Workshop on Dynamical Systems: Theory, Modelling and AI 19–21 November 2025 

Day-3 programme

Modern Computational Approaches

Computer Algebra Systems in DEs Research, Neural DEs and Physics-Informed Neural Networks, From Data to DEs Models and Reduced Order Modeling & Surrogate Modeling with AI

Workshop Abstracts

Speaker: Francesco Petruccione (Director of the NITheCS, Pro-VC: Quantum Technology and Artificial Intelligence, Stellenbosch University, Stellenbosch)

Bio: Francesco Petruccione serves as Professor of Quantum Computing and Director of the National Institute for Theoretical and Computational Sciences (NITheCS) at Stellenbosch University. He was born in Genova, Italy, and began his journey in the field of physics at the University of Freiburg, where he received both his PhD and “Habilitation” degree. Following his academic training, Francesco held various positions at the University of Freiburg until his promotion to adjunct Professor in 2001. His pursuit of quantum physics led him to South Africa, where he was appointed Professor of Theoretical Physics at the University of KwaZulu-Natal (UKZN) in 2004. Recognising his contribution to the field, he was granted the South African Research Chair for Quantum Information Processing and Communication in 2007. At UKZN, Francesco also served as the Pro-Vice-Chancellor for Big Data and Informatics and he still holds a fractional Professorship at the university. In 2020, he was tasked to set up the National Institute for Theoretical and Computation Sciences, serving as its interim Director until he was appointed director in 2024. Francesco is an elected member of various prestigious scientific societies, including the Academy of Sciences of South Africa, the Royal Society of South Africa, and the African Academy of Sciences. He has published around 250 papers in refereed scientific journals and has co-authored significant texts on quantum systems and machine learning with quantum computers.

Title: Introduction to matching flows 

Speaker: Shane Josias (Stellenbosch University, Stellenbosch)

Bio: Dr. Shane Josias is a Lecturer in Applied Mathematics at Stellenbosch University, South Africa. He recently completed his Ph.D. in Applied Mathematics, focusing on computational efficiency and likelihood reliability in continuous-time normalising flows. His research interests include deep generative models, particularly flow matching, as well as applications in computer vision. Before entering academia, he worked as a software engineer in the telecommunications sector, developing complex event-processing systems and contributing to database integration and optimisation.

Title:  Applications of machine learning for simulations

Speaker: Hugo Touchette (Stellenbosch University, Stellenbosch)

Bio: Prof. Hugo Touchette is a professor at Stellenbosch University. His research covers many areas of applied mathematics and theoretical physics, driven by an unhealthy interest in anything random and unpredictable: noisy dynamical systems, turbulence, coin tossing, finance, the weather, etc. He trained as a physicist but now work mostly at the interface of probability theory, statistics, simulation, optimization, control, and, more recently, machine learning. His main specialty is the theory of large deviations - a branch of probability theory used to estimate the probability of very rare events arising in random systems as diverse as gases, queues, random walks, information systems or nonequilibrium systems driven by noise and external forces. He has written a review article on the many applications of large deviation theory in statistical physics.

Abstract: The recent development of machine learning (ML) models and techniques has had a profound impact on the way that we now do research in science. Many ML models are available for modelling many different systems and, significantly, many different packages (e.g., PyTorch, Keras, TensorFlow, JAX, etc.) have been developed and adopted for implementing these models and other ML techniques. In this talk, I will discuss the use of these packages for modelling and simulating stochastic processes. One of the important features that is now available in many ML packages is the ability to apply automatic differentiation to programs and simulations directly without coding overhead. I will discuss the application of this feature for computing the response of stochastic models and for simulating rare events, two current projects I'm working on.

Title: Ultraspherical Spectral Methods

Speaker: Nick Hale (Stellenbosch University, Stellenbosch)

Bio: Prof. Nick Hale is an Associate Professor of Applied Mathematics at Stellenbosch University. His research focuses on fast algorithms for computations involving orthogonal polynomials, including spectral methods for differential and integral equations, efficient computation of Gauss quadrature nodes and weights, and fast polynomial transforms.

Abstract: Spectral methods based on orthogonal polynomials provide exceptional accuracy for differential equations, especially when very high precision is required. This talk introduces Chebyshev spectral methods for ordinary differential equations and outlines their basic principles before presenting the ultraspherical spectral method, which generalises the approach through systematic polynomial basis conversions. The resulting sparse, well-conditioned formulation offers a unified framework for ordinary, partial, integro-, and fractional differential equations. The talk concludes with recent advances in the Ultraspherical Element Method, which combines spectral accuracy with the geometric flexibility of finite elements for solving PDEs on complex domains.

