors when they are nearly orthogonal, however, the correlation loss fails when the dot product between the index vectors exceeds a certain threshold.

TGF-β, essential for development and immunity, is expressed as a latent complex (L-TGF-β) non-covalently associated with its prodomain and presented on immune cell surfaces by covalent association with GARP. Binding to integrin αvβ8 activates L-TGF-β1/GARP. The dogma is that mature TGF-β must physically dissociate from L-TGF-β1 for signaling to occur. Our previous studies discovered that αvβ8-mediated TGF-β autocrine signaling can occur without TGF-β1 release from its latent form. Here, we show that mice engineered to express TGF-β1 that cannot release from L-TGF-β1 survive without early lethal tissue inflammation, unlike those with TGF-β1 deficiency. Combining cryogenic electron microscopy with cell-based assays, we reveal a dynamic allosteric mechanism of autocrine TGF-β1 signaling without release where αvβ8 binding redistributes the intrinsic flexibility of L-TGF-β1 to expose TGF-β1 to its receptors. Dynamic allostery explains the TGF-β3 latency/activation mechanism and why TGF-β3 functions distinctly from TGF-β1, suggesting that it broadly applies to other flexible cell surface receptor/ligand systems.

Single-molecule experiments are a unique tool to characterize the structural dynamics of biomolecules. However, reconstructing molecular details from noisy single-molecule data is challenging. Simulation-based inference (SBI) integrates statistical inference, physics-based simulators, and machine learning and is emerging as a powerful framework for analysing complex experimental data. Recent advances in deep learning have accelerated the development of new SBI methods, enabling the application of Bayesian inference to an ever-increasing number of scientific problems. Here, we review the nascent application of SBI to the analysis of single-molecule experiments. We introduce parametric Bayesian inference and discuss its limitations. We then overview emerging deep-learning-based SBI methods to perform Bayesian inference for complex models encoded in computer simulators. We illustrate the first applications of SBI to single-molecule force-spectroscopy and cryo-electron microscopy experiments. SBI allows us to leverage powerful computer algorithms modeling complex biomolecular phenomena to connect scientific models and experiments in a principled way.

We introduce cppdlr, a C++ library implementing the discrete Lehmann representation (DLR) of functions in imaginary time and Matsubara frequency, such as Green's functions and self-energies. The DLR is based on a low-rank approximation of the analytic continuation kernel, and yields a compact and explicit basis consisting of exponentials in imaginary time and simple poles in Matsubara frequency. cppdlr constructs the DLR basis and associated interpolation grids, and implements standard operations. It provides a flexible yet high-level interface, facilitating the incorporation of the DLR into both small-scale applications and existing large-scale software projects.

Given an intractable target density p, variational inference (VI) attempts to find the best approximation q from a tractable family Q. This is typically done by minimizing the exclusive Kullback-Leibler divergence, KL(q||p). In practice, Q is not rich enough to contain p, and the approximation is misspecified even when it is a unique global minimizer of KL(q||p). In this paper, we analyze the robustness of VI to these misspecifications when p exhibits certain symmetries and Q is a location-scale family that shares these symmetries. We prove strong guarantees for VI not only under mild regularity conditions but also in the face of severe misspecifications. Namely, we show that (i) VI recovers the mean of p when p exhibits an \textit{even} symmetry, and (ii) it recovers the correlation matrix of p when in addition~p exhibits an \textit{elliptical} symmetry. These guarantees hold for the mean even when q is factorized and p is not, and for the correlation matrix even when~q and~p behave differently in their tails. We analyze various regimes of Bayesian inference where these symmetries are useful idealizations, and we also investigate experimentally how VI behaves in their absence.

Extracting consistent statistics between relevant free energy minima of a molecular system is essential for physics, chemistry, and biology. Molecular dynamics (MD) simulations can aid in this task but are computationally expensive, especially for systems that require quantum accuracy. To overcome this challenge, we developed an approach combining enhanced sampling with deep generative models and active learning of a machine learning potential (MLP). We introduce an adaptive Markov chain Monte Carlo framework that enables the training of one normalizing flow (NF) and one MLP per state, achieving rapid convergence toward the Boltzmann distribution. Leveraging the trained NF and MLP models, we compute thermodynamic observables such as free energy differences and optical spectra. We apply this method to study the isomerization of an ultrasmall silver nanocluster belonging to a set of systems with diverse applications in the fields of medicine and catalysis.