This workshop will bring together mathematicians working on computational dynamics, spanning a wide range from computability theory to data driven modelling. The workshop will ask questions when and how dynamical systems can be used to perform computation in the sense of Turing machines, including undecidability. Recent advances have shown that fluid flows can in principle be used to perform computations, implying that their future dynamics will be unpredictable. A probabilistic approach to dynamical systems which aims at understanding its statistical properties rather than the exact evolution of a single trajectory, offers an alternative to overcome the predictability barrier of certain dynamical systems. Set-oriented methods allow for the characterization of global statistical properties and detection of coherence in the flow, when the dynamical system is accessible only through some partial observations. The convergence properties of these methods have only recently been explored, giving rise to improved algorithms.
Organizers
University of Sydney
Universitat Politècnica
de Catalunya
Universidad Católica de Chile
Speakers
Semi-plenary speakers
ICMAT Madrid
University of Chicago
Invited speakers
Ecole Polytechnique
Jagiellonian University
University of Chicago
University of Augsburg
University of Cambridge
University of Augsburg
Pennsylvania State University
University of Algarve
MPI SWS Saarbrücken
Universidad Complutense de Madrid
University of Potsdam
IHES
Université Paul Sabatier
University of Turku
Mississippi State University
Santa Fe Institute
University of Sydney
Cornell University
Thursday, 16 July
![[HS 13 – 2nd Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/HS13.png){kind=link}
14:00-15:00 semi-plenary talk
Daniel Peralta-Salas (ICMAT Madrid)
I will survey the recent constructions of stationary solutions to the Euler and the Navier–Stokes equations on certain Riemannian 3-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. The proofs make use of tools from cosymplectic and contact geometry. This is based on joint works with Robert Cardona, Soren Dyhr, Angel Gonzalez-Prieto and Eva Miranda.
15:00-15:30
Angel González Prieto (Universidad Complutense de Madrid)
The possibility of encoding computation in continuous dynamical systems has fascinated mathematicians and computer scientists since the earliest models of computation. In recent years, this connection has attracted renewed attention, partly due to T. Tao’s programme of searching for blow-up solutions to the Navier-Stokes equations through the embedding of a computer into the dynamics.
In this talk, we will discuss what it means for a dynamical system to be Turing complete. We will survey recent advances on Turing-complete dynamical systems, including the existence of Turing-complete solutions to the Euler and Navier-Stokes equations on arbitrary 3-manifolds, and we will reflect on other systems that may potentially exhibit Turing completeness. Time permitting, we will also present a novel approach to computability in dynamics, inspired by Topological Quantum Field Theory, in which computable functions are represented by volume-preserving vector fields on bordisms.
15:30-16:00
David Wolpert (Santa Fe Institute)
16:00-16:30 Coffee Break
16:30-17:00
Matthew Colbrook (University of Cambridge)
Koopman operator theory offers a compelling route from nonlinear dynamics to linear spectral problems, but this transformation is infinite-dimensional and therefore delicate: finite-dimensional discretisations can converge to the wrong object, while finite data may be compatible with dynamically very different systems. In this talk I will discuss a foundations programme for Koopman learning at the interface of computational dynamics, numerical analysis, and computability theory. I will explain how adversarial dynamical systems and solvability-complexity ideas reveal intrinsic limits on what can be recovered from trajectory data, including spectral information relevant to prediction. I will then describe complementary positive results: residual and structure-preserving algorithms that provide verifiable spectral information, convergence guarantees, and a posteriori error bounds for forecasting and dictionary learning. The goal is to distinguish what is impossible, what is merely ill-conditioned, and what can be made trustworthy in practical data-driven modelling.
17:00-17:30
Daniel Graça (Universidade do Algarve)
The iteration of quadratic maps is well-known to originate complex dynamical behavior such as chaotic behavior. In the 1990s Moore asked whether the iteration of analytic maps would be rich enough to not only generate chaotic behavior, but also to be Turing universal on compact sets, and conjectured that the answer was negative. A recent breakthrough by Cardona, Miranda, and Peralta-Salas has shown that the answer to this question is surprisingly positive and that a Turing complete analytic diffeomorphism exists on the sphere S^{17}. In this talk we further explore this question. In particular we show that, under some mild conditions, analytic maps cannot simulate Turing machines with no slowdown in the compact space [0,1]^n with point-to-point encodings of configurations.
17:30-18:00
Kai Cieliebak (University of Augsburg)
This talk is about the following recent result: Given any contact 3-manifold and any computable function f, there exists a defining contact form and a Poincare section of its Reeb flow whose partially defined return map computes f.
18:00-18:30
Ville Salo (University of Turku)
A $d$-dimensional cellular automaton acts on $\mathbb{Z}^d$-configurations over a finite alphabet $A$, by repeatedly rewriting all the cells in parallel, by a fixed spatiotemporally homogenous local rule. Cellular automata form a class of finitely-presented dynamical systems on Cantor spaces $A^{\mathbb{Z}^d}$. Dynamical properties of these systems can be classified in the lightface hierarchy. We give a light survey of known classification results, touching on properties such as sensitivity, almost equicontinuity, reversibility, measures, and nilpotency.
Friday, 17 July
14:00-14:30
Urs Frauenfelder (University of Augsburg)
In the study of families of periodic orbits it is important to distinguish left from right. Retrograde periodic orbits have a completely different behaviour than direct orbits. However, the spectrum of a symplectic matrix cannot distinguish left from right. In joint work with Agustin Moreno we showed how to associate to a symmetric periodic orbit two B-signs. These B-signs lead to an enhancement of the classical Broucke diagram. For elliptic eigenvalues they correspond to the Krein sign. However, they cannot just distinguish left from right for elliptic eigenvalues, but even lead to a notion of left and right for hyperbolic eigenvalues. These B-signs play as well an important role to understand bifurcations of families of periodic orbits and in particular about the behaviour of Maslov indices at bifurcations. These Maslov indices for families are crucial in the recent global approach to networks of periodic orbits using bifurcation graphs due to Aydin and Bathkin.
14:30-15:00
Caroline Wormell (University of Sydney)
Transfer operators, Chebyshev interpolation and the Apollonian gasket
Many properties of chaotic dynamics and fractals can be studied in terms of the spectral properties of transfer operators, weighted composition operators with respect to the dynamics. These include mixing rates, statistical response to perturbations, and fractal dimensions. For relatively simple dynamics, the current best approach is via interpolation using Chebyshev polynomials. These methods are highly accurate and have a pleasingly straightforward theory.
More recently, it has become relatively simple to produce validated spectral computations from these calculations. We have recently applied these techniques to the 50-year problem of a rigorous calculation of the Hausdorff dimension of the Apollonian gasket, which is generated by an expanding map with an infinite number of pre-images. Many other well-known low-dimensional systems may be susceptible to similar approaches.
15:00-15:30
John Harlim (Pennsylvania State University)
In this talk, I will discuss a kernel ridge regression (KRR) framework with dynamics-aware validation for long-horizon prediction in complex dynamical systems. Our Diffusion Maps (DM) kernel adapts implicitly to the intrinsic geometry of the system’s invariant set, without explicit manifold reconstruction or attractor modeling, both of which often constrain predictive performance. Under a manifold assumption with uniformly sampled data, we prove that the RKHS induced by the DM kernel converges to the RKHS induced by the heat kernel of the Laplace-Beltrami operator, providing a theoretical explanation for its strong performance across systems with smooth underlying manifolds. Beyond this well-understood regime, including chaotic attractors and high-dimensional spatiotemporal flows, DM-KRR consistently outperforms state-of-the-art random-feature, neural-network, and other operator-learning methods in both accuracy and data efficiency, even under noisy observations
15:30-16:00
Yunan Yang (Cornell University)
Measures provide valuable insights into long-term and global behaviors across various dynamical systems. In this talk, I present our recent work that employs measure theory and optimal transport to address challenges in dynamical system identification. First, we adopt a PDE-constrained optimization perspective to learn ODEs and SDEs from slowly sampled trajectories, enabling stable forward models and uncertainty quantification. We use optimal transport to align physical measures for parameter estimation, even when time-derivative data is unavailable. We also extend the celebrated Takens’ time-delay embedding, a foundational result in dynamical systems, from state space to probability distributions. It establishes a robust theoretical and computational framework for state reconstruction that remains effective under noisy and partial observations. Finally, we show that by comparing invariant measures in time-delay coordinates, one can overcome identifiability challenges and uniquely recover the underlying dynamics. These works demonstrate the excellent research potential of measure-transport approaches for data-driven dynamical modeling.
16:00-16:30 Coffee Break
16:30-17:30 semi-plenary talk
Nisha Chandramoorthy (University of Chicago)
In any Generative Model, the generated samples have a different distribution than the data distribution, due to inevitable learning errors. Moreover, this discrepancy, and metrics for evaluating the generated samples, are hard to characterize in high dimensions, motivating the need to understand how learning errors affect the reproducibility of certain “features” of the generated distributions. A first question is whether generative models produce “physical” samples, i.e., samples whose support is close to that of the true target distribution, despite algorithmic errors. A second question concerns what we term a lazy generative model: given samples from the target, we apply an arbitrary random dynamical system such that the distribution at finite time is approximately Gaussian. In principle, this noising process cannot be exactly inverted to recover target samples—but under what conditions can we approximately recover samples from a nearby distribution?
The first part is joint work with Adriaan de Clercq (UChicago) and the second with Georg Gottwald (U Sydney).
17:30-18:00
Sebastian Reich (University of Potsdam)
Optimal control for fully observed diffusion processes is well established and has led to numerous numerical implementations based on, for example, Bellman’s principle, model free reinforcement learning, Pontryagin’s maximum principle, and model predictive control. On the contrary, much fewer algorithms are available for optimal control of partially observed processes.
However, this scenario is central to the digital twin paradigm where a physical twin is partially observed and control laws are derived based on a digital twin. In this talk, we contribute to this challenge by combining data assimilation in the form of the ensemble Kalman filter with the recently proposed McKean–Pontryagin approach to stochastic optimal control. The proposed methodology is suited for real time applications of digital twins. We present numerical results for a controlled Lorenz-63 system, a controlled Lorenz-96 model, and a controlled inverted pendulum.
18:00-18:30
Christian Wolf (Mississippi State University)
Computability in dynamical systems is a relatively young field that has attracted significant attention in recent years. One of its central questions is whether dynamically relevant objects can be algorithmically represented by a Turing machine. While this question has been extensively studied in symbolic dynamics, where computability results are known for various thermodynamical quantities such as entropy, pressure, equilibrium states and zero-temperature measures, a corresponding general theory for broader classes of topological and smooth dynamical systems is lacking.
In this talk, we present an approach to bridging this gap by introducing the concept of computable Markov partitions. This framework allows us to establish far-reaching computability results for several classes of topological and smooth dynamical systems. The results presented in this talk are joint work with Michael Burr and Tamara Kucherenko.
Saturday, 18 July
14:00-15:00
Cristóbal Rojas (Universidad Católica de Chile)
This talk concerns the computational abilities of continuous systems when regarded as computing devices. The subject has a long history, encompassing systems from classical dynamical systems to modern AI architectures such as Transformers—the neural architectures underlying contemporary large language models such as ChatGPT and Claude. Much of the literature asks whether a given class contains a system capable of universal computation. Yet universality is only a first, rather coarse, distinction. How prevalent is maximal computational power within a class? Which structural or dynamical features make it possible? And, in the case of a trained language model, what computation is actually performed when the model is prompted to solve a problem—and by what internal mechanism is it realized? I will present recent results addressing these questions, moving from the distribution of computational capacity within one parameter families of simple dynamical systems, to the resources and learned mechanisms that govern computation in modern language models.
15:00-15:30
Mathieu Sablik (Université Paul Sabatier)
A central question in computational dynamics is the following: Given the initial data and the infinitesimal evolution rule of a dynamical system, what can we determine about its long-term behavior? One of the fundamental objects that captures asymptotic dynamics is the non-wandering set, namely the collection of points that the system can never completely leave behind. In general, whenever a class of dynamical systems is capable of simulating a universal Turing machine, its non-wandering set is non-computable. In this talk, we present dynamical conditions, including Axiom A diffeomorphism, under which the non-wandering set becomes computable. We also show that, within certain classes of dynamical systems, a Baire-generic system has a computable non-wandering set.
15:30-16:00
Olivier Bournez (Ecole Polytechnique)
How hard is it to solve an ordinary differential equation? Several traditions bear on this question, each with its own answer.
Algorithmic methods can be extremely efficient: Newton-type approaches solve polynomial ODEs over Q[[X]] in quasi-linear time. Analog models of computation show that polynomial ODEs and Turing machines are two presentations of the same phenomenon, with solution length acting as time and precision as space. Computable analysis, by contrast, shows that ODEs can be intrinsically hard, even
PSPACE-complete, and in some cases undecidable over compact domains.
Comparing these traditions is natural and necessary, yet such comparisons routinely reduce to comparisons of encodings rather than of underlying algorithmic content.
We argue that reverse mathematics is a relevant tool in this setting: it provides a representation-invariant lens in which algorithmic content is compared directly. We prove that every level of the Big Five hierarchy of reverse mathematics is inhabited by a natural statement from classical ODE theory, as an exact equivalence rather than an analogy: the regularity of the vector field ff
f is an intrinsic algorithmic invariant placing the initial value problem y'(t)=f(t,y(t)),
y(t_0)=y_0 into one of several computational strata, ranging from polynomial-time solvability to transfinite computation.
