SPEAKERS

Abstract: The increasing complexity of modern engineering systems is driving an urgent need for faster, more reliable, and more adaptive computational models. Across many industrial domains, including advanced manufacturing, materials science, energy systems, and digital twins, high-fidelity simulations have become indispensable, yet their computational cost often prevents their use in real-time decision making, optimization, and large-scale design exploration.

At the same time, recent advances in artificial intelligence and machine learning are potentially creating new opportunities for accelerating scientific computation. However, these approaches frequently struggle with reliability, interpretability, physical consistency, and extrapolation beyond the training regime. This has led to the emergence of Scientific Machine Learning (SciML), a rapidly developing field at the interface of mathematics, scientific computing, engineering, and artificial intelligence.

In this lecture, we discuss how modern numerical mathematics can provide the foundations for the next generation of scientific machine learning methods. We present recent developments in hybrid approaches that combine physics-based modelling with data-driven learning, including reduced-order modelling, operator learning, structure-preserving methods, and physics-informed architectures. Particular emphasis will be placed on the role of mathematical structure in ensuring robustness, efficiency, and physical reliability.

We further discuss how ideas originating from scientific computing, such as localization, multiscale methods, model reduction, and structure preservation, are increasingly influencing the design of modern artificial neural networks and learning algorithms. These developments point towards a new generation of mathematically grounded AI methods that are better suited for complex engineering applications.

The central message of the talk is that the future of AI for science and engineering will not be driven by machine learning alone, but by a deep integration of mathematics, physical insight, and computational science. Building these bridges between disciplines is essential for transforming recent AI advances into reliable industrial technologies and scientific breakthroughs

Abstract:

The explosive advancement of Generative AI has accelerated its integration into society, challenging the conventional view that mathematics is merely the underlying theoretical foundation of AI algorithms. From the perspective of industrial and applied mathematics, this talk presents various emerging relationships between mathematics and AI in the Generative AI era. Rather than just a foundation, mathematics is acting as a dynamic catalyst that bridges advanced AI with the complex physical world, serving simultaneously as a translator, controller, and co-creator.

First, we illustrate the role of mathematics as a translator through my recent research on Topological Flow Data Analysis (TFDA). By utilizing mathematical classification theorems with topological invariants, TFDA converts complex continuous dynamics into discrete transition graphs via Topological Dynamic Mode Decomposition. This mathematical "tokenization" of physical flow patterns allows for seamless integration with Large Language Models (LLMs) and Transformer architectures, enabling novel, highly accurate predictive models that fundamentally surpass conventional raw-data machine learning.

Second, we explore mathematics as a controller in the context of planetary-scale challenges, specifically Moonshot Goal 8 for extreme weather control. While AI weather forecasting has made it possible to efficiently generate massive ensembles, it is the rigorous mathematical frameworks—such as Uncertainty Quantification (UQ), advanced data assimilation, and Model Predictive Control (MPC)—that guarantee the safety and optimality required to intervene in actual physical environments.

Finally, we discuss the role of mathematics as a co-creator by introducing the vision of the Center for Generative Science at Kyoto University. In an era where AI assists in automated theorem proving and conjecturing, we aim to rapidly implement these mathematical breakthroughs into society. Furthermore, through interdisciplinary collaboration with philosophy, literature, and law, we seek to understand the profound psychological and philosophical roots of mathematical invention by humans.

This talk envisions a future where mathematical sciences are not rendered obsolete by AI, but rather serve as the indispensable universal language to decode, control, and co-create reality alongside it.