Spring 2023

Abstract: Many spectral responses in materials science, physics, and chemistry experiments can be characterized as resulting from the superposition of a number of more basic individual spectra. In this context, unmixing is defined as the problem of determining the individual spectra, given measurements of multiple spectra that are spatially resolved across samples, as well as the determination of the corresponding abundance maps indicating the local weighting of each individual spectrum. Matrix factorization is a popular linear unmixing technique that considers that the mixture model between the individual spectra and the spatial maps is linear.  Given an input matrix A, Non-negative Matrix Factorization (NMF) is about determining two non-negative matrices called factors, corresponding to the rows and columns of A respectively,  such that the product of them closely approximates A. This is called “low rank approximation” as the rank of the factors are generally much smaller than the input matrix A.  Also, it is known as dimensionality reduction in Machine Learning Community and inverse problems in scientific community. In recent times, there is a huge interest for non-negative factorizations for data with more than two modes, which can be represented via a generalization of matrices known as tensors. There are wide applications of non-negative matrix and tensor factorization (NTF) in the scientific community such as spectral unmixing, compressing scientific data, scientific visualization, feature engineering for deep learning etc., as the factors are scientifically interpretable and an important cornerstone for explainable AI and representation learning. In this talk, we will look at various constraints on factors along with non-negativity such as symmetric, spatial smoothness, sparsity etc., and challenges in the realization of scalable algorithms. In the latter part of the talk, we will focus on determining the parameters of a forward problem for scientific experiments. We report on the efficacies of a variety of ML models, including convolutional neural networks, auto-encoders, random forests and combinations thereof, in addition to techniques such as transfer learning in predicting these structural parameters.  

Host person: Paul Atzberger

Abstract: While blood clot formation has been relatively well studied, little is known about the mechanisms underlying the subsequent structural and mechanical clot remodeling called contraction or retraction. Impairment of the clot contraction process is associated with both life-threatening bleeding and thrombotic conditions, such as ischemic stroke, venous thromboembolism, and others. Recently, blood clot contraction was observed to be hindered in patients with COVID-19. First, a multi-phase model of blood clot embolization will be described and discussed. Then, a novel three-dimensional multi-scale model will be described and applied to quantify biomechanical mechanisms of the kinetics of clot deformation and contraction driven by platelet filopodia-fibrin fiber pulling interactions. These results provide novel biological insights since filopodia have been thought of previously as performing mostly a sensory function. The biomechanical mechanisms and modeling approach to be described can potentially apply to studying other blood clotting processes as well as to other systems in which cells are embedded in a filamentous network and exert forces on the extracellular matrix modulated by the substrate stiffness. Samuel Britton, Oleg Kim, Francesco Pancaldi, Zhiliang Xu, Rustem I. Litvinov, John W.Weisel, Mark Alber [2019], Contribution of nascent cohesive fiber-fiber interactions to the non-linear elasticity of fibrin networks under tensile load, Acta Biomaterialia 94, 514-523. Shixin Xu, Zhiliang Xu, Oleg Kim, Rustem I. Litvinov, John W. Weisel and Mark Alber [2017], Model Predictions of Deformation, Embolization, and Permeability of Partially Obstructive Blood Clots under Variable Shear Flow, Journal of the Royal Society Interface 14: 20170441. Oleg V. Kim, Rustem I. Litvinov, Mark S. Alber and John W. Weisel [2017], Quantitative Structural Mechanobiology of Platelet-Driven Blood Clot Contraction, Nature Communications 8: 1274.

Host person: Paul Atzberger

Abstract: The more AI agents are deployed in scenarios with possibly unexpected situations, the more they need to be flexible, adaptive, and creative in achieving the goal we have given them. Thus, a certain level of freedom to choose the best path to the goal is inherent in making AI robust and flexible enough. At the same time, however, the pervasive deployment of AI in our life, whether AI is autonomous or collaborating with humans, raises several ethical challenges. AI agents should be aware and follow appropriate ethical principles and should thus exhibit properties such as fairness or other virtues. These ethical principles should define the boundaries of AI's freedom and creativity. However, it is still a challenge to understand how to specify and reason with ethical boundaries in AI agents and how to combine them appropriately with subjective preferences and goal specifications. Some initial attempts employ either a data-driven example-based approach for both, or a symbolic rule-based approach for both. We envision a modular approach where any AI technique can be used for any of these essential ingredients in decision making or decision support systems, paired with a contextual approach to define their combination and relative weight. In a world where neither humans nor AI systems work in isolation, but are tightly interconnected, e.g., the Internet of Things, we also envision a compositional approach to building ethically bounded AI, where the ethical properties of each component can be fruitfully exploited to derive those of the overall system. In this paper we define and motivate the notion of ethically-bounded AI, we describe two concrete examples, and we outline some outstanding challenges.

Host person: Paul Atzberger

Abstract: We will discuss several recent results for aggregation-diffusion equations related to partial concentration of the density of particles. Nonlinear diffusions with homogeneous kernels will be reviewed quickly in the case of degenerate diffusions to have a full picture of the problem. Most of the talk will be devoted to discuss the less explored case of fast diffusion with homogeneous kernels with positive powers. We will first concentrate in the case of stationary solutions by looking at minimisers of the associated free energy showing that the minimiser must consist of a regular smooth solution with singularity at the origin plus possibly a partial concentration of the mass at the origin. We will give necessary conditions for this partial mass concentration to and not to happen. We will then look at the related evolution problem and show that for a given confinement potential this concentration happens in infinite time under certain conditions. We will briefly discuss the latest developments when we introduce the aggregation term. This talk is based on a series of works in collaboration with M. Delgadino, J. Dolbeault, A. Fernández, R. Frank, D. Gómez-Castro, F. Hoffmann, M. Lewin, and J. L. Vázquez.

Host person: Katy Craig

Abstract: Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, we prove the existence of an alternative ODE limit, a stochastic differential equation, or neither of these. For each case, we also derive the limit of the backpropagation dynamics and address its adaptiveness issue. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.

When the gradient descent method is applied to the training of ResNets, we prove that it converges linearly to a global minimum if the network is sufficiently deep and the initialization is sufficiently small. In addition, the global minimum found by the gradient descent method has finite quadratic variation without using any regularization in the training. This confirms existing empirical results that the gradient descent method enjoys an implicit regularization property and is capable of generalizing to unseen data.

This is based on joint work with Rama Cont (Oxford), Alain Rossier (Oxford), and Alain-Sam Cohen (InstaDeep).

Host person: Ruimeng Hu

Abstract: In this talk, we will describe a class of algorithms for identifying unknown parameters of nonlinear PDEs. In the absence of observational errors, the convergence of these algorithms can be rigorously established under the assumption that sufficiently many scales of the solution are observed and that certain non-degeneracy conditions hold, which ensures identifiability of the parameters. This approach to parameter estimation is robust and can be applied not only to recover damping coefficients, but also external driving forces that are unknown apriori. Moreover, it is applicable to a large class of nonlinear equations, including many of those that arise in hydrodynamics, such as the 2D Navier-Stokes equations of incompressible flow, the 2D system for Rayleigh-Benard convection, the 3D primitive equations, or even dispersive-type models such as the 1D Korteweg-de Vries equation or 1D cubic nonlinear Schrödinger equation. We describe the derivation of these algorithms, address their convergence, and showcase the results of several computational experiments.

Host person: Quyuan Lin

Abstract: Kinetic equations describe the nonequilibrium dynamics of a complex particle system using a probability density function. Despite of their important role in multiscale modeling to bridge microscopic and macroscopic scales, numerically solving kinetic equations is computationally demanding as they lie in the six-dimensional phase space. Dynamical low-rank method is a dimension-reduction technique that has recently been applied to kinetic theory, yet most of the endeavor is devoted to linear or collisionless problems. In this talk, we introduce efficient dynamical low-rank methods for Boltzmann type collisional kinetic equations, building on certain prior knowledge about the low-rank structure of the solution.

Host person: Xu Yang

Abstract: We present several quantitative results that generalize known nonlocal rigidity relations for vector fields representing deformations of elastic media. We show that the distance in Lebesgue norm of a deformation from a rigid motion is bounded by a multiple of a strain energy associated to the deformation. This nonconvex energy is a nonlocal constitutive relation that represents the extent to which the deformation stretches and shrinks distances. This inequality can be thought of as a nonlinear fractional Poincare-Korn inequality. We linearize this inequality to obtain a fractional Poincare-Korn inequality for Lipschitz domains with an explicit universal bounding constant. This inequality allows for the establishment of a full, fractional Korn-type inequality. We apply these inequalities to obtain quantitative statements for solutions to variational problems arising in peridynamics and dislocation models.

