Nonlinear Phenomena in Complex Systems 2025 Conference

• Jacob Bedrossian, University of California Los Angeles
• Shibin Dai, University of Alabama
• Lou Kondic, New Jersey Institute of Technology
• Yifei Lou, University of North Carolina at Chapel Hill
• Stanley Osher, University of California at Los Angeles
• Hayden Schaeffer, University of California at Los Angeles
• Martin Short, Georgia Institute of Technology
• Dejan Slepcev, Carnegie Mellon University

Speaker: Anne Andrews

Institution: University of California, Los Angeles

Title: High-throughput ssDNA secondary structure classification to discover low-probability aptamers for biosensing applications

Abstract: Aptamers are rare single-stranded DNA or RNA molecules selected from billions of randomized sequences based on their ability to bind specific targets. We developed a high-throughput workflow (GMfold) that calculates thousands of oligonucleotide secondary structures in real-time within an interactive setting, when combined with selections that produce candidate pools too large for experimental testing. Each secondary structure is determined by identifying the lowest-energy bipartite subgraph matching the DNA graph to itself—modern machine learning algorithms cluster sequences according to similarities in secondary structural motifs. We conducted two selections for the small-molecule neurotransmitter and hormone norepinephrine. Using GMfold, we identified new norepinephrine aptamers, including some with exceptional affinity and selectivity from low-sequence-count populations that would otherwise be ignored. GMfold enables exploration of experimentally challenging sequence spaces and guides searches for aptamers designed for specific applications, including wearable sensors for personalized stress management.

Speaker: Jacob Bedrossian

Institution: University of California, Los Angeles

Title: 

Abstract:

Speaker: Mark Bowen

Institution: Waseda University

Title: Cauchy-Dirichlet Problems for the Porous Medium Equation

Abstract: We consider the porous medium equation subject to zero-Dirichlet conditions on a variety of two-dimensional domains. We employ a formal analysis of the intermediate-asymptotic solutions of these Cauchy-Dirichlet problems supported by computational results. Self-similar solutions play an important role, alongside the identification of suitable conserved quantities. We also describe some remaining open problems.

Speaker: Dino Di Carlo

Institution: University of California, Los Angeles

Title: Leveraging Nonlinear Microfluidics for Biological Applications

Abstract: Nonlinear fluid dynamics underpins a surprising range of phenomena in microfluidic systems, from the inertial lift forces that focus particles into precise streamlines, to the instabilities that govern droplet formation and breakup. In this talk, I will highlight how our group has harnessed these nonlinear effects to advance biological applications, with a particular emphasis on inertial microfluidics. By exploiting finite Reynolds number flows in microchannels, we create predictable particle trajectories and equilibrium positions that enable high-throughput, label-free manipulation of cells and biomolecules. I will also reflect on collaborative work with Andrea Bertozzi that tackled fundamental problems in interfacial fluid mechanics—examining the surface-tension-driven assembly of particle-associated droplets (“dropicles”) and the phase separation dynamics within temperature induced aqueous two-phase systems. These studies revealed how capillarity, particle geometry, and compositional instabilities interact to produce complex morphologies, including the spontaneous formation of micro- and nanoparticle structures. Together, these efforts demonstrate how nonlinear microfluidic phenomena can be transformed from fundamental curiosities into powerful tools for life science research, diagnostics, and materials synthesis.

Speaker: Selim Esedoglu

Institution: University of Michigan

Title: Vectorial median filters and curvature motion of networks

Abstract: The median filter is a standard tool in image processing (a typical application is removing salt and pepper noise from digital pictures). It is also well known as a monotone discretization of the level set formulation of two-phase motion by mean curvature.

In fact, median filters turn out to be closely connected, in a precise sense, to another class of algorithms for curvature motion: threshold dynamics. This precise connection between two disparate families of algorithms — level set methods and threshold dynamics — allows porting over to one (the level set) world what we have learned in recent years in the context of the other (threshold dynamics).

In particular, we give a variational (minimizing movements) formulation of the median filter (and thus exhibit a Lyapunov function, implying unconditional energy stability). Furthermore, we discuss extending median filters to multiphase, weighted motion by mean curvature of networks. The resulting level set methods are new: they are vectorial median filters. They avoid the issues that plagued previous attempts in the level set literature for this challenging evolution, and can accommodate a much wider range of surface tensions.

