October 8, 2024
Ziming Liu [MIT & IAIFI]

▦ Towards Unification of Artificial Intelligence and Science ▦

A major challenge of AI + Science lies in their inherent incompatibility: today's AI is primarily based on connectionism, while science depends on symbolism. In the first part of the talk, I will talk about Kolmogorov-Arnold Networks (KANs) as a solution to synergize both worlds. Inspired by Kolmogorov-Arnold representation theorem, KANs are more aligned with symbolic representations than MLPs, and demonstrate strong accuracy and interpretability. In the second part, I will talk about more broadly the intersection of AI and Science, including science for AI (Poisson Flow Generative Models), science of AI (understanding grokking), and AI for Science (AI scientists).

October 15, 2024
Tim Roith [DESY]

▦ Gullible networks and the mathematics of adversarial attacks ▦

With the increasing incentive to employ deep learning techniques in real-life scenarios, one naturally questions their reliability. For image classification, a well-known phenomenon called adversarial examples shows how small, humanly imperceptible input-perturbations can change the output of a neural network completely. This insight formed the field of adversarial robustness, which we explore in this talk. We discuss how regularizing the standard training objective with Lipschitz and TV regularization terms can lead to resilient neural networks. Furthermore, we also explore the adversarial attack problem. We derive an associated gradient-flow for the so-called fast gradient sign method, which is commonly used to find malicious input-perturbations. Here, we work in an abstract metric setting, where we then highlight the distributional Wasserstein case, which relates back to the robustness problem. Finally, we also consider the attack problem in a realistic closed-box scenario, where we employ gradient-free optimizers.

October 22, 2024
Pranava Jayanti [University of Southern California]

▦ Avoiding vacuum in two models of superfluidity ▦

At low pressures and very low temperatures, Helium-4 is composed of two interacting phases: the superfluid and the normal fluid. We discuss some recent mathematical results in the analysis of two distinct models of superfluidity. Micro-scale model: The nonlinear Schrödinger equation is coupled with the incompressible inhomogeneous Navier-Stokes equations through a bidirectional nonlinear relaxation mechanism that facilitates mass and momentum exchange between phases. For small initial data, we construct solutions that are either global or almost-global in time, depending on the strength of the superfluid's self-interactions. The primary challenge lies in controlling inter-phase mass transfer to prevent vacuum formation within the normal fluid. Two approaches are employed: one based on energy estimates alone, and another combining energy estimates with maximal regularity. These results are part of joint work with Juhi Jang and Igor Kukavica. Macro-scale model: Both phases are governed by the incompressible Euler equations, coupled through a nonlinear and singular interaction term. We construct unique local-in-time analytic solutions. To address the singularity in the coupling, we ensure the absence of vorticity vacuum, while the derivative loss due to the nonlinearity is offset by trading regularity for dissipation.   

November 5, 2024
Yannick Sire [Johns-Hopkins University ]

▦ Some problems in the flow of Liquid Crystals ▦

"I will describe some recent results related to some simplified models of Liquid Crystals, with a view towards geometric free boundaries. A simplified version of the Ericksen-Leslie system has been introduced by FH Lin in the 80's. After describing the state of the art for this system, I will introduce a new one involving a free boundary. Though the mathematical analysis of the system is still very preliminary, some results are still available in 2 dimensions and I will mainly motivate the introduction of geometric variational problems with free boundaries and how one can deal with them, thanks to recent advances in compensated-compactness in odd dimension. I will mention several open questions and possible further generalizations."

  

November 12, 2024
Nicolas Boulle [Imperial College]

▦ Operator learning without the adjoint ▦

There is a mystery at the heart of operator learning: how can one recover a non-self-adjoint operator from data without probing the adjoint? Current practical approaches suggest that one can accurately recover an operator while only using data generated by the forward action of the operator without access to the adjoint. However, naively, it seems essential to sample the action of the adjoint for learning time-dependent PDEs. In this talk, we will first explore connections with low-rank matrix recovery problems in numerical linear algebra. Then, we will show that one can approximate a family of non-self-adjoint infinite-dimensional compact operators via projection onto a Fourier basis without querying the adjoint.  

