Matthias Möller (TU Delft)

IgaNets: Physics-Informed Machine Learning Embedded Into Isogeometric Analysis

Many engineering problems of practical interest are modelled by systems of partial differential equations equipped with initial and boundary conditions and complemented by problem-specific constitutive laws. For decades, numerical methods like the finite element method have been the method of choice for computing approximate solutions to problems that cannot be solved analytically. Starting with the seminal paper on physics-informed neural networks (PINNs) a new paradigm has entered the stage: learning the behavior of the differential operator instead of discretizing it and solving the resulting linear(ized) systems of equations brute-force. Next to PINNs, several alternative approaches like DeepONets and Fourier neural networks have been proposed in recent years. Their ease of implementation and fast response time, once training is completed, makes learning-based methods particularly attractive for engineering applications as they offer the opportunity to combine computer simulations with real-world data, e.g., stemming from measurements of real physical systems.

In this talk we propose a novel approach to embed the PINN paradigm into the framework of Isogeometric Analysis to combine the best of both worlds. IGA is an extension of the finite element method that integrates the simulation-based analysis into the computer-aided geometric design pipeline. In short, the same mathematical formalism, namely (adaptive) B-splines or NURBS, that is used to model the geometry is adopted to represent the approximate solution, which is computed following the same strategy as in classical finite elements.

In contrast to classical PINNs, which predict point-wise solution values to (initial-)boundary-value problems directly, our IGA-PINNs learn solutions in terms of their expansion coefficients relative to a given B-Spline or NURBS basis. This approach is furthermore used to encode the geometry and other problem parameters such as boundary conditions and parameters of the constitutive laws and feed them into the neural network as inputs. Once trained, our IGA-PINNs make it possible to explorer various designs from a family of similar problem configurations efficiently, without the need to recompute every new problem configuration brute force.

Wouter Koolen (Centrum Wiskunde & Informatica)

Pure Exploration Problems: Information theory and Equilibria

Pure exploration is a class of problems in statistical learning theory, focused around the active collection of data for the purpose of hypothesis testing. Pure exploration systems aim to collect only data that are most useful to answer a particular question quickly and confidently.

In this talk we will introduce pure exploration problems in the stylised multi-armed bandit setting, and provide motivation by reviewing the practical examples of best arm, best CVaR arm and arms-better-than-control.

We will then highlight the modern approach for constructing efficient learning algorithms. We will present information-theoretic lower bounds, and the path to matching instance-optimal algorithms by means of sequential saddle point computation.

Erik Bekkers (University of Amsterdam)

Geometric and Physical Quantities improve E(3) Equivariant Message Passing

Including covariant information, such as position, force, velocity or spin is important in many tasks in computational physics and chemistry. In this talk I will introduce Steerable E(3) Equivariant Graph Neural Networks (SEGNNs) that generalise equivariant graph networks, such that node and edge attributes are not restricted to invariant scalars, but can contain covariant information, such as vectors or tensors. This model, composed of steerable MLPs, is able to incorporate geometric and physical information in both the message and update functions. Through the definition of steerable node attributes, the MLPs provide a new class of activation functions for general use with steerable feature fields. We discuss ours and related work through the lens of equivariant non-linear convolutions, which further allows us to pin-point the successful components of SEGNNs: non-linear message aggregation improves upon classic linear (steerable) point convolutions; steerable messages improve upon recent equivariant graph networks that send invariant messages. The effectiveness of our method is demonstrated -with extensive ablation studies- on several tasks in computational physics and chemistry.