Title: Laguerre spectral collocation

Speaker: Emma Nel (Stellenbosch University, Stellenbosch)

Bio: Emma Nel is a doctoral researcher in Applied Mathematics at Stellenbosch University and a recipient of the Wilhelm Frank Scholarship. Her research focuses on numerical methods for solving differential equations, with particular emphasis on developing fast and stable spectral and pseudospectral methods. In her PhD, she works on the formulation of stable Laguerre collocation methods and their adaptation to problems on the half-line, where the choice of nodes, scaling, and implementation plays a critical role.

Abstract: In this work, we investigate the use of Laguerre spectral collocation methods for solving differential equations on the semi-infinite line. While spectral methods are known for their high accuracy, the unbounded domain and the exponential weighting associated with Laguerre polynomials present notable challenges. For example, the direct computation of the entries in the Laguerre spectral differentiation matrix leads to underflow and overflow issues in IEEE-754 arithmetic for large degrees N, even though the entries themselves are well-behaved. We discuss recent progress in addressing these numerical difficulties and explore approaches for improving convergence. By employing these techniques, one can achieve significantly higher values of N in Laguerre spectral methods, ultimately leading to better numerical results.

Title: Explainable Models for Dynamical Systems in STEM — Work in Progress (Concept & Early Scan)

Speaker: Anass Bayaga (Stellenbosch University, Stellenbosch)

Bio: Prof. Anass Bayaga is an internationally recognised scholar and innovator in the Faculty of Education at Stellenbosch University, where he serves in the Department of Curriculum Studies. His research spans Cognitive (Neuro) STEM Enhancement, Human–Computer Interaction, and emerging educational technologies, including AI, VR, AR, and machine learning. His work is dedicated to transforming learning experiences, particularly in under-resourced communities, through advanced, technology-driven pedagogical innovation.

Abstract: This work‑in‑progress sets out a simple plan for models that both predict how systems change and explain why. Objective: learn from small datasets while obeying basic rules (for example, energy is conserved). Method: pair a small neural “change‑over‑time” model with clear rules (maths/logic), then translate patterns into short equations. Evidence base: a scan of recent studies plus small pilots. Examples: a swinging pendulum, a filling‑and‑draining water tank, predator–prey cycles, and a cart–pole balance task. Expected outcomes: (1) readable plots that answer “what if we change X?” (2) equations that match the plots and known rules; (3) good accuracy with less data. Evaluation: three checks—do results respect the rules, do “what‑if” runs behave sensibly, and do models trained on one topic help on another (algebra → calculus → mechanics). Contribution: a short, practical template—models, rules, and metrics—plus code and a class‑ready checklist for data‑scarce settings.

Title: Heteroclinic Networks Meet the Themes of this Workshop

Speaker:  Gray Manicom (Stellenbosch University, Stellenbosch)

Bio: Gray Manicom is a researcher in Digital Transformation within the School for Data Science and Computational Thinking at Stellenbosch University. He is also affiliated with the Deputy Vice-Chancellor (Research, Innovation and Postgraduate Studies) research group.

Abstract: A useful tool for modelling intermittent behaviour using systems of ODEs is a heteroclinic network, consisting of invariant sets connected by trajecto- ries called heteroclinic connections. For example, Figure 1 represents a het- eroclinic network containing four saddle equilibria (A, B, C and D) which make two heteroclinic cycles: an A-B-C cycle and an A-D-C cycle. While these networks are typically not robust, in systems with symmetries invari- ant subspaces may appear which contain robust heteroclinic connections. Conveniently, many real-world systems have physical constraints resulting in these types of symmetries, so that heteroclinic cycles and networks have been used to model systems in thermal dynamics, ecology, population dynamics, climate change, psychology and more. A key feature of these systems is that local dynamics at one saddle may effect the long-term dynamics of the network. For example, if we associate the states of a stochastic process with a trajectory being “near an invariant set” (so that we may observe the stochastic process ABCADC from the heteroclinic network in Figure 1), then changing the eigenvalues at a saddle equilibrium may change the transition probabilities of the stochastic process (for example, the probability of observing from AB or AD) and the amount of time each state is active. In this presentation we introduce heteroclinic networks and their appli- cations. We discuss how machine learning techniques have been used to fit the dynamics near heteroclinic networks to target data. For example, re- inforcement learning has been used for agents moving in a 2d environment by associating movement states with saddle equilibria in a heteroclinic net- work and dynamically optimising the eigenvalues. We propose that these techniques can be developed so that, given an arbitrary directed graph (with no 1-loops) with known transition probabilities and residence times, one can construct a system of ODEs containing a heteroclinic network and optimise the parameters such that the associated stochastic process dynamics with matching transition probabilities and residence times.