  

November 19, 2024
Vincent Martinez [The City University of New York - Hunter College]

▦ On reconstructing unknown state and parameters in hydrodynamic systems from time-series data ▦

This talk will describe a basic approach to the problem of simultaneous state and parameter reconstruction from low-dimensional time-series data in the context of hydrodynamic systems. We present theorems identifying conditions under which these approaches are guaranteed to succeed in an idealized setting that give some clarity to the general issue of reconstructability. Ultimately, the success of these algorithms rely on a crucial nonlinear mechanism common to these systems, the exact role of which will be discussed.  

January 21, 2025
Alasdair Hastewell [NSF-Simons NITMB]

▦ Inferring the dynamics of locomotion using spectral mode representations ▦

Recent advances in automated experimental imaging allow for high-resolution tracking of locomotion across biological scales, from whole animal behavior to single-cell trajectories during embryogenesis. Inferring distinct dynamical states from these high-dimensional data and transforming them into effective low-dimensional models is an essential challenge to characterize and compare dynamics within and across species. Spectral mode representations have been used successfully across physics, from quantum mechanics to fluid turbulence, to compress dynamical data. We develop a computational framework that combines geometry-aware spectral mode representations with wavelet analysis and dynamical systems inference to realize a generic procedure for characterizing the dynamics of locomotion. We demonstrate the framework by applying it to tracked centerlines of Pterosperma flagella. Finally, I will discuss ongoing work generalizing this approach to datasets with graphical structures, such as limbed animals.  

February 18, 2025
Antonin Della Noce [Inria and Ecole des Ponts ParisTech]

▦ Derivative-free Bayesian Inversion using Multiscale Dynamics: Weak convergence and Invariant Distribution ▦

Bayesian Inversion consists of deriving the posterior distribution of unknown parameters or functions from partial and indirect observations of a system. When the dimension of the search space is high or infinite, methods leveraging local information, such as derivatives of different orders, of the target probability measure have the advantages to converge faster than Monte-Carlo sampling techniques. Nevertheless, many applications are characterized by posterior distributions with low regularity or gradients that are intractable to compute. An interesting research direction consists in using interacting particle systems to explore the potential landscape, and Ensemble Kalman Sampler (EKS) is one of those. In this talk, we consider a simplified EKS dynamics, where the gradient of the potential is approximated by finite differences using independent Ornstein-Uhlenbeck processes that explore the neighborhood of the candidate parameter. We will characterize the invariant distribution of this system and compare its dynamics to the overdamped Langevin process.   

February 25, 2025
Urbain Vaes [Inria and Ecole des Ponts ParisTech]

▦ Mean field limits for Consensus-Based Optimization and Sampling ▦

For algorithms based on interacting particle systems that admit a mean-field description, convergence analysis is often more accessible at the mean-field level. In order to translate convergence results obtained at the mean-field level to the finite ensemble setting, it is desirable to obtain estimates on the distance, in an appropriate metric, between the particle dynamics and the corresponding mean-field dynamics. In this talk, we present quantitative mean-field limit results for two related interacting particle systems: Consensus-Based Optimization and Consensus-Based Sampling. Our approach extends Sznitman's classical argument: in order to circumvent issues related to the lack of global Lipschitz continuity of the coefficients, we discard an event of small probability, the contribution of which is controlled using moment estimates for the particle systems.   

March 4, 2025
Fred Hickernell [Illinois Institute of Technology]

▦ Adaptive Algorithms where Input Functions Lie in Cones ▦

Adaptive numerical algorithms expend computational effort necessary to meet the error tolerance requirements of the user. Besides constructing approximate solutions, adaptive algorithms also use function data to determine the computational effort required to obtain a satisfactory solution and how best to allocate that computational effort. The challenge is knowing what can be reliably learned from function data. We contend that successful adaptive algorithms should be constructed for non-convex cones of input functions. We illustrate via some simple one-dimensional problems. Subsequently, we survey our work constructing adaptive (quasi-)Monte Carlo algorithms. Finally, we propose some research directions for adaptive algorithms based on cones of inputs.   

March 13, 2025
Edriss S. Titi [Texas A&M University / University of Cambridge]

▦ Is dispersion a stabilizing or destabilizing mechanism? ▦

In this talk I will present a unified approach for the effect of fast rotation and dispersion as an averaging mechanism for regularizing and stabilizing certain evolution equations, such as the Euler, Navier-Stokes and Burgers equations. On the other hand, I will also present some results in which large dispersion acts as a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. In addition, I will present some results concerning two- and three-dimensional turbulent flows with high Reynolds numbers in periodic domains, which exhibit enhanced dissipation mechanism due to large spatial average in the initial data - a phenomenon which is similar to the "Landau-damping" effect.  

