Scientific Program

NUMRAD26 will take place in the week of June 2nd-5th. The mornings will be devoted to introductory lectures, while more advanced topics will be presented in the afternoon. According to the availability of the participants, poster sessions will also be organized.

Topics

• High-Dimensional Approximation
• Gaussian Processes
• Reduced-Order Models
• Mathematical Finance

Confirmed speakers

• Alexandra Gessner (AstraZeneca, Spain)
• David Ginsbourger (Universität Bern, Switzerland)
• George Haller (ETH Zürich, Switzerland)
• Benjamin Jourdain (École Nationale des Ponts et Chaussées, France)
• Motonobu Kanagawa (EURECOM, France)
• Damiano Lombardi (Inria, COMMEDIA, France)
• Andrea Manzoni (Politecnico di Milano, Italy)
• Anthony Nouy (Centrale Nantes – Nantes Université, France)
• Christoph Reisinger (University of Oxford, UK)
• Gianluigi Rozza (SISSA, Italy)
• Olivier Zahm (Inria, Laboratoire Jean Kuntzmann, France)

Timetable

Lecture titles & Abstracts

Tuesday June 2nd – High-Dimensional Approximation

“Manifold approximation and optimal sampling” (Prof. Anthony Nouy, Centrale Nantes (FR))

Approximating functions in high dimension is at the heart of many learning and scientific computing problems.  
To overcome the curse of dimension, it is necessary to exploit particular structures of functions, beyond classical regularity. This requires the introduction of dedicated approximation tools capable of capturing these structures, and algorithms for constructing these approximations.

In a first part of this course, we will first introduce benchmarks for function approximation, defining what we can expect from optimal algorithms. These benchmarks depend on how the quality of the approximation is measured, the nature of the available information (point evaluations of functions, gradient evaluations…) and the properties of the algorithms (linearity, stability…). We will present manifold approximation (or dimension reduction) methods which aim at constructing approximation tools dedicated to a certain task. 

High-dimensional approximation tools are in general extremely nonlinear.  
The use of generic approaches from statistical learning and optimization generally lead to very high complexity, excessive data consumption, and accuracies that are not up to the expectations of scientific computing.   
A real challenge lies in generating or exploiting optimal data for a given task, thereby considerably reducing the complexity of algorithms. Data generation must be specific to the approximation tool used, and to the nature of the available  information (function evaluations, residual of a partial differential equation in physics-informed learning…).  In a second part of this course, we will give a short introduction to optimal sampling methods for approximation, and provide an overview of recent developments in the field.

“Gradient-based dimension reduction: the functional inequality way” (Rs. Olivier Zahm, INRIA (FR))

Many approximation tools suffer from poor scalability as the dimensionality of a model inputs and outputs increases. A powerful approach to handling high-dimensional variables is to reduce their dimensionality, allowing the problem to be reformulated in a lower-dimensional space. In this presentation, we will present a line of work that leverages gradient information to identify important low-dimensional subspaces within the input parameters of a given function. We will also demonstrate that these same gradients reveal critical low-dimensional subspaces in the output space, enabling data reduction in Bayesian inference and optimal sensor placement. These subspaces emerge from error bounds derived using functional inequalities to approximate the input-output mapping. The error analysis not only provides a priori guarantees for the dimension reduction process but also offers criteria for selecting the dimensions of the reduced variables. We will illustrate the advantages of this linear dimension reduction technique and further extend the methodology to perform nonlinear dimension reduction.

“On adaptivity and structure preservation for high-dimensional problems” (DR Damiano Lombardi, INRIA (FR))

In this talk we will discuss about several aspects of the resolution of high-dimensional problems. In particular, with a particular focus on problems defined by a system of parametric Partial Differential Equations (PDEs).

First, a motivation on a real industrial case will be presented, aiming at illustrating the challenges that certain computations and tasks involve. Most of these practical challenges are inherently related to the curse of dimensionality, which we will examine through the lens of the Kolmogorov metric entropy and entropy numbers.

In the first part of the talk we will describe low-rank approximations and discuss two possible strategies to adapt them, and to start introducing more ‘flexible’ representations. The first one considers sums of tensor trains in which we do not fix a priori the order of the variables. The second one consists in a piece-wise tensor approximation in which we do not fix a priori the domain decomposition. In both, the adaptation is based on error and parsimony criteria. Solving the problems with non-linear approximations at hand poses undeniable challenges. We will describe several ways to precondition multilinear problems and will detail a method combining CUR factorisation and Krylov subspace iterations.

In the last part of the talk we will focus on structure preservation, i.e. the ability to construct numerical approximations which respect geometrical and physical properties of the system. We will restrict to PDEs which can be written as a sum of a Hamiltonian flow and a dissipation mechanism which is defined by a gradient flow. A mixed formulation will be introduced. This makes it possible to provably conserve several quantities of interest (Casimirs, invariants, equilibria) and have the correct dissipation rates if dissipation occurs. We will extend the formulation to dynamical discretisations (such as Dynamical Low Rank, neural Galerkin, shape morphing functions) and illustrate the method on some test-cases.

