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.

Title: How to make sense of sonic boom and other discontinuities in fluid mechanics

Abstract: The compressible Euler equation can lead to the emergence of shock discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier-Stokes solutions with evanescent viscosities. The mathematical study of this problem is however very difficult because of the destabilization effect of the viscosities.

Bianchini and Bressan proved the inviscid limit to small BV solutions using the so-called artificial viscosities in 2004. However, until very recently, achieving this limit with physical viscosities remained an open question.

In this presentation, we will provide the basic ideas of classical mathematical theories to compressible fluid mechanics and introduce the recent method of a-contraction with shifts. This method is employed to describe the physical inviscid limit in the context of the barotropic Euler equation, and to solve the Bianchini and Bressan conjecture in this special case.

Title: More unlikely intersections on elliptic surfaces
(w/ G. Urzua and F. Voloch)

Abstract: In arithmetic geometry, one encounters situations where an intersection is “unlikely”, for example on dimension grounds, and where one wants to give the unlikely thing infinitely many chances to happen (say by varying some parameter). A statement that the intersection happens only finitely many times is then interesting (and a bound is even better). Families of elliptic curves host many examples of this type of problem. We discuss one such situation where the main result is a beautiful bound of topological nature, but whose proof, somewhat surprisingly, requires us to briefly leave the world of algebraic geometry for a real-analytic context. Time permitting, I’ll also mention a connection to a famous homomorphism of Manin and point out some inaccuracies in the literature about it.

Title: Physics-Informed Gaussian Process for ODE/PDE Simulation and Calibration

Abstract: Parameter estimation for nonlinear dynamic system models, represented by ordinary differential equations (ODEs) or partial differential equations (PDEs), is a fundamental challenge across science and engineering. Traditional methods rely heavily on numerical integration, which is computationally expensive and often ill-suited for noisy and sparse data. We propose a new method, the Physics-Informed Gaussian Process (PIGP), that combines statistical rigor with physics-based constraints. Specifically, we model system components as Gaussian processes, explicitly conditioned on the governing ODE/PDE system. This approach bypasses traditional numerical integration entirely, leading to substantial computational savings while providing uncertainty quantification and enabling inference even for unobserved system components. Our framework offers a principled Bayesian alternative to physics-informed neural networks, with unique strengths in transparency, interpretability, and robust uncertainty awareness.

At a high level, many real-world scientific and engineering systems, ranging from disease spread to weather patterns, from fluid flows to acoustic waves, are described by differential equations. Our method introduces a way to apply uncertainty-aware machine learning directly to these equations using limited data. In plain terms, we can use small amounts of experimental observations, combined with known physics, to make accurate predictions while quantifying confidence in the results. This ability is especially valuable in high-stakes applications where uncertainty must be explicitly managed.

Title: A Universal Description of Stochastic Oscillators

Abstract: Many systems in physics, chemistry and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. In joint work with Alberto Perez, Benjamin Lindner, and Boris Gutkin, we have introduced a nonlinear transformation of stochastic oscillators via a complex-valued function, Q, that greatly simplifies and unifies the mathematical description of the oscillator’s spontaneous activity, its response to external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The Q function is the eigenfunction of Kolmogorov’s backward operator (also called the stochastic Koopman operator) with the least negative (but non-vanishing) eigenvalue $\lambda_1=\mu_1+i\omega_1$. The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency $\omega_1$ and half-width $\mu_1$; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around $\omega_1$; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of both weakly and strongly coupled stochastic oscillators.

Joint work with:
Alberto Perez-Cervera (Universitat Politècnica de Catalunya, Barcelona)
Boris Gutkin (Ecole Normale Supérieure, Paris)
Benjamin Lindner (Humboldt University, Berlin)
Max Kreider (Penn State University)