The resulting stratification acts as a diagnostic common to the three traditions. By abstracting from representation, it separates fundamental barriers from the technical shortcomings of solvers, the representation artefacts of analog encodings, and the effectivity constraints of computable analysis, and identifies the intrinsic parameters (length bounds, radii of convergence, moduli of continuity) under which feasibility is restored.
16:00-16:30 Coffee Break
16:30-17:00
Semon Rezchikov (IHES)
What can differentiable dynamical systems compute? How does one formalize the idea that a class of differentiable dynamical systems, which typically include smooth maps which often exhibit properties that are not themselves computable, can nonetheless only solve problems that lie in a certain classical complexity class due to comparatively “simple” dynamics? In order to give meaning to such questions, one must consider an encoding/decoding scheme between classical computational systems and the differentiable system, and it becomes evident that such questions are only meaningful when one constrains the complexity of the encoding/decoding scheme itself. I will explain a formalism which achieving this goal, some analogies to classical complexity theory, some results about Axiom A and generic 1 dimensional systems that can be obtained in this formalism, as well as several different directions for further investigation. Based on joint work with Jordan Cotler.
17:00-17:30
Joël Ouaknine (MPI SWS Saarbrücken)
It has been described as the most important problem whose decidability is still open: the Skolem Problem asks how to determine algorithmically whether a given integer linear recurrence sequence (such as the Fibonacci numbers) has a zero term. This deceptively simple question arises across a wide range of topics in computer science and mathematics, from program verification and automata theory to number theory, dynamical systems, and logic. This talk traces the history of the Skolem Problem: from the early 1930s to the current frontier of one of the most enduring open questions in computer science.
17:30-18:00
Nicanor Carrasco (Jagiellonian University)
A countable group has the strong topological Rokhlin property when the topological space of its actions/dynamical systems over the Cantor space admits a residual conjugacy class. Kechris and Rosendal established that the group of integers has the this property, and a classification of countable groups having the same property remains an open problem. The goal of this talk is presenting a sufficient condition for the failure of this property in terms of algorithmically complex subshifts of finite type.
Posters
![[PZ13 – 2nd Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/poster_zones_2-stock.png){kind=link}
Irene Erazo Estrada
(Tulane University)
This study models the dynamic behavior of small spherical particles immersed in a viscous fluid under the influence of thermal fluctuations. Using the Regularized Stokes Images outside a sphere, we perform theoretical and numerical analyses of particle diffusion to characterize their motion across varying particle sizes. The objective is to determine whether the fluid–particle coupling for rigid spheres adheres to the Stokes Einstein scaling relation.
Samuel Everett
(University of Chicago)
We present a method for studying dynamical systems using concepts from computational complexity theory, and conversely. The technique centers on associating of families of decision problems, called telic problems, to dynamical systems of a certain class. These decision problems formalize finite-time reachability questions for the dynamics. We find that the computational complexity of solving telic problems communicates information about the entropy of the dynamical system: exponential runtime lower bounds imply positive topological entropy. In the other direction, the behavior of a dynamical system constrains the type of algorithm that can solve its telic problems: certain “natural” approaches can be ruled out.
Isao Ishikawa
(Kyoto University)
We develops a functional-analytic framework for approximating the push-forward induced by an analytic map from finitely many samples. Instead of working directly with the map, we study the push-forward on the space of locally analytic functionals and identify it, via the Fourier–Borel transform, with an operator on the space of entire functions of exponential type. This yields finite-dimensional approximations of the push-forward together with explicit error bounds expressed in terms of the smallest eigenvalues of certain Hankel moment matrices. Moreover, we obtain sample complexity bounds for the approximation from i.i.d.~sampled data. As a consequence, we show that linear algebraic operations on the finite-dimensional approximations can be used to reconstruct analytic vector fields from discrete trajectory data. In particular, we prove convergence of a data-driven method for recovering the vector field of an ordinary differential equation from finite-time flow map data under fairly general conditions.
Bernd Krauskopf
(University of Auckland)
Periodic forcing is a common phenomenon that is generally described by models in the form of a vector field with a periodically varying term. As a specific example, we consider a periodically forced planar vector field for (scaled) temperature and salinity in the North Atlantic Ocean, which is a conceptual model for the crucial vertical mixing process between warmer surface water and cold, deep water. The observed behaviour depends on three parameters: the density threshold η, the virtual salinity flux μ, and the strength c of the forcing due to seasonal variation in glacial meltwater. When c ⩾ 0 there is dynamics on an invariant torus, which may be phase-locked or quasiperiodic. To find the structure of resonance tongues with locked periodic orbits, we present an algorithm to compute the rotation number ρ over the corresponding region of the (c, μ)-plane; moreover, we find and continue individual curves of saddle-node bifurcations that bound selected (lower-order) resonance tongues. Since ρ is locally constant at rational values (where the dynamics is locked), its graph consists of devil’s terraces with steps between them along curves of irrational values of ρ. We present and discuss generic bifurcations of resonance tongues — that is, local changes to devil’s terraces — that explain how the overall resonance structure changes with the parameter η.