March 25, 2025
David Ginsbourger [University of Bern/Institute of Mathematical Statistics and Actuarial Science]

▦ Modelling invariances and equivariances with GP models ▦

Gaussian Process models offer elegant possibilities to encode invariances and equivariances. Choosing adapted mean and covariance functions enables going around data augmentation or making it somehow implicit. Furthermore, resulting models do not only propagate invariances and equivariances via the predictive mean but also via posterior simulations. We review related results and illustrations pertaining to invariances, and tackle recent work pertaining to equivariances, introducing in turn via examples some recent research about computationally efficient equivariant GP modelling. Based on several collaborations to be further detailed during the presentation.  

April 1, 2025
Jennifer J. Sun [Cornell University]

▦ AI for Scientists: Perception, Reasoning, & Discovery ▦

Artificial intelligence (AI) holds tremendous promise to accelerate scientific discovery. However, significant gaps exist in translating these models to complex, real-world challenges in science, including integrating domain knowledge, improving data efficiency, and tailoring solutions to the unique needs of each lab. My work focuses on collaborative AI systems designed to bridge these gaps, in order for scientists to extract insights from high-dimensional data (e.g. animal behavior videos). First, we demonstrate how general-purpose video foundation models, by leveraging Internet-scale datasets, enable new ways to tackle domain-specific problems in behavior analysis. Next, we explore how recent advancements, such as large language models (LLMs), facilitate neurosymbolic approaches for analyzing complex scientific data through techniques such as library learning and external knowledge integration. Looking ahead, we envision AI agents collaborating with scientists throughout the scientific process to understand the world around us.

April 15, 2025
Varun Shankar [University of Utah]

▦ Kernels in Numerical Methods and Operator Learning ▦

In this talk, I will give an overview of kernel methods and their applications to both the numerical solution of partial differential equations (PDEs) and for surrogate modeling via operator learning. As PDE solvers, kernel-based finite differences enable the high-order accurate solution of PDEs on point clouds. I will show applications of this to PDEs on moving domains and on manifolds. In the context of operator learning, I will show recent work on two neural operators that leverage the strengths of closed-form kernels: (1) the Kernel Neural Operator (KNO), a particular generalization of the well-known Fourier Neural Operator (FNO); and (2) Ensemble and Partition-of-Unity Deep Operator Networks (DeepONets), which leverage localized kernel-based approximation to enhance accuracy near solution features of interest.

April 22, 2025
Yuqing Wang [Johns-Hopkins University]

▦ The Mechanism Behind the Implicit Biases of Large Learning Rates: Edge of Stability, Balancing, and Catapult ▦

Large learning rates, when applied to gradient descent for nonconvex optimization, yield various implicit biases, including edge of stability, balancing, and catapult. There are a lot of theoretical works trying to analyze these phenomena, while the high level idea is still missing: it is unclear when and why these phenomena occur. In this talk, I will show that these phenomena are actually various tips of the same iceberg. They occur when the objective function of optimization has some good regularity. This regularity, together with the effect of large learning rate on guiding gradient descent from sharp regions to flatter ones, leads to the control of the largest eigenvalue of Hessian, i.e., sharpness, along the GD trajectory, which results in various phenomena. The result is based on the convergence analysis under large learning rate on a family of nonconvex functions of various regularities without Lipschitz gradient which is usually a default assumption in nonconvex optimization. Neural network experiments will also be presented to validate this result.

May 6, 2025
Yannick Sire [Johns-Hopkins University]

▦ Some problems in the flow of Liquid Crystals ▦

I will describe some recent results related to some simplified models of Liquid Crystals, with a view towards geometric free boundaries. A simplified version of the Ericksen-Leslie system has been introduced by FH Lin in the 80's. After describing the state of the art for this system, I will introduce a new one involving a free boundary. Though the mathematical analysis of the system is still very preliminary, some results are still available in 2 dimensions and I will mainly motivate the introduction of geometric variational problems with free boundaries and how one can deal with them, thanks to recent advances in compensated-compactness in odd dimension. I will mention several open questions and possible further generalizations.

