Applied and Computational Mathematics Seminar

Sea ice, which covers a significant portion of the earth's surface, is an interestingly complicated material consisting of a mixture of solid ice and liquid brine phases which are coupled by thermodynamic considerations, Among other things, sea ice plays an important role in regulating macroscale heat transport between the ocean and the atmosphere. It also is a platform for microbial life, lots of it in fact, that uses the ice as a sort of shelter though eventually becoming part of the local food chain. A model will be presented that hypothesizes that, in turn, the resident microbial population might impact sea ice structure and, in particular, its transport properties including heat transport.

Phylogenetic networks are a generalization of phylogenetic trees allowing for the representation of speciation and reticulate evolutionary events such as hybridization or horizontal gene transfer. The inference of phylogenetic networks from biological sequence data is a challenging problem, with many theoretical and practical questions still unresolved. In this talk, I will give an overview of the state of the art in phylogenetic network inference. I will then discuss a novel divide-and-conquer approach for inferring level-1 networks under the network multispecies coalescent model. I will end by discussing some open problems and avenues for future research.
Parts of this talk are based on joint work with Elizabeth Allman, Hector Baños, and John Rhodes.

Dynamical systems can undergo qualitative, topological changes in their orbit structure called bifurcations when underlying parameters cross a threshold: the "shape" of their behavior alters fundamentally. The development of data-driven tools for modeling these changes holds special promise in the life sciences, from the design of gene regulatory networks to the prediction of catastrophic oscillations in neural circuits. In this talk, I describe an ongoing research program which tackles this challenge by focusing on the realistic case where governing equations are unknown and dynamical behavior must be predicted from prior knowledge given noisy, sparse data. Building on classical work in so-called model manifold theory, our approach learns a shared feature landscape where diverse systems coalesce within a unified embedding space, revealing their underlying qualitative structure. I first describe work which uses such learned universal embeddings of low-dimensional dynamical systems to classify circuits by their function. Next, I demonstrate how a simple autoencoder can learn an implicit notion of topological conjugacy which functions as a robust detector of Hopf bifurcations in single-cell RNA sequencing data from the pancreas. Finally, we generalize to the case of spatiotemporal dynamics, where I outline recent work on building reduced-order parametric models ofpartial differential equations with applications to spatial patterning in the ocellated lizard. We conclude with some future directions, notably extensions to high-dimensional systems and applications to synthetic biology, where engineered organisms and tissues could be designed for stable, predictable functions in dynamic environments.

Most physiological mechanisms exhibit high variability due to the fundamental stochasticity of biochemical reaction pathways. Quantifying the impact of stochastic effects is necessary for a deeper understanding of physiological processes and their regulation, and helps in the choice of an efficient approach for their computational modeling. This is especially true in the case of synaptic neurotransmitter release, which is caused by the fusion of the secretory vesicle membrane with the cell membrane in response to calcium ion binding. Although stochastic calcium channel gating is one of the primary source of this stochasticity, it can be implemented in a computationally inexpensive way in combination with deterministic simulation of the downstream calcium diffusion and binding reactions. Another fundamental reason for the high variability of synaptic response is that only a small number of calcium ions enter the synaptic terminal through a single channel during an action potential. This fact entails large fluctuations due to calcium diffusion and its binding to calcium buffers and vesicle release sensors, leading to a widely-held view that solving continuous deterministic reaction-diffusion equations does not provide high accuracy when modeling calcium-dependent cell processes. 

However, several comparative studies show a surprising close agreement between deterministic and trial-averaged stochastic simulations of calcium dynamics, as long as calcium channel gating is not calcium-dependent. This result deserves careful investigation. This talk will focus on further analysis and comparison of stochastic and mass-action modeling of vesicle release, showing that the discrepancy between deterministic and stochastic approaches remains small even when only as few as 40-50 ions enter per single channel-vesicle complex. The reason for the close agreement between stochastic and mass-action simulations is that the discrepancy between the two approaches is determined by the size of the correlation between the local calcium concentration and the state of the vesicle release sensor, rather than fluctuation amplitude. Whereas diffusion and buffering increases fluctuation size, the same processes appear to de-correlate fluctuations in calcium concentration from fluctuations in calcium sensor binding state. Finally, contrary to naïve intuition, the mass action / mean-field reaction-diffusion description allows an accurate estimate of the probability density of vesicle release latency (first-passage time), rather than providing information about trial-averaged quantities only. These results may help in the choice of appropriate and efficient tools for the modeling of this and other fundamental biochemical cell processes.

We study the problem of noisy data fitting and spectral measure recovery in the classes Stieltjes and Completely monotone functions. The analytical setting is least squares over the convex or conical hull of a space curve in IR^n. I will begin with the case of least squares over convex hulls where the convergence analysis is simple. Here, I will introduce our method of analyzing the support of the optimal spectral measure. I will then move to the case of conical hulls (non-negative least squares). In this context, convergence analysis is not well understood. We introduce a simple assumption on the problem which allows us to overcome the convergence problem. While our assumption does not initially hold in either our motivating examples, I will show they can be rescaled so that is does hold. This rescaling allows certain algorithms for convex minimization to be extended to minimization problems with non-negativity constraints. I will provide several numerical examples which show convergence of algorithms as well as highlight our spectral measure analysis at certain noise levels.