Wednesday June 3rd – Gaussian Processes

“Gaussian Processes and Reproducing Kernel Hilbert Spaces: Connections and Equivalences” (MdC Motonobu Kanagawa, EURECOM (FR))

This talk discusses the relations between two approaches using positive definite kernels: probabilistic methods using Gaussian processes, and non-probabilistic methods using reproducing kernel Hilbert spaces (RKHS). They are widely studied and used in machine learning, statistics, and numerical analysis. Connections and equivalences between them are reviewed for fundamental topics such as regression, interpolation, numerical integration, distributional discrepancies, and statistical dependence, as well as for sample path properties of Gaussian processes. A unifying perspective for these equivalences is established, based on the equivalence between the Gaussian Hilbert space and the RKHS. This serves as a foundation for many other methods based on Gaussian processes and reproducing kernels, which are being developed in parallel by the two research communities.

“The Many Lives of Gaussian Processes” (Prof. David Ginsbourger, University of Bern, (CH))

Beyond their foundational roots in statistical and stochastic process theory, Gaussian processes (GPs) are now central to a versatile and widely used framework spanning applied mathematics and probabilistic machine learning. Despite well-known limitations, GP-based methods continue to show remarkable adaptability, often outlasting shifting methodological trends. In this talk, I will highlight developments in modeling, prediction, and active learning with GPs, offering perspectives on their enduring role as a powerful building block for function approximation and uncertainty quantification.

“Multi-objective Bayesian optimization for drug design” (AI Sc. Alexandra Gessner, AstraZeneca (ES))

Multi-objective optimization seeks solutions that balance competing objectives, leading not to a single optimum but to a set of Pareto-optimal trade-offs. This tutorial introduces the main ideas behind multi-objective Bayesian optimization (MOBO), a framework for solving expensive black-box optimization problems with multiple objectives using probabilistic surrogate models. We will discuss different approaches to designing active learning schemes for selecting new observations in the multi-objective setting. Finally, we will see how MOBO can make a difference in industrial applications on the example of therapeutic antibody design.

Thursday June 4th – Reduced-Order Modelling

“A 2 hours trip in Reduced Order Modeling for parametrized systems: from projection-based methods to deep learning techniques… and back” (Prof. Andrea Manzoni, Politecnico di Milano (IT))

In this presentation I will cover the latest development in reduced order modeling for parametrized systems described in terms of partial differential equations (PDEs), ranging from classical reduced basis methods to more recent deep learning based techniques. After reviewing fundamental tools in reduced order modeling like, e.g., proper orthogonal decomposition and Galerkin reduced basis methods, I will introduce a series of strategies taking advantage of (i) neural networks to surrogate the reduced problem and to perform dimensionality reduction, (ii) sparse regression methods for discovery of latent dynamics models, (iii) shallow decoder architectures for the construction of ROMs capable of leveraging sensor data, and (iv) generative models to embed uncertainty quantification. Examples from applied sciences and engineering will be presented to showcase the accuracy, the efficiency, and the reliability of the proposed techniques.

Nonlinear Model Reduction and Control via Spectral Submanifolds (Prof .George Haller, ETH Zürich (CH))

A long-standing objective in applied science and engineering has been to reduce complex nonlinear differential equations and data sets to simple, low-dimensional models. The main topic of this lecture, model reduction to spectral submanifolds (SSMs), introduces a mathematically well-founded solution to this objective. SSMs are very low-dimensional attracting invariant manifolds tangent to eigenspaces (spectral subspaces) of linearizations of nonlinear systems at steady states. The internal dynamics of SSMs provide simple reduced-order models with which typical system trajectories synchronize exponentially fast. We will discuss applications of SSM reduction include accelerated finite-element computations, reduced-order modeling of fluid-structure interactions from experimental data, derivation of equations of motion from videos, prediction of transitions to turbulence in pipe flows, model-predictive control of soft robots and other physical problems.

Friday June 5th – Mathematical Finance

“Monte-Carlo methods for financial applications” (Prof. Benjamin Jourdain, École Nationale des Ponts et Chaussées, (FR))

We will first discuss adaptive variance reduction techniques, in particular, for functions of Gaussian random vectors. We will next present the regression techniques that permit to approximate conditional expectations in view of American options pricing. The remaining of the lecture will be devoted to the discretization of stochastic differential equations : Euler-Maruyama and Milstein schemes, strong and weak errors, MultiLevel Monte-Carlo methods, schemes with high order of weak convergence like the Ninomiya-Victoir scheme. We will finally discuss the simulation of stochastic Volterra equations by combining Markovian approximations with time-discretization.

“Numerical approximation of financial problems with machine learning” (Prof. Christoph Reisinger, University of Oxford, UK)

We will survey the strength and challenges of current machine learning approaches to financial modeling and computation on three examples: optimal investment, hedging, and trade execution. For these case studies, we will investigate the intertwined problems of (synthetic) data generation and optimisation of (over-)parametrised strategies. In the first case, we will theoretically examine the generalisation error achieved by optimal investment strategies learnt from finitely many samples and demonstrate good stability properties for entropy regularised mean-field neural networks. In the hedging context, we will propose an ambiguity-averse modification of a deep hedger which achieves more robustness to shifts in the market regime by minimising nested model and market risk measures constructed through feature clustering. Finally, we will exhibit synthetic data generators for limit order books using, alternatively, generative adversarial networks and nearest-neighbour resampling, and discuss implications on the construction of execution strategies.