Abstracts

I. Scientific Computing and Machine Learning

Weinan E, Peking University

Lecture 1: Deep Learning-based Algorithms for Solving High Dimensional PDEs and Control Problems
There has been a growing interest on deep learning-based algorithms for solving high dimensional PDEs and control problems. But how do these algorithms perform in practice? In this lecture I will give a survey of this active field, focusing on the practical issues for deep learning-based algorithms. I will discuss adaptive sampling and the initial value problem-enhanced sampling method that seem to help improving drastically the performance of deep learning-based algorithms for computing optimal closed loop controls.

Lecture 2:  The Random Feature Method for PDEs
When solving low dimensional problems, it remains unclear whether deep learning-based methods have a real advantage over traditional algorithms. I will discuss the random feature method (RFM) that seems to bridge nicely deep learning-based algorithms and traditional algorithms.  RFM competes well with traditional methods in terms of both accuracy and efficiency, and at the same time, it inherits the flexibility of  deep learning-based algorithms and is particularly suited for problems with complex geometry.


Petros Koumoutsakos, Harvard University

Computing - AI Alloys for prediction and control in fluid mechanics
Human minds have produced laws to describe complex systems along with numerical methods and algorithms that harness the powers of modern supercomputers. Simulations and data processing have provided our generation with unprecedented physical insight. Despite progress in methods, software and hardware we realize that we will never be able to rely solely on this mode of inquiry to understand, predict and control complex systems. Artificial intelligence (AI) offers new modes of inquiry but questions remain on whether it replaces exiting modes of inquiry, complements them and how ? In this talk I will present algorithms formulated on a fusion of computational science and AI for the prediction and control of diverse physical systems. I will present the Remember and Forget Experience Replay (ReFer) algorithm for reinforcement learning, a multiscale approach to Learning the Effective Dynamics (LED) of complex systems and a fusion of scientific computing and multi-agent reinforcement learning (SciMARL) for developing closures for unresolved dynamics of complex systems. Examples will include benchmark problems from physics engines to high fidelity simulations of complex systems that range from molecular systems to fish schooling. I will discuss successes and failures and hope for a dialogue on how the integration of AI and Computational science may lead to new forms of Computational Intelligence.


II. Quantum Computing

Oleksandr Kyriienko, University of Exeter

Talk 1: "Quantum computing in the near-term: introduction to software and hardware"
This talk will cover an introduction to the field, discussing state-of-the-art, and concentrating on quantum machine learning strategies.

Talk 2: "Quantum computing in the near-term: a path towards quantum scientific machine learning"
In the second part I will concentrate on prospective tasks of quantum machine learning for solving differential equations, generative modelling, and classification.


Ashley Montanaro, University of Bristol

Talk1: An overview of quantum algorithms.
Quantum computers are designed to outperform their classical counterparts by running quantum algorithms. In this talk I will give a brief general overview of quantum algorithms and their applications in fields including cryptography, search and optimisation, and simulation of quantum systems.

Talk 2: Quantum algorithms for solving differential equations.
One application domain where quantum computers could achieve a significant speedup over classical computers is solving differential equations. In this talk I will discuss two recent developments in this area. First, a characterisation of the level of acceleration that could be achieved by quantum algorithms for solving a prototypical PDE, the heat equation; second, new quantum algorithms for speeding up the solution of stochastic differential equations with applications in mathematical finance.


III. Probabilistic Numerical Methods for ODE's and PDE's

Mark Girolami, University of Cambridge

Talk 1: The Statistical Finite Element Method: Methodology and Theory
The finite element method (FEM) is one of the great triumphs of applied mathematics, numerical analysis and software development. Recent developments in sensor and signalling technologies enable the phenomenological study of engineered and natural systems. The connection between sensor data and FEM is restricted to solving inverse problems placing unwarranted faith in the fidelity of the mathematical description of the system. If one concedes mis-specification between generative reality and the FEM then a framework to systematically characterise this uncertainty is required. This talk will present a statistical construction of the FEM which systematically blends mathematical description with observations.

Talk 2: The Statistical Finite Element Method: Developments and Applications
The Statistical Finite Element Method is developed and applied to Reaction Diffusion systems and a full scale deployment on an operational rail bridge is described along with other methodological developments of the methodology.

 

Catherine Powell, University of Manchester

Intrusive Methods for Forward UQ in PDE Models

Part 1 - Basic Stochastic Galerkin Approximation

Stochastic Galerkin (SG) approximation, also known as intrusive polynomial chaos approximation, can be used to facilitate forward uncertainty quantification (UQ) in models consisting of partial differential equations (PDEs) with uncertain (or parameter-dependent) inputs. Unlike conventional sampling methods such as Monte Carlo, SG schemes yield approximations that are functions of the uncertain inputs, leading to surrogate models. Standard SG schemes use simple tensor product approximation spaces and give rise to huge but highly structured linear systems whose coefficient matrices have a characteristic Kronecker product structure. The number of equations can easily run into the hundreds of millions, even for relatively simple physical models. Memory-efficient solvers that exploit structure are essential. In this first talk, we will illustrate the use of basic SG approximation for common models arising in engineering applications such as heat diffusion and linear elasticity, focusing on computational aspects.

Part 2 - Adaptive Multilevel Stochastic Galerkin Approximation

While tensor product approximation spaces are simple to work with, it is well known that approximation spaces with a more complex multilevel structure lead to superior convergence rates.  Multilevel SG finite element methods (SGFEMs) provide surrogates with spatial coefficients that reside in potentially different finite element spaces. For elliptic PDEs with diffusion coefficients represented as affine functions of countably many uncertain parameters, well-established theoretical results state that such methods can achieve rates of convergence independent of the number of input parameters, thereby breaking the curse of dimensionality. Moreover, for nice enough test problems, it is even possible to prove convergence rates afforded to the chosen finite element method for the associated deterministic PDE.  Achieving these rates in practice using automated computational algorithms remains highly challenging. However, a key advantage of working in the intrusive setting is that we have a rigorous and well developed framework for performing error analysis and deriving provably accurate error estimators. In this second talk, we discuss a state-of-the-art hierarchical error estimation strategy and use it to drive an adaptive multilevel SGFEM for scalar elliptic PDEs. All the results presented can be reproduced with our MATLAB software. Extensions to more complex problems will also be briefly discussed.


Laura Scarbosio, Radboud University

Quantifying the effects of geometric uncertainties
We consider a partial differential equation (PDE) on a domain whose geometric features are subject to variations, and our goal is to quantify how these variations affect the solution to the PDE or some functionals of it. The changes in the geometry might represent intrinsic variability in the system (for example, manufacturing defects in fabricated objects) or some lack of knowledge (for example, because we extract an approximate geometry from a low resolution image). In both cases, we adopt the point of view of uncertainty quantification (UQ), by modeling the geometry variations using random variables. We will see how to handle the realization-dependent geometry in a computationally efficient way, and why so-called non-intrusive UQ methods are more
suited for this setting. We will discuss pros and cons of some of these for different applications.