Rachel Morris
(Concordia University)
From fluid flows to chaotic systems, the Koopman operator has emerged as a powerful tool to analyze dynamical systems and forecast data. Modern data-driven methods such as Extended Dynamic Mode Decomposition (EDMD) provide a Galerkin approximation of the Koopman operator on a finite dictionary of user-defined functions. However since the quality of the approximation depends greatly upon the dictionary choice, a more recent approach to approximating the Koopman operator trains a deep autoencoder neural network to learn the basis functions from data. Universal approximation theorems justify the use of deep neural networks (DNNs) to learn the dictionary, but the network architecture, training plan, and amount of data required for high-quality approximations all merit further investigation. In this poster/presentation, I will provide rigorous performance guarantees for learning the Koopman operator by DNNs via a sparsity-promoting training program. In particular, I will give insight into the accuracy and sample complexity of neural network Koopman approximations by providing generalization bounds based on recent function approximation techniques. Furthermore, I will highlight a DNN architecture and training routine that achieves these theoretical benchmarks in practice.
Isaac Ramos
(ETH Zurich)
Billiard systems span the full spectrum of dynamical behavior, from integrable motion to chaos. This remarkable diversity raises a natural question: can billiard dynamics realize the power of computation? In this talk, we will answer this question by showing that two-dimensional billiard systems are Turing complete. Our approach adapts the machinery from Topological Kleene Field Theories, allowing us to encode arbitrary Turing machines into billiard dynamics. As a consequence, we obtain billiard models exhibiting undecidable dynamics properties, such as the existence of certain types of periodic orbits. Beyond abstract billiards, this result has also implications for billiard-type natural systems, including collision dynamics in the n-body problem. Joint work with Eva Miranda.
Madalena Roque
(University of Manchester)
The stochastic Euler dynamics are a continuous-time Markov process introduced by [Latz 2025; IMA J. Numer. Anal., draf096] that is obtained when linearly interpolating a forward Euler trajectory with independent, exponentially distributed waiting times. Random timestep Euler methods have applications in chaotic dynamics, Carathéodory differential equations, and optimisation. The goal of the present work is to prove strong convergence of the Stochastic Euler Dynamics to the underlying ODE, complementing previous weak convergence proofs. Indeed, for an ODE of type u’ = f(u) we show local convergence if f is locally Lipschitz continuous and global convergence if f is globally Lipschitz continuous. We obtain error estimates for local and global error bounds. We also extend a previous stability analysis using Foster-Lyapunov criteria and illustrate our results in numerical experiments.
Yuanchao Xu
(Kyoto University)
We propose Koopman Spectral Wasserstein Gradient Descent (KSWGD), a particle-based generative modeling framework that uses Koopman spectral analysis to construct a data-driven approximation of the inverse Langevin generator and then deploys this approximation within Wasserstein gradient descent. The key observation is that, for overdamped Langevin dynamics with invariant law $\pi$, the Koopman generator equals the negative Langevin generator. Hence, leading spectral information of the target distribution can be estimated from trajectory data without explicit access to the potential $V$. This constructs a deterministic particle sampler that is particularly useful when high-dimensional observations are governed by lower-dimensional dominant dynamics. We prove exponential convergence and error bounds for the continuous-time population dynamics, separating the contributions of spectral truncation and Koopman approximation error. We also give a Feynman–Kac interpretation of the Koopman construction. Experiments on compact manifolds, metastable multi-well systems, and a high-dimensional stochastic partial differential equation show that KSWGD improves convergence speed and sample quality relative to other tested baselines.
Štěpán Zapadlo
(University of Graz)
We consider the problem of identifying an ODE from data sampling values of the state variables. The de facto standard tool for this task, the Sparse Identification of Nonlinear Dynamics (SINDy), in its original formulation, requires measurements of derivatives of the state variables of the system of interest. While various techniques to estimate these derivatives exist in the literature, all exhibit significant sensitivity to input noise. We propose a novel formulation of SINDy based on optimal control theory, which eliminates the requirement for the derivative measurements at the cost of higher computational complexity. In particular, we compare special cases of our framework to existing methods addressing the same issue. Lastly, we examine various properties of this reformulation both analytically and via numerical experiments.