May 13, 2025
Oliver Tse [Eindhoven University of Technology]

▦ TBA ▦

TBD

May 20, 2025
Jinghao Cao [Caltech]

▦ TBA ▦

TBD

January 7, 2025
• CMX Special Seminar •

ANB 213
4:00 pm

Juan Toscano [Brown University]

▦ Inferring turbulent velocity and temperature fields and their statistics from Lagrangian velocity measurements using physics-informed Kolmogorov-Arnold Networks (PIKANs) ▦

We propose the Artificial Intelligence Velocimetry-Thermometry (AIVT) method to infer hidden temperature fields from experimental turbulent velocity data. This physics-informed machine learning method enables us to infer continuous temperature fields using only sparse velocity data, eliminating the need for direct temperature measurements. Specifically, AIVT is based on physics-informed Kolmogorov-Arnold Networks (not neural networks) and is trained by optimizing a combined loss function that minimizes the residuals of the velocity data, boundary conditions, and governing equations. We apply AIVT to a unique set of experimental volumetric and simultaneous temperature and velocity data of Rayleigh-Bénard convection (RBC) acquired by combining Particle Image Thermometry and Lagrangian Particle Tracking. This allows us to directly compare AIVT predictions with measurements. We demonstrate the ability to reconstruct and infer continuous and instantaneous velocity and temperature fields from sparse experimental data at a fidelity comparable to direct numerical simulations (DNS) of turbulence. This, in turn, enables us to compute important quantities for quantifying turbulence, such as fluctuations, viscous and thermal dissipation, and QR distribution. This paradigm shift in processing experimental data using AIVT to infer turbulent fields at DNS-level fidelity offers a promising approach for advancing quantitative understanding of turbulence at high Reynolds numbers, where DNS is computationally infeasible.

January 8, 2025
• CMX Special Seminar •

ANB 213
4:30 pm

Ziming Liu [MIT]

▦ Physics of AI - What can "spherical cow" models tell us? ▦

As John Hopfield eloquently stated in his Nobel lecture, "Physics is not subject matter, but a point of view." Physicists are renowned for their boldness in simplifying complex systems-modelling even a cow as a "spherical cow in a vacuum" when other details are irrelevant. While such abstractions may appear overly simplistic, this approach has proven instrumental in unraveling many mysteries of deep learning. In this talk, I will explore how "spherical cow" models can illuminate phenomena such as grokking, neural scaling laws, and emergent skills. Furthermore, I will discuss how these models can offer valuable insights into designing more efficient next-generation AI systems.

February 27, 2025
• CMX Special Seminar •

ANB 213
12:00 pm

Molei Tao [Georgia Institute of Technology]

▦ Where do all the scores come from? - generation accuracy of diffusion model, and multimodal sampling via denoising annealing ▦

Diffusion model is a prevailing Generative AI approach. It uses a score function to characterize a complex data distribution and its evolution toward an easy distribution. This talk will report progress in two different topics, both closely related to the origins of the score function. The first topic, which will take most time of the talk, will be on a quantification of the generation accuracy of diffusion model. The importance of this problem already led to a rich and substantial literature; however, most existing theoretical investigations assumed that an epsilon-accurate score function has already been oracle-given, and focused on just the inference process of diffusion model. I will instead describe a first quantitative understanding of the actual generative modeling protocol, including both score training (optimization) and inference (sampling). The resulting full error analysis will elucidate (again, but this time theoretically) how to design the training and inference processes for effective generation. The second topic will no longer be about generative modeling, but sampling instead. The goal is leverage the fact that diffusion model is very good at handling multimodal distributions, and extrapolate it to the holy grail problem of efficient sampling from multimodal density. There, one needs to rethink about how to get the score function, as no more data samples are available and one instead has unnormalized density. A new sampler that is insensitive to metastability, with performance guarantee, and not even requiring continuous density, will be presented.

October, 2024
• CMX Student/Postdoc Seminar •

TBA [Caltech]
▦ TBD ▦

TBD   

Inverse Problems: Algorithms, Analysis and Applications (Feb 6-7, 2020)

Southern California Applied Mathematics Symposium (SOCAMS 2019)

• Meeting Poster