Advances in Computational Mathematics with Machine Learning
Monday October 6th – 12:30 – 4:15 pm – Hosted by Dr. John Harlim – HUB 132 Flex Theater
Workshop on Advances in Computational Mathematics with Machine Learning
In this workshop, we have four distinguished speakers who will discuss recent developments in Computational Mathematics. Specifically, they will address how Machine Learning has emerged as an important part of scientific computational tools for solving long standing problems in physical and biological models. Topics range from transport information geometry, neural attention operator for foundational modeling of physical systems, Digital twins, generative AI, to time series analysis for denoising psychological signals arising in healthcare application.
Agenda:
12:30 PM – Welcome & Introduction – Dr. John Harlim
12:35 PM – Speaker 1: Prof. Hau-Tieng Wu, Courant Institute, New York University
1:20 PM – Speaker 2: Prof. Wuchen Li, Dept. of Mathematics, University of South Carolina
2:05 PM – Break
2:15 PM – Speaker 3: Prof. Yue Yu, Dept. of Mathematics, Lehigh University
3:00 PM – Speaker 4: Prof. Harbir Antil, Dept. of Mathematics, George Mason University
3:45 PM – Panel Discussion
4:15 PM – We encourage you to attend the ICDS Symposium Poster session in 129 Alumni Hall
Speaker 1: Prof. Hau-Tieng Wu, Courant Institute, New York University.
Title: Manifold Denoising of Nonstationary Physiological Signals for Machine Learning in Healthcare
Abstract: Recent advances in wearable devices and monitoring technologies now enable continuous acquisition of rich, multimodal physiological waveforms, moving well beyond traditional pointwise clinical measurements. These data streams are inherently nonstationary, often exhibiting time-varying periodic structure, nonlinear dynamics, and cross-scale interactions, which pose fundamental challenges for statistical modeling and machine learning. To address this, we introduce a manifold denoising framework, a data sharpening technique, that suppresses noise under a manifold model and transforms raw nonstationary time series into mathematically tractable representations for learning. Our approach leverages random matrix theory and spectral geometry to provide rigorous theoretical guarantees and practical algorithms. A clinical application in brain wave analysis demonstrates the effectiveness of the method, highlighting its potential to enhance healthcare data analysis and inform medical decision-making.
Speaker 2: Prof. Wuchen Li, Dept. of Mathematics, University of South Carolina.
Title: Transport Information Geometric Computations
Abstract: Current technology innovations rely on information (entropy), transportation (differential equations), and their algorithms (AI). This talk reviews the research update of the project: Transport Information Geometric Computations (TIGC). The project designs and analyzes numerical algorithms for statistical physics from generative AI methods, including normalization flows, generative adversarial networks, and transformers. Through algorithms and numerical examples, the TIGC project builds on mathematical foundations of AI with various applications in sampling algorithms and statistical physics-related simulations.
Speaker 3: Prof. Yue Yu, Dept. of Mathematics, Lehigh University
Title: Nonlocal Attention Operator: Understanding Attention Mechanism for Physical Responses
Abstract: While foundation models have gained considerable attention in core AI fields such as natural language processing (NLP) and computer vision (CV), their application to learning complex responses of physical systems from experimental measurements remains underexplored. In physical systems, learning problems are often characterized as discovering operators that map between function spaces, using only a few samples of corresponding function pairs. For instance, in the automated discovery of heterogeneous material models, the foundation model must be capable of identifying the mapping between applied loading fields and the resulting displacement fields, while also inferring the underlying microstructure that governs this mapping. While the former task can be seen as a PDE forward problem, the later task frequently constitutes a severely ill-posed PDE inverse problem.
In this talk, we will explore the attention mechanism towards a foundation model for physical systems. Specifically, we show that the attention mechanism is mathematically equivalent to a double integral operator, enabling nonlocal interactions among spatial tokens through a data-dependent kernel that characterizes the inverse mapping from data to the hidden PDE parameter field of the underlying operator. Consequently, the attention mechanism captures global prior information from training data generated by multiple systems and suggests an exploratory space in the form of a nonlinear kernel map. Based on this theoretical analysis, we introduce a novel neural operator architecture, the Nonlocal Attention Operator (NAO). By leveraging the attention mechanism, NAO can address ill-posedness and rank deficiency in inverse PDE problems by encoding regularization and enhancing generalizability. To demonstrate the applicability of NAO to material modeling problems, we apply it to the development of a foundation constitutive law across multiple materials, showcasing its generalizability to unseen data resolutions and system states. Our work not only suggests a novel neural operator architecture for learning an interpretable foundation model of physical systems, but also offers a new perspective towards understanding the attention mechanism.
Speaker 4: Prof. Harbir Antil, Dept. of Mathematics, George Mason University
Title: Digital Twins and Generative AI: A PDE–Constrained Optimization Perspective
Abstract: Digital Twins (DTs) are real-time, adaptive virtual replicas of physical systems, integrating physics-based simulation, sensor data, and intelligent decision-making. This talk presents a unifying PDE–constrained optimization (PDECO) framework for DTs, in which state estimation and control are solved within a moving-horizon loop using adjoint-based methods. We establish a novel connection between PDECO and Generative AI, demonstrating how score-based generative models can be rigorously interpreted as backward-in-time PDE problems—linking ill-posed inverse problems, stability analysis, and modern machine learning. This perspective bridges physics-informed modeling and data-driven synthesis, potentially enabling score-based Digital Twins. Applications span structural and biomedical systems—from bridges and dams to aneurysm modeling and neuromorphic computing—illustrating a path toward predictive, adaptive, and trustworthy Digital Twins.