We construct a model of the feedback mechanisms that regulate the abundance of ribosomes in E.coli, a prototypical prokaryotic organism. The translation process contains an important feedback loop: ribosomes are made up of proteins, which need to be translated by ribosomes. The model accounts for the main feedback loops that control abundance of ribosomes in response to external conditions. It includes the concentrations of free ribosomes, ribosomal RNA (rRNA), and ribosomal proteins. We include direct negative feedback loops where ribosomal proteins, when in excess, slow down their own translation. The effect of the signaling molecule ppGpp is also included as a negative feedback mechanism, along with the effect of the abundance of building blocks for mRNA and rRNA synthesis. The model consists of a system of six differential equations parameterized by 23 parameters. An equlibrium analysis shows that for all values of parameters, the model system has either one equilibrium S, or two equilibria S and P in the biologically feasible region of parameters.

We are entering a new era of electrification, marked by the widespread adoption of lithium-ion batteries across critical sectors such as transportation, renewable energy, and grid infrastructure. Despite their advantages, lithium-ion batteries still grapple with notable performance and safety issues, casting doubt on their long-term role in future electrification efforts. This raises an urgent question: how can we improve batteries to meet the diverse demands of real-world applications? Systems theory, control strategies, and learning-based approaches have emerged as powerful tools in the search for effective solutions. We aim to make the batteries more efficient by creating more accurate models that are battery specific and adjustable. Specifically, we introduce an interpretable, physics-inspired, data-driven approach for discovering governing equations and estimating the state-of-charge (SOC) and voltage dynamics of Li-ion batteries. SOC estimation is a key challenge in battery management systems, particularly for high-demand applications like electric vehicles, where errors in low and high SOC regions can limit performance. The proposed method leverages sparse identification, using a physics-based library of electrochemical functions to uncover governing equations that accurately capture battery dynamics. This approach ensures interpretability and physical consistency, addressing common issues in purely data-driven models, such as overfitting and lack of generalizability.

Over the past twenty years, the field of plasmonics has been revolutionized with the isolation and utilization of two-dimensional materials, particularly graphene. Consequently there is significant interest in rapid, robust, and highly accurate computational schemes which can incorporate such materials. Standard volumetric approaches can be contemplated, but these require huge computational resources. Here we describe an algorithm which addresses this issue for nonlocal models of the electromagnetic response of graphene. Our methodology not only approximates the graphene layer with a surface current, but also reformulates the governing volumetric equations in terms of surface quantities using Dirichlet-Neumann Operators. We have recently shown how these surface equations can be numerically simulated in an efficient, stable, and accurate fashion using a High-Order Perturbation of Envelopes methodology. We extend these results to the nonlocal model mentioned above, and using an implementation of this algorithm, we study absorbance spectra of TM polarized plane-waves scattered by a periodic grid of graphene ribbons.

The application of neural networks to solving partial differential equations has experienced a tumultuous journey since the early 1990s, culminating in the development of Physics-Informed Neural Networks (PINNs) that have generated both excitement and skepticism within the scientific computing community. In this talk we will discuss why early approaches showed promise but failed to gain widespread adoption, and why PINNs themselves have faced significant criticism despite their practical appeal. A critical gap in the field has been the failure to recognize that PINNs operate under fundamentally different assumptions than conventional supervised learning paradigms -- unlike traditional machine learning tasks with abundant labeled data, PINNs must simultaneously approximate unknown functions without direct supervision while satisfying physical constraints. This creates fundamentally different algorithmic requirements where standard deep learning assumptions, from network initialization and architecture design to optimization strategies, must be reconsidered. We present an overview of recent methodological advances that address these challenges, including advanced optimization algorithms, improved loss balancing techniques, and specialized architectures designed for physical problems. We demonstrate that properly designed PINNs can successfully perform high-fidelity simulation of complex three-dimensional turbulent flows -- a notoriously challenging setting that has long resisted both classical and ML approaches. These results suggest that with proper understanding of their unique characteristics and careful algorithmic design, PINNs can indeed fulfill their promise as a transformative tool for scientific computing, with the tantalizing possibility that approaches free from resolution constraints may one day enable nearly exact solutions to fundamental problems like Navier-Stokes turbulence.

Among the many concepts in numerical analysis for differential equations, "stiffness" appears one of the least straightforward ones to define and characterize. This talk contrasts various forms of stiff limits and associated model problems, and establishes how these different shapes of stiffness manifest in the numerical solution of both ordinary and partial differential equations. Then it is presented which hierarchies of additional order conditions can be formulated that ensure that various classes of Runge-Kutta methods retain their desired convergence rates in the eye of different forms of stiffness.

Multiphysics models couple two or more physical processes together in a single simulation. These combinations may include systems of differential equations with different type (parabolic, hyperbolic, etc.), with different degrees of nonlinearity, and that evolve on disparate time scales. As a result, such simulations prove challenging for "monolithic" time integration methods that treat all processes using a single approach.

In this talk, I will discuss recent work on time integration methods that allow the flexibility to apply different techniques to distinct physical processes. While such techniques have existed for some time, including additive Runge-Kutta implicit-explicit (ImEx), multirate (a.k.a., multiple time stepping), and operator-splitting methods, there have been comparably few that combine these types of flexibility into a single family, while also supporting high orders of accuracy and temporal adaptivity. In this talk, I focus on newly developed implicit-explicit families of methods for multirate problems, along with novel techniques for time adaptivity in multirate infinitesimal time integration methods.

We introduce approximations of ab-initio molecular dynamics derived from quantum mechanics. Molecular dynamics simulations are often used to approximate canonical quantum correlation observables in complex nuclei-electron systems. We present shallow random feature neural networks and provide an analysis of their approximation properties. Furthermore, we describe an adaptive sampling strategy that ensures a near-optimal distribution of features, thus enabling controlled approximation of inter-atomic potentials for molecular dynamics simulations. Finally, we demonstrate that the resulting molecular dynamics accurately approximate correlation observables with quantifiable error estimates.