SPEAKERS

Title: Towards a New Mathematical Foundation for Scientific Machine Learning:

Bridging Mathematics, Engineering, and Industrial Innovation

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

Title: Our changing climate: the challenge of obtaining data and the role of conceptual modelling

Abstract: Climate Science is a field that concerns a wide range of time scales, from millennia for glaciation cycles all the way to months when it comes to the prediction of the next El Niño event. This range goes hand-in-hand with a massive difference in the available data. Long-term data is available only indirectly from ice and sediment cores, while shorter-term data is now becoming increasingly available from different types of measurement systems, such as ocean buoys and satellite observations. 

A crucial issue is that there is only one experiment: the actual development of the climate on Earth. This is why the simulation of climate models plays the very important role of providing options for experimentation and the study of different scenarios. There exists a hierarchy of models that stretches all the way to sophisticated high-resolution global circulation models (GCMs) that run on supercomputers. With the presently available and ever increasing computing power there has been an explosion of available data from such models. 

Large climate models and especially GCMs are effectively black boxes that do not allow one to ‘look under the bonnet’. So what can one say about the accuracy of climate simulations? We will discuss two complementary approaches to this question with the example of the El Niño Southern Oscillation system. 

Conceptual models capture only certain core properties and are mathematically tractable, and they have a vital role to play in identifying underlying mechanisms behind observed behaviour, such as the irregular occurrence of El Niño events. Such insights are not only of great interest at a fundamental level, but they also meant to filter back into every level of the climate model hierarchy.

At `the other end' are machine learning and AI techniques. They are able to work with enormous data sets, and this has made it possible to study the physics of the Earth's climate system in novel ways and in considerably more detail. These techniques have been used to extended and refine observational data records, and to overcome `predictability barriers' when forecasting phenomena on a human timescale, including when the next El Niño will take place and how strong it will be. 

Title: Beyond the Foundation: The New Roles of Mathematical Sciences in the Generative AI Era

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.