Winter 2025
Time: Fridays, 2pm-3pm, Pacific time
Location: Hybrid - South Hall 4607 and in Zoom (link provided upon request)
Please contact Mingsong Yan (mingsongyan@ucsb.edu), Xin Su (xsu2@ucsb.edu ), Ruimeng Hu (rhu@ucsb.edu), or Sui Tang (suitang@ucsb.edu) to reserve a slot.
Upcoming Seminar Schedule:
(Click the event below to see the title and abstract)
Title: Data-Driven Methods for PDE Solutions and Model Discovery
Abstract: In this talk, I will present our recent advancements in solving partial differential equations (PDEs) and discovering models in complex systems. The first part focuses on our approaches for approximating PDE solutions with rigorous guarantees, addressing challenges such as rough and noisy forcing terms. This includes the development of novel loss functions, such as the Negative Sobolev Norm for Gaussian Process (GP) methods and physics-informed neural networks (PINNs), along with a sparse GP approach to enhance computational efficiency. The second part highlights our work on uncovering unknown parameters and dynamics in scientific systems, utilizing hypergraph-based models and GP frameworks.
Host: Ruimeng Hu
Title: Gradient flow structure for some nonlocal diffusion equations
Abstract: Nonlocal diffusion equations describe the continuum limit of particle systems where the paths have jumps, and have applications in the analysis of Markov processes as well as in stochastic optimization and sampling. Here, I study a class of nonlocal diffusions from the optimal transport point of view. By analogy with the result of Jordan, Kinderlehrer, and Otto which showed that the Fokker-Planck equation is the gradient flow of the relative entropy with respect to the 2-Wasserstein metric, our results likewise show that a "nonlocal Fokker-Planck equation" can be viewed as the gradient flow of the relative entropy with respect to a certain "nonlocal Wasserstein metric". This variational structure allows us to derive existence, uniqueness, and stability of solutions, consistency of a finite volume scheme, and indicates the right nonlocal analog of the log-Sobolev inequality.
Host: Katy Craig
Title: Modelling rare rupture and adhesion of nanoscale thin films with fluctuating hydrodynamics
Abstract: Nanoscale thin liquid films are crucial in applications like solar cell manufacturing, insulation layer coating, and mediating cell-cell interactions. At this scale, random particle movements (Brownian motion) become significant, requiring fluctuating hydrodynamics to describe fluids. To simplify modelling, the high aspect ratio of thin films is exploited, and the lubrication approximation is applied to derive stochastic thin film equations (STFE), describing film height evolution. In this talk, we demonstrate STFE usage in two scenarios: i) rare rupture of thin films on solid substrates, ii) adhesion and phase separation of cell membrane adhesion patches. In the first scenario, we solve STFE numerically with free surface boundary conditions and Van der Waals forces, observing film ruptures in the linearly stable regime induced by thermal fluctuations. The STFE can be rearranged into a gradient flow form, allowing for the application of rare event theory to predict average rupture time theoretically. Molecular dynamics simulations show good agreement with numerical and theoretical results. In the second scenario, the cell membranes are modelled as elastic bending sheets, with STFE describing the fluid flow between them. Numerical simulations reveal that for membrane adhesion (protein bonds to form), thermal fluctuations must bring membranes close enough for protein bonds to form, aligning with rare event theory. Coarsening is observed during later stage of adhesion patch phase separation, though similar to the Ostwald ripening, revealing a different power law.
Host: Paul Atzberger
Title: Pattern-forming instabilities of planar interfaces in reaction-diffusion-advection equations
Abstract: Motivated by the appearance of desertification fronts between bare soil and vegetation in dryland ecosystems, we consider the dynamics of planar interfaces between stable homogeneous rest states in 2-component reaction-diffusion-advection equations. On sloped terrain, one can find stable traveling interfaces, while on flat ground, one finds that long wavelength instabilities along the interface can lead to labyrinthine Turing-like patterns. To explore this behavior, using singular perturbation methods, we analyze instability criteria for planar interfaces in reaction diffusion systems, and we examine the effect of terrain slope on the stability of the interfaces.
Host: Paul Atzberger
Title: The Weak form is Stronger than you Think
Abstract: Recent advances in data-driven modeling approaches have proven highly successful in a wide range of fields in science and engineering. In particular, learning governing equations via mimizing an equation error criteria,offers a powerful and explainable scientific machine-learning framework. However, the first generation of these methods has proven poorly suited to noise-corrupted data. In this talk, I will present our weak form approach and briefly discuss how it addresses several ubiquitous challenges within conventional model development, discretization, parameter inference, and model refinement. In particular, I will describe our equation learning (WSINDy) and parameter estimation (WENDy) algorithms. Our approach has exhibited surprising performance, accuracy, and robustness properties. In many applications, the method is an order of magitude more accurate, robust to orders of magnitude more noise, and multiple orders of magnitude faster than conventional approaches. I will demonstrate these performance properties via applications to several benchmark problems in ordinary, partial, and stochastic differential equations, as well as coarse-graining and reduced-order modeling.
Host: Paul Atzberger
Title: Arbitrary Lagrangian--Eulerian finite element method for lipid membranes
Abstract: Biological membranes are unique two-dimensional materials in which lipids flow in-plane as a Newtonian fluid, while the entire membrane bends out-of-plane as an elastic sheet. Though the dynamical equations governing lipid membranes are known, they are analytically intractable: membrane dynamics are highly nonlinear and involve spatial derivatives on a surface which is itself arbitrarily curved and deforming over time. The challenges in analytically solving the membrane equations extend to their numerical solution as well: standard computational techniques from fluid and solid mechanics cannot model a two-dimensional material with arbitrarily large shape deformations, in-plane flows, and out-of-plane elasticity. We address this issue by developing an arbitrary Lagrangian–Eulerian (ALE) finite element method for lipid membranes. The membrane surface is endowed with a mesh whose in-plane motion can be specified independently of the material velocity; the out-of-plane mesh motion is required to be Lagrangian such that the mesh and material always overlap. A new in-plane mesh motion is implemented, in which the mesh evolves according to the dynamical equations governing a two-dimensional area-compressible viscoelastic fluid film. A Lagrange multiplier constrains the out-of-plane membrane and mesh velocities to be equal, such that the mesh and material always overlap. An associated numerical inf–sup instability ensues, and is removed by adapting established techniques in the finite element analysis of fluids. In our implementation, the aforementioned Lagrange multiplier is projected onto a discontinuous space of piecewise linear functions. The new mesh motion is compared to established Lagrangian and Eulerian formulations by investigating a preeminent numerical benchmark of biological significance: the pulling of a membrane tether from a flat patch, and its subsequent lateral translation.
Host: Paul Atzberger
Title: Tensor Iterative Methods for Large-Scale Linear Systems
Abstract: Solving linear systems is a crucial subroutine and challenge in the large-scale data setting. In this presentation, we introduce an iterative method for approximating the solution of large-scale multi-linear systems, represented in the form A*X=B under the tensor t-product. Unlike previously proposed randomized iterative strategies, such as the tensor randomized Kaczmarz method (row slice sketching) or the tensor Gauss-Seidel method (column slice sketching), which are natural extensions of their matrix counterparts, our approach delves into a distinct scenario utilizing frontal slice sketching. In particular, we explore a context where frontal slices, such as video frames, arrive sequentially over time, and access to only one frontal slice at any given moment is available. This talk will present our novel approach, shedding light on its applicability and potential benefits in approximating solutions to large-scale multi-linear systems.
Host: Paul Atzberger
Title: Deciphering Stochastic Gene Expression Dynamics from Static Imaging through Point Process Inference
Abstract: The molecules of life operate with behavior dominated by randomness and disorder. Yet, from these ingredients, robust cellular function emerges. Understanding this paradox requires both mechanistic models that capture molecular-scale stochasticity and statistical approaches to extract meaningful patterns from noisy, heterogeneous data. This talk presents a case study in bridging these gaps to infer gene expression dynamics from static spatial patterns of mRNA molecules in cells. The approach links spatial point processes for individual molecule locations with tractable solutions to (stochastic) partial differential equations. This framework combines the strengths of mechanistic modeling and machine learning, enabling new discoveries from challenging large-scale biological datasets. I will discuss recent advances and future directions, including the incorporation of additional biological complexities, the development of more sophisticated computational methods, and expanding the framework to address a range of other biological questions.
Host: Paul Atzberger
Title: Nodal set and observability estimate for Gevrey regular parabolic equations
Abstract: In this talk, I will discuss our recent work on the size of nodal set and an observability estimate for Gevrey regular parabolic equations. We provide an upper bound of the nodal set as a function of time, and the dependence agrees with a sharp upper bound when the coefficients are analytic. Secondly, we prove an observability inequality from measurable set for general Gevrey regular functions. Moreover, we provide applications to the sum of Laplace eigenfunctions on a Riemannian manifold that belongs to the Gevrey class, as well as the solution of a one-dimensional parabolic equation of arbitrary order with Gevrey coefficients.
Host: Ruimeng Hu
Title: Operator Learning Neural Scaling and Distributed Applications
Abstract: In this talk, we will explore mathematical and scientific machine learning, with a particular focus on operator learning—a framework for approximating mappings between function spaces that has broad applications in PDE-related problems. We will begin by discussing the mathematical foundations of operator approximation, which inform the design of neural network architectures and provide a basis for analyzing the performance of trained models on test samples. Specifically, we will introduce the neural scaling law, which characterizes error convergence in relation to network size and generalization error in relation to training dataset size. Building on these theoretical insights, we will present a distributed learning algorithm designed to address a key computational challenge: efficiently handling heterogeneous problems where input functions exhibit vastly different properties. Such multiscale problems typically demand significant computational resources to capture fine-scale details, but our distributed approach enables efficient training with improved accuracy.
Host: Sui Tang
Title: Gradient flow structure for some nonlocal diffusion equations
Abstract: Nonlocal diffusion equations describe the continuum limit of particle systems where the paths have jumps, and have applications in the analysis of Markov processes as well as in stochastic optimization and sampling. Here, I study a class of nonlocal diffusions from the optimal transport point of view. By analogy with the result of Jordan, Kinderlehrer, and Otto which showed that the Fokker-Planck equation is the gradient flow of the relative entropy with respect to the 2-Wasserstein metric, our results likewise show that a "nonlocal Fokker-Planck equation" can be viewed as the gradient flow of the relative entropy with respect to a certain "nonlocal Wasserstein metric". This variational structure allows us to derive existence, uniqueness, and stability of solutions, consistency of a finite volume scheme, and indicates the right nonlocal analog of the log-Sobolev inequality.
Host: Katy Craig
Title: Community Detection with the Bethe-Hessian
Abstract: The Bethe-Hessian matrix, introduced by Saade, Krzakala, and Zdeborová (2014), is a Hermitian matrix designed for applying spectral clustering algorithms to sparse networks. Rather than employing a non-symmetric and high-dimensional non-backtracking operator, a spectral method based on the Bethe-Hessian matrix is conjectured to also reach the Kesten-Stigum detection threshold in the sparse stochastic block model (SBM). We provide the first rigorous analysis of the Bethe-Hessian spectral method in the SBM under both the bounded expected degree and the growing degree regimes. Joint work with Ludovic Stephan.
Host: Sui Tang