Publications

The Laplace–Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle deformation). Using classical potential theory, the Laplace–Beltrami operator can be pre-/post-conditioned with an integral operator whose kernel is translation invariant, resulting in well-conditioned Fredholm integral equations of the second-kind. These equations have the standard 1/r kernel from potential theory, and therefore the equations can be solved rapidly and accurately using a combination of fast multipole methods (FMMs) and high-order quadrature corrections. In this work we detail such a scheme, presenting two alternative integral formulations of the Laplace–Beltrami problem, each of whose solution can be obtained via FMM acceleration. We then present several applications of the solvers, focusing on the computation of what are known as harmonic vector fields, relevant for many applications in electromagnetics. A battery of numerical results are presented for each application, detailing the performance of the solver in various geometries.

Coarsening of two-phase systems is crucial for the stability of dense particle packingssuch as alloys, foams, emulsions, or supersaturated solutions. Mean field theoriespredict an asymptotic scaling state with a broad particle size distribution. Aqueousfoams are good model systems for investigations of coarsening-induced structures,because the continuous liquid as well as the dispersed gas phases are uniform andisotropic. We present coarsening experiments on wet foams, with liquid fractionsup to their unjamming point and beyond, that are performed under microgravity toavoid gravitational drainage. As time elapses, a self-similar regime is reached wherethe normalized bubble size distribution is invariant. Unexpectedly, the distributionfeatures an excess of small roaming bubbles, mobile within the network of jammedlarger bubbles. These roaming bubbles are reminiscent of rattlers in granular materials(grains not subjected to contact forces). We identify a critical liquid fraction흓∗, abovewhich the bubble assembly unjams and the two bubble populations merge into a singlenarrow distribution of bubbly liquids. Unexpectedly,흓∗is larger than the randomclose packing fraction of the foam흓rcp. This is because, between흓rcpand흓∗, the largebubbles remain connected due to a weak adhesion between bubbles. We present modelsthat identify the physical mechanisms explaining our observations. We propose a newcomprehensive view of the coarsening phenomenon in wet foams. Our results shouldbe applicable to other phase-separating systems and they may also help to control theelaboration of solid foams with hierarchical structures

The forces which orient the spindle in human cells remain poorly understood due to a lack of direct mechanical measurements in mammalian systems. We use magnetic tweezers to measure the force on human mitotic spindles. Combining the spindle’s measured resistance to rotation, the speed it rotates after laser ablating astral microtubules, and estimates of the number of ablated microtubules reveals that each microtubule contacting the cell cortex is subject to ∼1 pN of pulling force, suggesting that each is pulled on by an individual dynein motor. We find that the concentration of dynein at the cell cortex and extent of dynein clustering are key determinants of the spindle’s resistance to rotation, with little contribution from cytoplasmic viscosity, which we explain using a biophysically based mathematical model. This work reveals how pulling forces on astral microtubules determine the mechanics of spindle orientation and demonstrates the central role of cortical dynein clustering.

The rapid progress in machine learning in recent years has been based on a highly productive connection to gradient-based optimization. Further progress hinges in part on a shift in focus from pattern recognition to decision-making and multi-agent problems. In these broader settings, new mathematical challenges emerge that involve equilibria and game theory instead of optima. Gradient-based methods remain essential -- given the high dimensionality and large scale of machine-learning problems -- but simple gradient descent is no longer the point of departure for algorithm design. We provide a gentle introduction to a broader framework for gradient-based algorithms in machine learning, beginning with saddle points and monotone games, and proceeding to general variational inequalities. While we provide convergence proofs for several of the algorithms that we present, our main focus is that of providing motivation and intuition.

Surface tension at cavity walls can play havoc with the mechanical properties of perforated soft solids when the cavities are filled with a fluid. This study is an investigation of the macroscopic elastic properties of elastomers embedding spherical cavities filled with a pressurized liquid in the presence of surface tension, starting with the linearization of the fully nonlinear model and ending with the enhancement properties of the linearized model when many such liquid filled cavities are present.

Incomplete labels are common in multi-task learning for biomedical applications due to several practical difficulties, e.g., expensive annotation efforts by experts, limit of data collection, different sources of data. A naive approach to enable joint learning for partially labeled data is adding self-supervised learning for tasks without ground truths by augmenting an input image and forcing the multi-task model to return the same outputs for both the input and augmented images. However, the partially labeled setting can result in imbalanced learning of tasks since not all tasks are trainable with ground truth supervisions for each data sample. In this work, we propose a multi-task curriculum learning method tailored for partially labeled data. For balanced learning of tasks, our multitask curriculum prioritizes less performing tasks during training by setting different supervised learning frequencies for each task. We demonstrate that our method outperforms standard approaches on one biomedical and two natural image datasets. Furthermore, our learning method with partially labeled data performs better than the standard multi-task learning methods with fully labeled data for the same number of annotations.

