### EXPOSÉS

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

Limits of learning dynamical systems

## Résumé

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

## Résumé

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

## Résumé

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

## Résumé

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

## Résumé

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 to discretize the state space of the system and construct a combinatorial representation of the model. This combinatorial representation is used to extract the dynamics of the system. The dynamics is validate via algebraic topological methods. The predictive distribution of the Gaussian process model is used to provide lower bounds on the probabilistic confidence level of the computed dynamics. We present examples where we can achieve high confidence levels on the computed dynamics from sparse data.

#### Dimitris Giannakis (Dartmouth College, USA)

Tensor network approximation of Koopman operators

## Résumé

A key structural property of Koopman (composition) operators of dynamical systems is that they are multiplicative on spaces of observables with a product structure. This property implies, for instance, that the point spectrum is an abelian group and that the corresponding eigenfunctions also form a group under pointwise function multiplication. Computational approximation techniques for Koopman operators typically do not preserve multiplicativity in a strict sense, yet one might expect that for approximations that are sufficiently « close » to the original operator products of eigenvalues and eigenfunctions should be useful for building models for the evolution of observables. Using this as a working hypothesis, in this talk we present a scheme for approximating the Koopman evolution of observables that is based on a lift to a Fock space where regularized Koopman operators are multiplicative with respect to the tensor product. This Fock space is generated by a reproducing kernel Hilbert space of observables built such that it has the structure of a coalgebra with respect to the tensor product, and the structure of a Banach algebra with respect to the pointwise product of functions. The resulting approximation scheme can be cast in the form of a tree tensor network allowing for efficient computation in high-dimensional spaces generated multiplicatively from a modest number of approximate Koopman eigenfunctions. We illustrate this approach with applications to measure-preserving ergodic flows on tori.

#### Yasuaki Hiraoka (Kyoto University, Institute for Advanced Study, Japan)

scEGOT: Single-cell trajectory inference framework based on entropic Gaussian mixture optimal transport

## Résumé

We present scEGOT, a comprehensive single-cell trajectory inference framework based on entropic Gaussian mixture optimal transport. The main advantage of scEGOT allows us to go back and forth between continuous and discrete problems, and it provides a versatile trajectory inference method including reconstructions of the underlying vector fields at a low computational cost. Applied to the human primordial germ cell-like cell (PGCLC) induction system, scEGOT identified the PGCLC progenitor population and bifurcation time of segregation. Our analysis shows TFAP2A is not sufficient for identifying PGCLC progenitors, but requiring NKX1-2.

#### Oliver Junge (TUM, Germany)

Discretization of transfer operators from trajectory data

## Résumé

We propose a new concept for the approximation of transfer operators in dynamical systems using trajectory data. Using eigenfunctions of that operator, information about the macroscopic dynamics can be computed. Our approach is based on the entropically regularized optimal transport between two probability measures. In particular, we use optimal transport plans in order to construct a finite- dimensional approximation of the transfer operator which can be analysed computationally. We show that the spectrum of the discretized operator converges to the one of the regularized original operator and report on corresponding numerical experiments, including one based on the raw trajectory data of a small biomolecule from which its dominant conformations are recovered.

#### Miro Kramar (University of Oklahoma, USA)

Detecting shadowing events in turbulent flows

## Résumé

The idea that turbulence can be described as a deterministic walk through a repertoire of patterns goes back to Eberhard Hopf. Over the years it was established that these patterns closely correspond to exact coherent structures (ECS) which are often formed by unstable (relative) periodic orbits. Over recent years, a large body of numerical simulations and experiments indicated that the turbulent trajectory moves through the phase space from one ECS to another. The turbulent trajectory approaches the ECS along its stable manifold and leaves along its unstable manifold. It is therefore natural to ask whether these results are coincidental or whether some collection of ECSs can in fact provide a dynamical and statistical description of fluid turbulence. In order to properly answer this question one needs to be able to identify when the turbulent trajectory follows (shadows) a given ECS. However, in systems with continuous symmetries, detecting when the turbulent trajectory shadows a given ECS remains challenging and computationally very expensive. In this talk, we present a novel and computationally efficient approach for detecting the shadowing based on persistent homology. To demonstrate the potential of our method we apply it to the Kuramoto-Sivashinsky equation, which serves as a simple model that mimics some of the properties of fluid turbulence, such as spatiotemporal chaos, in a more accessible setting.

#### Rachel Kusle (Georgia Institute of Technology, USA)

Dynamical insights in identifying new data-driven mechanistic microbial models

## Résumé

We consider some recent model identification tools, together with complementary computations of dynamical characteristics that can often be necessary to isolate relevant biological mechanisms based on data. In recent studies of microbial dynamics, specifically community behavior of bacteria and dynamics under antibiotic treatment, the available data limits the extensive use of these tools. Nevertheless, we illustrate their utility, together with critical dynamical features, in identifying new biologically-relevant models that allow for heterogeneity and state-dependent features that are ubiquitous in the ecology and evolution of microbial dynamics.

