After three successful TMS Seminars related to the theory of RPs with various perspectives, we wish again to invite several leading experts working in RPs theory as well as related algebraic structures and the mathematical theory of machine learning, to discuss recent trends, present novel results, and to encourage collaboration across the disciplines.

This year's topic will be "Rough paths, algebraic structures and machine learning", with a central focus on discussions and exchanges of ideas between researchers working on algebraic and analytic sides of RPs theory and researchers working with machine learning algorithms and applications of such.

• Harald Oberhauser - Oxford University, UK

• Joscha Diehl - Greifswald University, Germany

• Elena Celledoni - NTNU

• Fred E. Benth - University of Oslo

• Hans Zanna Munthe-Kaas - University of Bergen

• Subbarao Venkatesh Guggilam - NTNU

• Michele Coghi - University of Trento

• Silvia Lavagnini - Univeristy of Verona

• Darrick Lee - University of Oxford

Hans Zanna Munthe-Kaas:

Title: Introduction to postLie algebras and planarly branched rough paths

Abstract: PostLie algebras originated (among other works) in the study of integration on manifolds. Planarly BRP are rough paths adapted to homogeneous spaces.

We give an overview of this work. Joint work with Ebrahimi-Fard, Manchon and Curry.

Michele Coghi

Title: Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering

Abstract: Motivated by the challenge of incorporating data into misspecified and multiscale dynamical models, we study a McKean-Vlasov equation that contains the data stream as a common driving rough path. This setting allows us to prove well-posedness as well as continuity with respect to the driver in an appropriate rough-path topology. The latter property is key in our subsequent development of a robust data assimilation methodology: We establish propagation of chaos for the associated interacting particle system, which in turn is suggestive of a numerical scheme that can be viewed as an extension of the ensemble Kalman filter to a rough-path framework. Finally, we discuss a data-driven method based on subsampling to construct suitable rough path lifts and demonstrate the robustness of our scheme in a number of numerical experiments related to parameter estimation problems in multiscale contexts.

Fred Espen Benth

Title: Pricing options on commodity flow forwards by neural networks in Hilbert space

Abstract: We propose a methodology for pricing options on commodity flow forwards by applying infinite-dimensional neural networks. We recast the option pricing problem as an optimization problem in a Hilbert space of real-valued function on the positive real line, which is the state space for the forward price term structure dynamics. This optimization problem is solved by facilitating a feedforward neural network architecture designed for approximating continuous functions on the state space. The proposed neural net is built upon the basis functions of the Hilbert space. We present a case study showing numerical efficiency of the approach, with an improved performance over classical neural net trained on discretely sampling the term structure curves.

This is joint work with Nils Detering (University of California at Santa Barbara) and Luca Galimberti (Norwegian University of Science and Technology)

Silvia Lavagnini

Title: Deep Quadratic Hedging

Abstract: We present a novel computational approach for quadratic hedging in a high-dimensional incomplete market. This covers both mean-variance hedging and local-risk minimization. In the first case, the solution is linked to a system of BSDEs, one of which being a backward stochastic Riccati equation (BSRE); in the second case, the solution is related to the Fölmer-Schweizer decomposition and is also linked to a BSDE. We apply (recursively) a deep neural network-based BSDE solver. Thanks to this, we solve high-dimensional quadratic hedging problems, providing the entire hedging strategies paths, which, in alternative, would require the numerical solution of high dimensional PDEs. We test our approach with a classical Heston model and with a multi-dimensional generalization of it. Due to the unboundedness of the variance process, existence and uniqueness results for the BSRE must be considered. This is a joint work with Alessandro Gnoatto and Athena Picarelli

Harald Oberhauser

Title: Capturing Graphs with Hypo-elliptic Diffusions

Abstract: Convolutional layers within graph neural networks operate by aggregating information about local neighbourhood structures; one common way to encode such substructures is through random walks. The distribution of these random walks evolves according to a diffusion equation defined using the graph Laplacian. We extend this approach by leveraging classic mathematical results about hypo-elliptic diffusions. This results in a novel tensor-valued graph operator called the hypo-elliptic graph Laplacian. We provide theoretical guarantees and efficient low-rank algorithms. In particular, this gives a structured approach to capturing long-range dependencies on graphs. Joint work with Csaba Toth, Darrick Lee, and Celia Hacker.

Darrick Lee

Title: Mapping Space Signatures

Abstract: We introduce the mapping space signature, a generalization of the path signature for maps from higher dimensional cubical domains, which is motivated by the topological perspective of iterated integrals by K. T. Chen. We show that the mapping space signature shares many of the analytic and algebraic properties of the path signature; in particular it is universal and characteristic with respect to Jacobian equivalence classes of cubical maps. This is joint work with Chad Giusti, Vidit Nanda, and Harald Oberhauser.

Subbarao Venkatesh Guggilam

Title: Nonlinear System Identification for Multivariable Control via Discrete-Time Chen-Fliess Series

Abstract: The system identification problem especially in the context of nonlinear systems is an age-old problem and is very rich in literature. However the same system identification problem is looked into in this talk in the context of a multivariable nonlinear input-output system that can be represented in terms of a Chen-Fliess functional series or equivalently, a weighted Chen signature series. The problem is stated in the discrete-time setting and hence the talk begins with the description of discrete-time Chen Fliess series. The identification problem is then formulated in terms of a regression which is linear in the parameters but has a nonlinear regressor. An inductive implementation of the nonlinear regressor is developed using the underlying algebraic and combinatorial structure on non-commutative words equipped with a partial ordering. The method is demonstrated in an adaptive control application involving a two-input, two- output Lotka-Volterra dynamical system where the identification module is used to learn the dynamics of error between the plant and the model available to the user. The talk shall conclude with the description of further research directions. This is a joint work with Prof. W. Steven Gray (Old Dominion University) and Prof. Luis. A. Duffaut Espinosa (University of Vermont) as a part of NSF project from 2019–2020.

Participation is free of charge and open for everyone.

To participate you need to register here .

This year the workshop will be located at the University of Agder, in Kristiansand, Norway.

Kristiansand is a beautiful city on the south coast of Norway. To travel there one can either fly directly to Kristiansand Airport Kjevik, or travel by bus or train from Oslo. The bus trip from Oslo takes approximately 4 hours.

From the Airport to the hotel there is a bus leaving approximately every hour.

The university is reachable with a 30 minutes walk or 15 minutes bus-ride from the hotel.

• Kurusch Ebrahimi-Fard - NTNU, Trondheim - webpage

• Torstein K. Nilssen - University of Agder (UiA) - webpage

• Fabian A. Harang - BI Norwegian Business School - webpage

• TMS - Trond Mohn Stiftelsen through Pure Mathematics Norway.

• CODYSMA: Computational Dynamics and Stochastics on Manifolds