Blastomere instance segmentation is important for analyzing embryos’ abnormality. To measure the accurate shapes and sizes of blastomeres, their amodal segmentation is necessary. Amodal instance segmentation aims to recover an object’s complete silhouette even when the object is not fully visible. For each detected object, previous methods directly regress the target mask from input features. However, images of an object under different amounts of occlusion should have the same amodal mask output, making it harder to train the regression model. To alleviate the problem, we propose to classify input features into intermediate shape codes and recover complete object shapes. First, we pre-train the Vector Quantized Variational Autoencoder (VQ-VAE) model to learn these discrete shape codes from ground truth amodal masks. Then, we incorporate the VQ-VAE model into the amodal instance segmentation pipeline with an additional refinement module. We also detect an occlusion map to integrate occlusion information with a backbone feature. As such, our network faithfully detects bounding boxes of amodal objects. On an internal embryo cell image benchmark, the proposed method outperforms previous state-of-the-art methods. To show generalizability, we show segmentation results on the public KINS natural image benchmark. Our method would enable accurate measurement of blastomeres in In Vitro Fertilization (IVF) clinics, potentially increasing the IVF success rate.

Function approximation has been an indispensable component in modern reinforcement learning algorithms designed to tackle problems with large state spaces in high dimensions. This paper reviews recent results on error analysis for these reinforcement learning algorithms in linear or nonlinear approximation settings, emphasizing approximation error and estimation error/sample complexity. We discuss various properties related to approximation error and present concrete conditions on transition probability and reward function under which these properties hold true. Sample complexity analysis in reinforcement learning is more complicated than in supervised learning, primarily due to the distribution mismatch phenomenon. With assumptions on the linear structure of the problem, numerous algorithms in the literature achieve polynomial sample complexity with respect to the number of features, episode length, and accuracy, although the minimax rate has not been achieved yet. These results rely on the $L^∞$ and UCB estimation of estimation error, which can handle the distribution mismatch phenomenon. The problem and analysis become substantially more challenging in the setting of nonlinear function approximation, as both $L^∞$ and UCB estimation are inadequate for bounding the error with a favorable rate in high dimensions. We discuss additional assumption necessary to address the distribution mismatch and derive meaningful results for nonlinear RL problems.

Developing robust constitutive models is a fundamental and longstanding problem for accelerating the simulation of complicated physics. Machine learning provides promising tools to construct constitutive models based on various calibration data. In this work, we propose a neural operator to develop nonlocal constitutive models for tensorial quantities through a vector-cloud neural network with equivariance (VCNN-e). The VCNN-e respects all the invariance properties desired by constitutive models, faithfully reflects the region of influence in physics, and is applicable to different spatial resolutions. By design, the model guarantees that the predicted tensor is invariant to the frame translation and ordering (permutation) of the neighboring points. Furthermore, it is equivariant to the frame rotation, i.e., the output tensor co-rotates with the coordinate frame. We evaluate the VCNN-e by using it to emulate the Reynolds stress transport model for turbulent flows, which directly computes the Reynolds stress tensor to close the Reynolds-averaged Navier--Stokes (RANS) equations. The evaluation is performed in two situations: (1) emulating the Reynolds stress model through synthetic data generated from the Reynolds stress transport equations with closure models, and (2) predicting the Reynolds stress by learning from data generated from direct numerical simulations. Such a priori evaluations of the proposed network pave the way for developing and calibrating robust and nonlocal, non-equilibrium closure models for the RANS equations.

Electron-electron (e-e) and electron-phonon (e-ph) interactions are challenging to describe in correlated materials, where their joint effects govern unconventional transport, phase transitions, and superconductivity. Here we combine first-principles e-ph calculations with dynamical mean field theory (DMFT) as a step toward a unified description of e-e and e-ph interactions in correlated materials. We compute the e-ph self-energy using the DMFT electron Green's function, and combine it with the e-e self-energy from DMFT to obtain a Green's function including both interactions. This approach captures the renormalization of quasiparticle dispersion and spectral weight on equal footing. Using our method, we study the e-ph and e-e contributions to the resistivity and spectral functions in the correlated metal Sr