#### Simon Mak (Duke, USA)

Recent developments in Bayesian surrogate modeling for accelerating scientific computing

## Résumé

Scientific computing is at a critical crossroad. While breakthroughs in computing technology enable the reliable simulation of complex phenonema (e.g., rocket dynamics), the expensive cost of simulations prohibits the exploration of such phenomena over large parameter spaces. Bayesian surrogate modeling provides a promising solution: it uses the expensive simulated data to train a probabilistic predictive model, which can be used to emulate and quantify uncertainty over the parameter space. We present two recent developments on this front. The first, called the Gaussian process subspace (GPS) model, targets the flexible probabilistic prediction of subspace-valued functions. The GPS can greatly accelerate the numerical simulation of dynamical systems via Galerkin projections over a broad parameter space. The second, called the Hierarchical Shrinkage Gaussian process (HierGP) model, targets the probabilistic prediction of a black-box simulator with highly limited samples. The HierGP embeds the principles of effect hierarchy, heredity, and smoothness widely used for analysis of experiments, which enable the accurate prediction of structured response surfaces with limited data. We demonstrate the effectiveness of these models in a suite of applications in aerospace engineering and dynamical systems recovery.

#### Elizabeth Munch (Michigan State University, USA)

Data driven homological approaches for detecting changes in dynamical systems

## Résumé

#### Nina Otter (INRIA)

A topological perspective on climate science

## Résumé

n this talk I will give an overview of some of the current challenges in climate science, and talk about how topology can help in addressing them. The talk is based on joint work with Kristian Strommen, Matthew Chantry, Joshua Dorrington, Hannah Christensen and António Leitão.

#### Jose Perea (Northeastern University, USA)

tba

#### Michael Schatz (Georgia Tech, USA)

Using exact coherent structures to describe dynamics and statistics of intermittent Taylor-Couette flow

## Résumé

A dynamical systems approach to understanding turbulence suggests that the complicated motion of turbulent flow is shaped by special solutions to the Navier-Stokes equations known as exact coherent structures (ECSs). In this picture, turbulent flow co-evolves with, or « shadows », ECSs for some period of time. This qualitative picture has recently been confirmed quantitatively at certain Reynolds numbers in experimental turbulent Taylor-Couette flow. Here we apply a similar methodology to an experimental Taylor-Couette flow that has the property of intermittency, wherein the flow exhibits a high degree of regularity for certain intervals (quiescence) punctuated at other times by significant increase in spatial and temporal activity of the flow. A collection of ECSs is found that describes the behavior of the flow for almost the entire duration of the flow, in both experimental and numerical investigations, for both quiescent and active behaviors of the flow. It is also demonstrated that the transition of flow behavior between quiescent and active regions is mediated by a specific ECS that connects both regions. Finally, observables of the flow are shown to be well-represented by weighted averages of ECS observables, thereby demonstrating the connection between dynamical shadowing of ECSs and average observables in a 3-dimensional experimental flow.

#### Shihao Yang (Georgia Tech, USA)

PIGP: Physics-Informed Gaussian Process

## Résumé

Partial differential equations (PDEs) are widely used for the description of physical and engineering phenomena. Some key parameters involved in PDEs, which represent certain physical properties with important scientific interpretations, are difficult or even impossible to measure directly. Estimating these parameters from noisy and sparse experimental data of related physical quantities is an important task. Many methods for PDE parameter inference involve a large number of evaluations for numerical solutions to PDE through algorithms such as the finite element method, which can be time-consuming, especially for nonlinear PDEs. In this paper, we propose a novel method for the inference of unknown parameters in PDEs, called the PDE-Informed Gaussian Process (PIGP) based parameter inference method. Through modeling the PDE solution as a Gaussian process (GP), we derive the manifold constraints induced by the (linear) PDE structure such that, under the constraints, the GP satisfies the PDE. For nonlinear PDEs, we propose an augmentation method that transforms the nonlinear PDE into an equivalent PDE system linear in all derivatives, which our PIGP-based method can handle. The proposed method can be applied to a broad spectrum of nonlinear PDEs. The PIGP-based method can be applied to multi-dimensional PDE systems and PDE systems with unobserved components. Like conventional Bayesian approaches, the method can provide uncertainty quantification for both the unknown parameters and the PDE solution. The PIGP-based method also completely bypasses the numerical solver for PDEs. The proposed method is demonstrated through several application examples from different areas.