Department of Mathematics

Title: From centralized to federated learning of neural operators: Accuracy, scalability, and reliability

Abstract: As an emerging paradigm in scientific machine learning, deep neural operators pioneered by us can learn nonlinear operators of complex dynamic systems via neural networks. In this talk, I will present the vanilla deep operator network (DeepONet) and several extensions of DeepONet, such as DeepONet with Fourier decoder layers and manifold operator learning. I will demonstrate their effectiveness on diverse multiphysics and multiscale 3D problems, such as geological carbon sequestration, full waveform inversion, and topology optimization. Deep learning models are usually limited to interpolation scenarios, and I will quantify the extrapolation complexity and develop a complete workflow to address the challenge of extrapolation for deep neural operators. Moreover, I will present the first operator learning method that requires only one PDE solution, i.e., one-shot learning, by introducing a new concept of local solution operator based on the principle of locality of PDEs. I will also present the first systematic study of federated scientific machine learning (FedSciML) for approximating functions and solving PDEs with data heterogeneity. Lastly, I will present FunDiff, a novel framework of diffusion models over function spaces for physics-informed generative modeling.

Title: Semi-retractions and Ramsey degrees

Abstract: Ramsey degrees and their transfer principles form a branch of combinatorics that has received much recent interest. As a brief illustration of definitions I will present in the talk, the ordered pair $a<b$ has small Ramsey degree 1 in the class of all finite linear orders by Ramsey's classical theorem for finite sets. In contrast, the ordered pair $a<b$ has big Ramsey degree 2 in the rational order $(\mathbb{Q},<)$: if we color ordered pairs from $\mathbb{Q}$ some finite number of colors, there is a suborder isomorphic to the rational order whose pairs take on at most two colors; however, there is a 2-coloring of pairs from $\mathbb{Q}$ such that any suborder isomorphic to the rational order has pairs of each color.

In a 2021 paper, I introduced the concept of a “semi-retraction'', which is a pair of maps $(g,f)$ between infinite mathematical objects that has the necessary architecture to preserve facts about Ramsey degrees. This notion was further developed in a 2024 paper joint with Dana Bartošová at University of Florida. In this talk, I will survey some results from this paper and extensions of these results.

Title: Policy iteration for inverse mean field games

Abstract: We propose a policy iteration method to solve an inverse problem for a mean f ield game (MFG) model, specifically to reconstruct the obstacle function in the game from the partial observation data of value functions, which represent the optimal costs for agents. The proposed approach decouples this complex inverse problem, which is an optimization problem constrained by a coupled nonlinear forward and backward PDE system in the MFG, into several iterations of solving linear PDEs and linear inverse problems. This method can also be viewed as a fixed-point iteration that simultaneously solves the MFG system and inversion. We prove its linear rate of convergence and present some numerical examples to demonstrate the effectiveness of the method. This is based on a joint work with Nathan Soedjak and Shanyin Tong.

Title: From Spheres to Bands: New Rigidity Phenomena in Scalar Curvature

Abstract: When does a lower bound on scalar curvature force a space to be “as round as a sphere”? A landmark theorem of Llarull says: if a closed spin manifold maps onto the unit sphere without increasing areas and with nonzero degree, then having the same scalar‑curvature lower bound as the sphere already forces the map to be an isometry—so the manifold itself is a round sphere. I will describe a new band version of this phenomenon for compact manifolds with boundary that map into a spherical band (the sphere with two caps removed—the region between two latitudes). We obtain sharp inequalities controlling how “long” such bands can be and we characterize the equality case. Two payoffs: (1) a proof of Llarull’s theorem in dimension four without the spin assumption, and (2) a rigidity theorem for manifolds with conical ends mapping into punctured spheres. I’ll emphasize the geometric mechanisms behind these results and keep technicalities to a minimum.