Host person:  Davit Harutyunyan

Abstract: The principle of developing structure-conforming numerical algorithms widely exists in scientific computing, such as conforming finite element methods, finite element exterior calculus, and energy/mass conservation schemes, which all have well-known advantages. In this work, following this principle, we propose a structure-conforming operator learning method for solving a class of geometric inverse problems. The constructed architecture is conforming to the structure of the underlying inverse operator and is also closely related to the Transformer, one state-of-art architecture for many scientific computing tasks. Numerical examples demonstrate that the proposed architecture outperforms many existing operator learning methods in the literature.

Host person:  Sui Tang

Abstract: Sampling and optimization are fundamental tasks in data science. While the literature on optimization for data science has developed widely in the past decade, with fine convergence rates for some methods, the literature on sampling remained mainly asymptotic until recently. We study the proximal sampler introduced recently by Lee, Shen, and Tian. This sampling algorithm can be seen as a proximal point algorithm for the purpose of sampling. We will discuss the connection with the standard proximal point algorithm in optimization, and how the proximal sampler can be seen as an optimization algorithm over a space of probability measures. Then, we will review an existing convergence guarantee relying on strong convexity, and show new convergence guarantees under weaker assumptions such as convexity and isoperimetry, which allow for nonconvexity. With these results, we obtain new state-of-the-art sampling guarantees for several classes of target distributions.

Host person: Katy Craig

Abstract: Compensated compactness, originally due to Murat and Tartar, states that the dot product of two weakly convergent in L^2 sequences of vector fields converges to the dot product of their weak limits, provided one of the sequences is curl-free, and the other is divergence-free. I will show how to generalize this result to a much larger class of differential operators and then use it to prove a homogenization theorem for a large class of elliptic systems of PDEs.

Host person: Davit Harutyunyan

Abstract: In this talk, we present a novel method for joint sparse recovery called orthogonally weighted $\ell_{2,1}$ regularization, which takes into account the rank of the solution matrix. Our method differs from existing regularization-based approaches by exploiting the full rank of the row-sparse solution matrix. We prove the rank-awareness of our method, establish the existence of solutions to the optimization problem, and provide an efficient algorithm for solving it. We analyze the convergence of our algorithm and present numerical experiments to demonstrate the effectiveness of our method.

Host person: Sui Tang

Abstract: Transportation of measure underlies many contemporary methods in machine learning and statistics. Sampling, which is a fundamental building block in computational science, can be done efficiently given an appropriate measure-transport map. We ask: what is the effect of using approximate maps in such algorithms? 

We propose a new framework to analyze the approximation power of measure transport. This framework applies to existing algorithms, but also suggests new ones. At the core of our analysis is the theory of optimal transport regularity, approximation theory, and an emerging class of inequalities, previously studied in the context of uncertainty quantification (UQ).

Host person: Katy Craig

Abstract: Bose-Einstein Condensates (BECs) can be described by a nonlinear partial differential equation known as the Nonlinear Schroedinger (NLS) equation. Understanding the behavior  of BECs will enable the design of quantum computers that can vastly outperform current  digital computers. A mathematical and computational analysis of the NLS equation will  be presented using Newton's method for nonlinear equations (and systems thereof) together  with penalty methods. These deflation techniques allow the possibility of finding inconspicuous  solutions to nonlinear equations by using a single initial guess. We have developed  the Deflated Continuation Method (DCM) to trace disconnected branches of solutions by  factoring out previously computed ones. This approach led to the discovery of previously unknown  solutions to the 2D and 3D NLS equation, and their bifurcations, thus elucidating the details of the  rich configuration space of this infinite-dimensional dynamical system. Motivated by innovative  experiments on BECs performed by Prof. David Hall's group (Physics & Astronomy, Amherst College),  we are embarking on an ambitious project to explore multi-component 3D NLS systems. 

Host person: Jimmie Adrizola

Abstract: In this talk we show how to learn the underlying geometry of data using heat diffusion, which can represent each point by a distribution of transition probabilities that are proportional to manifold distance. This distribution renders point cloud data into a statistical manifold. Then we show how to derive low dimensional visualizations and embeddings of such data using divergences between such datapoint transition probabilities. I will then cover recent work which learns a continuous model of such a statistical manifold using a neural network which is then used to learn the infinitesimal analog of such a divergence: a Fisher information metric. Next we show how to compare many such distributions using multiscale diffusion distances for optimal transport. FInally we move from static to dynamic optimal transport using neural ODEs in order to learn dynamic trajectories from static snapshot data—a key problem in inference from single cell data. Throughout the talk, we present examples of such techniques being applied to massively high throughput and high dimensional datasets from biology and medicine. 

Host person: Paul Atzberger