### RESEARCH TALKS

#### Tyrus Berry (George Mason, USA)

Limits of learning dynamical systems

## Abstract

A dynamical system is a transformation of a phase space, and the transformation law is the primary means of defining as well as identifying the dynamical system. It is the object of focus of many learning techniques. Yet there are many secondary aspects of dynamical systems – invariant sets, the Koopman operator, and Markov approximations, which provide alternative objectives for learning techniques. Crucially, while many learning methods are focused on the transformation law, we find that forecast performance can depend on how well these other aspects of the dynamics are approximated. These different facets of a dynamical system correspond to objects in completely different spaces – namely interpolation spaces, compact Hausdorff sets, unitary operators and Markov operators respectively. Thus learning techniques targeting any of these four facets perform different kinds of approximations. We examine whether an approximation of any one of these aspects of the dynamics could lead to an approximation of another facet. Many connections and obstructions are brought to light in this analysis. Special focus is put on methods of learning of the primary feature – the dynamics law itself. The main question considered is the connection of learning this law with reconstructing the Koopman operator and the invariant set. The answers are tied to the ergodic and topological properties of the dynamics, and reveal how these properties determine the limits of forecasting techniques.

#### Bree Cummins (Montana State, USA)

Matching data from multiple yeast cell cycle experiments to a genetic network model

## Abstract

Modeling biological systems holds great promise for speeding up the rate of discovery in systems biology by predicting experimental outcomes and suggesting targeted interventions. However, this process is dogged by an identifiability issue, in which network models and their parameters are not sufficiently constrained by coarse and noisy data to ensure unique solutions. In this work, we evaluated the capability of a simplified yeast cell-cycle network model to reproduce multiple observed transcriptomic behaviors under genomic mutations. We matched time-series data from both cycling and checkpoint arrested cells to model predictions using an asynchronous multi-level Boolean approach. We showed that this single network model, despite its simplicity, is capable of exhibiting dynamical behavior similar to the datasets in most cases, and we demonstrated the drop in severity of the identifiability issue that results from leveraging multiple datasets.

#### Pawel Dlotko (Polish Academy of Sciences, Poland)

Topological data analysis methods for understanding dynamical systems

## Abstract

In this talk, I will present a survey of topological data analysis (TDA) techniques that are particularly useful for understanding dynamical data, especially when dealing with sampled dynamics. We will begin with an introduction to persistent homology, a standard tool for distinguishing state spaces of different dynamical systems. From there, we will explore more advanced approaches, such as Euler characteristic curves and profiles, combined with goodness-of-fit tests. Finally, I will introduce conjugacyTests, a family of methods designed to assess the conjugacy of two finitely sampled trajectories. This talk will cover the fundamental tools and showcase examples of their application in the theory of dynamical systems.

#### Gary Froyland (University of New South Wales, Australia)

Emergence of macrophenomena in complex dynamics

## Abstract

Complex dynamics can display emergence. There may be simple (e.g. physics-based) microrules that describe how individual states interact and evolve, but collectively, at the level of many states, complicated macrophenomena may emerge. It is typically these macrophenomena that are observable and/or affect our daily lives. Macrophenomena can often be characterised as large collections of individual states that evolve as a group for a substantial time duration. To access these macrophenomena, we will use linear operators induced by the nonlinear microdynamics.

#### Marcio Gameiro (Rutgers, USA)

Computing dynamics of data with probabilistic guarantees

## Abstract

We present a combinatorial topological method to compute the dynamics of data. The method constructs a Gaussian process from the data and uses combinatorial methods