Abstract:

Abstract: The classical calculus of pseudodifferential operators extends differential operators in a way that is suited to the construction of parametrices (pseudo-inverses) for elliptic operators. A fundamental consequence is that elliptic operators are Fredholm between appropriate Sobolev spaces on compact manifolds.

On manifolds with boundary, this implication no longer holds. Melrose’s calculus of b-pseudodifferential operators is the analogous class of operators which leads to Fredholm properties for elliptic operators satisfying a certain condition related to the boundary. In this talk, I will compare the classical case and the boundary case, emphasizing the new features introduced by the boundary—most notably the b-stretched product and the indicial operator—and explain how these lead to Fredholmness on weighted b-Sobolev spaces.

Abstract: In chemical kinetics, accurately modeling the dynamic behavior of chemical systems is essential for predicting reaction outcomes and optimizing processes. However, the challenge known as the “curse of dimensionality” has posed significant difficulties for conventional techniques employed in addressing the chemical master equation (CME). This predicament arises when the state space of the Markov chain expands exponentially with the number of species, rendering the CME computation practically unsolvable.

In this talk, I will discuss some methods of solving the CME, including Gillespie’s algorithm, the Chemical Langevin Equation, and the Method of Moments, along with an overview of tensor train and machine learning-based methods, which offer promising strategies for gaining insights into complex biological systems.

Abstract:

Abstract:

Abstract:

Abstract:

Abstract:

Abstract:

Abstract:

Abstract:

Abstract:

Abstract:

Abstract:

Abstract: Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) with broad applications in physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is to learn the associated Green’s function. However, Green’s functions are notoriously difficult to approximate due to their poor regularity, often requiring large neural networks and long training times.

In this talk, we address these challenges by leveraging multiscale neural networks to learn Green’s functions efficiently. Through theoretical analysis based on multiscale Barron space techniques, together with numerical experiments, we show that the multiscale approach significantly reduces the required network size and accelerates training. We then extend this framework to operator learning, enabling neural networks to efficiently and accurately learn the mapping from coefficient functions to Green’s functions.

Abstract: