Date and time: Feb 14 at 2:00 pm (Parker 328)

Title: Fast solvers for radiative transfer and beyond

Abstract: Despite the tremendous developments in recent years, constructing efficient numerical solution methods for the radiative transfer equation (RTE) is still challenging in scientific computing. In this talk, I will present a simple yet fast computational algorithm for solving the RTE in isotropic media in steady-state and time-dependent settings. The algorithm we developed has two steps. In the first step, we solve a volume integral equation for the angularly averaged solution using iterative schemes such as the GMRES method. The computation in this step is accelerated with a variant of the fast multipole method (FMM). In the second step, we solve a scattering-free transport equation to recover the angular dependence of the solution. The algorithm does not require the underlying medium to be homogeneous. We present numerical simulations under various scenarios to demonstrate the performance of the proposed numerical algorithm for both homogeneous and heterogeneous media. Then I will extend the formulation to the time-domain and anisotropic scattering media and analyze the possibility of applying the fast algorithm.

Date and time: Mar 28 at 2:00 pm (Parker 328)

Title: Numerical Understanding of Neural Networks

Abstract: In this talk, I will talk about a couple of recent works on neural networks. The motivation is to see whether neural networks are suitable for general scientific computing. Our study of shallow neural networks demonstrates that shallow neural networks are in general low-pass filters from different perspectives. Based on this observation, we proposed to make use of the composition of shallow networks to construct deep neural networks, which demonstrates better performance over the vanilla fully connected neural networks of comparable parameters.

Date and time: Feb 21 at 2:00 pm (Parker 328)

Title: Symmetry-Preserving Machine Learning: Theory and Applications

Abstract: Symmetry underlies many machine learning and scientific computing tasks, from computer vision to physical system modeling. Models designed to respect symmetry often perform better, but several questions remain. How can we measure and maintain approximate symmetry when real-world symmetries are imperfect? How much training data can symmetry-based models save? And in non-convex optimization, do these models truly converge to better solutions? In this talk, I’ll share my work on these challenges, revealing that the answers are sometimes surprising. The approach draws on applied probability, harmonic analysis, differential geometry, and optimization, but no specialized background is required. 

Date and time: Mar 21 at 2:00 pm (Parker 328)

Title: Structure-preserving machine learning for learning dynamical systems

Abstract: I will present our recent work on structure-preserving machine learning (ML) for dynamical systems. First, I introduce a structure-preserving neural ODE framework that accurately captures chaotic dynamics in dissipative systems. Inspired by the inertial manifold theorem, our model learns the ODE’s right-hand side by combining a linear and a nonlinear term, enabling long-term stability on the attractor for the Kuramoto-Sivashinsky equation. This framework is further enhanced with exponential integrators. Next, I discuss ML for singularly perturbed systems, leveraging the Fenichel normal form to simplify fast dynamics near slow manifolds. A fast-slow neural network is proposed that enforces the existence of a trainable, attractive invariant slow manifold as a hard constraint.

Date and time: Apr 4 at 2:00 pm (Parker 328)

Title: Optimization, Sampling, and Generative Modeling in Non-Euclidean Spaces

Abstract: Machine learning in non-Euclidean spaces have been rapidly attracting attention in recent years, and this talk will give some examples of progress on its mathematical and algorithmic foundations. A sequence of developments that eventually leads to the generative modeling of data on Lie groups will be reported. Such a problem occurs, for example, in the Gen-AI design of molecules.

More precisely, I will begin with variational optimization, which, together with delicate interplays between continuous- and discrete-time dynamics, enables the construction of momentum-accelerated algorithms that optimize functions defined on manifolds. Selected applications, such as a generic improvement of Transformer, and a low-dim. approximation of high-dim. optimal transport distance, will be described. Then I will turn the optimization dynamics into an algorithm that samples from probability distributions on Lie groups. This sampler provably converges, even without log-concavity condition or its common relaxations. Finally, I will describe how this sampler can lead to a structurally-pleasant diffusion generative model that allows users to, given training data that follow any latent statistical distribution on a Lie group manifold, generate more data exactly on the same manifold that follow the same distribution. If time permits, applications such as molecule design and generative innovation of quantum processes will be briefly discussed.

Date and time: Apr 11 at 2:00 pm (Parker 328)

Title: Nonlocal Boundary Value Problems with Local Boundary Conditions

Abstract: We state and analyze nonlocal problems with classically-defined, local boundary conditions. The model takes its horizon parameter to be spatially dependent, vanishing near the boundary of the domain. We establish a Green's identity for the nonlocal operator that recovers the classical boundary integral, which permits the use of variational techniques. Using this, we show the existence of weak solutions, as well as their variational convergence to classical counterparts as the bulk horizon parameter uniformly converges to zero. In certain circumstances, global regularity of solutions can be established, resulting in improved modes and rates of variational convergence. Generalizations of these results pertaining to models in continuum mechanics and Laplacian learning will also be presented.

Date and time: Apr 18 at 2:00 pm on Zoom

Title: An energy stable, second-order time-stepping method for two phase flow in porous media

Abstract: We propose and analyze a second-order partitioned time-stepping method for a two phase flow problem in porous media. The algorithm is based on a refactorization of Cauchy’s one-legged θ-method. The main part consists of the implicit backward Euler method, while part two uses a linear extrapolation. In the backward Euler step, the decoupled equations are solved iteratively. We prove that the iterations converge linearly to the solution of the coupled problem, under some conditions on the data. When θ=1/2, the algorithm is equivalent to the symplectic midpoint method. Similar to the continuous case, we also prove a discrete Helmholtz free energy balance, without numerical dissipation. We compare this midpoint method with the classic backward Euler method, and two implicit-explicit time-lagging schemes. The midpoint method outperforms the other schemes in terms of rates of convergence, long-time behaviour and energy approximation, for small and large values of the time step.

Date and time: Apr 25 at 2:00 pm (Parker 328)

Title: TBA

Abstract: TBA

Date and time: May 2 at 2:00 pm (Parker 328)

Title: TBA

Abstract: TBA