Speaker: Yanghong Huang

Institution: University of Manchester

Title: Finite Difference Methods for fractional Laplacian of radial functions

Abstract: Numerical evaluation of nonlocal operators like the fractional Laplacian is more computationally intensive because of the dependence on the underlying function on the whole space. On the other hand, most solutions to the fractional counterparts of classical semi-linear PDEs, usually obtained from variational methods, are radial. In this talk, fractional Laplacian of radial functions will be considered with a quadrature kernel represented by a Gauss hypergeometric function of the radial variable.  The singular part of the kernel is isolated and then treated with effective methods well studied in the one-dimensional context, while the regular part can be evaluated with by classical quadrature. The method can be extended to general functions that can be expanded using spherical harmonics.

Speaker: Hangjie Ji

Institution: North Carolina State University

Title: Mean-field control of thin film droplet dynamics

Abstract: Interfacial instabilities in volatile liquid thin films on a hydrophobic substrate can lead to complex droplet dynamics, such as droplet merging, splitting, and transport. MControlling these fundamental droplet behaviors is essential for digital microfluidics in biomedical and electric applications. In this talk, I will present our recent work on mean-field control of thin film droplets. We design an optimal control problem by formulating droplet dynamics as gradient flows of free energies in modified optimal transport metrics with nonlinear mobilities. As an example, we consider a thin volatile liquid film laden with an active suspension, where control is achieved through its activity field. Numerical examples, including droplet transport, bead-up/spreading, and merging/splitting, demonstrate the effectiveness of the proposed control mechanism.  

Speaker: Alice Koniges

Institution: University of Hawaii

Title: Arbitrary Lagrangian-Eulerian Driven Adaptive Mesh Refinement for Modeling 3D Non-Linear Phenomena on HPC Platforms

Abstract:This presentation describes the development and application of the PISALE code, a HPC optimized, 3D Arbitrary Lagrangian-Eulerian (ALE) code with Adaptive Mesh Refinement (AMR) capabilities. The PISALE code included development in collaboration with Andrea, and has benefited from her expert guidance and contributions from her students. Numerical and mathematical challenges of combining ALE with AMR in a scalable fashion are discussed. Initially focused on modeling National Ignition Facility (NIF) targets, the PISALE code has evolved to tackle a wide range of complex, non-linear phenomena across various application areas. Notably, the code has been applied to simulate the effects of raindrops on hypersonic vehicles for Department of Defense (DoD) projects, and x-ray free electron laser experiments at SLAC and EUV lithography designs for chip manufacturing. New efforts are exploring applications in geothermal energy, as part of the Department of Energy’s (DOE) Earthshot program. We also describe student programs and workshops in Hawaiʻi, and opportunities for collaboration and give examples of other broad-based collaborations with Andrea.

Speaker: Thomas Laurent

Institution: Loyola Marymount University

Title: Feature Collapse

Abstract: I will discuss some theoretical results that shed light on how good local features can be learned in the early layers of a neural network. Our analysis, in particular, highlights the importance of normalization layers. If time permits, I will also present some ongoing work on material discovery using generative AI.

Speaker: Yifei Lou

Institution: University of North Carolina at Chapel Hill

Title: Nearly Blind Hyperspectral Unmixing via Graph Regularizations and Active Learning

Abstract: Hyperspectral unmixing (HSU) is an effective tool to ascertain the material composition of each pixel in a hyperspectral image with typically hundreds of spectral channels. In this talk, I will present two approaches for identifying the pure spectra of individual materials (i.e., endmembers) and their proportions (i.e., abundances) at each pixel. The first approach is built on graph total variation (gTV) regularization, which involves the alternating direction method of multipliers (ADMM) and the Merriman-Bence-Osher (MBO) scheme to find a model solution. The second approach relies on active learning, strategically selecting training pixels, which leads to significant improvement with minimal supervision. Specifically, the experiments demonstrate that active learning can improve the state-of-the-art blind unmixing approaches by 50% or more using only 0.4% of training pixels.

Speaker: Stanley Osher

Institution: University of California, Los Angeles

Title: A Characteristic-based Deep Learning Framework for Hamilton-Jacobi Equations with Application to Opitmal Transport

Abstract:

Speaker: Nancy Rodriguez

Institution: University of Colorado, Boulder

Title: Numerical Methods for Kinetic Social Models with Singular Transition Rates

Abstract: Kinetic equations are increasingly used to model social systems, often featuring Dirac delta transition rates to capture abrupt behavioral shifts. In the first part of this talk, I’ll present a mass-preserving collocation scheme for solving equations with transitions of the form T(x,y,u)=δϕ(x,y)−uT(x, y, u) = \delta_{\phi(x, y) – u}, designed for systems with multiple interacting subsystems. We assess its accuracy and efficiency, and compare it to Gillespie, tau-leaping, and hybrid agent-based methods.  In the second part, I introduce a kinetic framework connecting individual behavior to macro-level crime patterns using data from Seattle and Tacoma (2018–2023). This multiscale model shows how violence evolves under different enforcement strategies, highlighting shifts not only in crime rates but also in the spatial and severity patterns of violence.

Speaker: Carola-Bibiane Schönlieb

Institution: University of Cambridge

Title: Some topics in structure preserving deep learning

Abstract: I will discuss some of our recent works on structure preserving deep learning for the design of neural networks with specific properties – such as non-expansiveness or mass conservation – and their application to imaging and to the solution of partial differential equations.

Speaker: Li Wang

Institution: University of Minnesota Twin Cities

Title: Measure theoretic approaches for uncertainty propagation 

Abstract: Uncertainty is ubiquitous: both data and physical models inherently contain uncertainty. Therefore, it is crucial to identify the sources of uncertainty and control its propagation over time. In this talk, I will introduce two approaches to address this uncertainty propagation problem—one for the inverse problem and one for the forward problem. The main idea is to work directly with probability measures, treating the underlying PDE as a pushforward map. In the inverse setting, we will explore various variational formulations, focusing on the characterization of minimizers and their stability. In the forward setting, we aim to propose a new approach to tackle high-dimensional uncertainties.

Speaker: Thomas Ward

Institution: University of Virginia, Charlottesville

Title: Singular perturbation expansion of the equations governing room temperature liquid-bridge evaporation 

Abstract: Here we examine the isothermal-evaporation of an axisymmetric liquid bridge confined between parallel-planar substrates using both theory and experiments. We perform a singular perturbation analysis of the equations that govern its motion.  The expanded equations of motion produce both an inner and outer problem.  Here, we discuss the inner and outer problems and show that the interface mechanics for either one can be described using a one-dimensional unsteady PDE with time varying coefficients.  Experiments were performed using water to estimate transient liquid-bridge volume and contact angles via image analysis at fixed time intervals. The results from analyzing experiments were compared with the results from profiles that were computed using the one-dimensional equation model. 

Speaker: Jack Xin

Institution: University of California, Irvine

Title: Stochastic Interacting Particle Methods and Generative Learning of Nonlinear PDEs 

Abstract: Time dependent nonlinear partial differential equations (PDEs) are challenging to compute by mesh based methods especially when their solutions develop large gradients or concentrations at unknown locations.  We discuss stochastic interacting particle (SIP) methods for advection-diffusion-reaction PDEs based on probabilistic representations of solutions,  and show their self-adaptivity and efficiency in several space dimensions.  Using SIP solutions as training data, we compare generative models (such as optimal transport, diffusion, flow-matching and one-step diffusion) in learning, interpolating and predicting solutions as physical parameters vary.

Speaker: Shibin Dai

Institution: The University of Alabama

Title: Limiting behavior of the De Gennes-Cahn-Hilliard Energy

Abstract: The degenerate de Gennes-Cahn-Hilliard (dGCH) equation is a model for phase separation which may more closely approximate surface diffusion than others in the limit when the thickness of the transition layer approaches zero. We find that its Gamma–limit is a constant multiple of the interface area, where the constant is determined by the de Gennes coefficient together with the double well potential. In contrast, the transition layer profile is solely determined by the double well potential. We will also talk about the force convergence and the classification of the minimizers of the dGCH energy.