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While a real-world research program in mathematics may be guided by a motivating question, the process of mathematical discovery is typically open-ended. Ideally, exploration needed to answer the original question will reveal new…

Over the past years, machine learning has emerged as a powerful computational tool to tackle complex problems over a broad range of scientific disciplines. In particular, artificial neural networks have been successfully deployed to…

Machine Learning (ML) has the potential to accelerate discovery of new materials and shed light on useful properties of existing materials. A key difficulty when applying ML in Materials Science is that experimental datasets of material…

Machine learning interatomic potentials have revolutionized complex materials design by enabling rapid exploration of material configurational spaces via crystal structure prediction with ab initio accuracy. However, critical challenges…

Reasoning about the physical world requires models that are endowed with the right inductive biases to learn the underlying dynamics. Recent works improve generalization for predicting trajectories by learning the Hamiltonian or Lagrangian…

Most deep neural networks are considered to be black boxes, meaning their output is hard to interpret. In contrast, logical expressions are considered to be more comprehensible since they use symbols that are semantically close to natural…

Simulations of complex physical systems are typically realized by discretizing partial differential equations (PDEs) on unstructured meshes. While neural networks have recently been explored for surrogate and reduced order modeling of PDE…

In the past few years, the area of Machine Learning (ML) has witnessed tremendous advancements, becoming a pervasive technology in a wide range of applications. One area that can significantly benefit from the use of ML is Combinatorial…

Deep learning has received considerable empirical successes in recent years. However, while many ad hoc tricks have been discovered by practitioners, until recently, there has been a lack of theoretical understanding for tricks invented in…

Recognition of defects in concrete infrastructure, especially in bridges, is a costly and time consuming crucial first step in the assessment of the structural integrity. Large variation in appearance of the concrete material, changing…

Neural networks have been very successful in many applications; we often, however, lack a theoretical understanding of what the neural networks are actually learning. This problem emerges when trying to generalise to new data sets. The…

Biased regularization and fine-tuning are two recent meta-learning approaches. They have been shown to be effective to tackle distributions of tasks, in which the tasks' target vectors are all close to a common meta-parameter vector.…

Machine learning is making substantial progress in diverse applications. The success is mostly due to advances in deep learning. However, deep learning can make mistakes and its generalization abilities to new tasks are questionable. We ask…

We identify configurational phases and structural transitions in a polymer nanotube composite by means of machine learning. We employ various unsupervised dimensionality reduction methods, conventional neural networks, as well as the…

Machine learning models are a powerful theoretical tool for analyzing data from quantum simulators, in which results of experiments are sets of snapshots of many-body states. Recently, they have been successfully applied to distinguish…

Machine learning techniques for the solution of inverse problems have become an attractive approach in the last decade, while their theoretical foundations are still in their infancy. In this chapter we want to pursue the study of…

Computational intractability has for decades motivated the development of a plethora of methodologies that mainly aimed at a quality-time trade-off. The use of Machine Learning techniques has finally emerged as one of the possible tools to…

Programs to solve so-called constraint problems are complex pieces of software which require many design decisions to be made more or less arbitrarily by the implementer. These decisions affect the performance of the finished solver…

Data-driven algorithms, in particular neural networks, can emulate the effect of sub-grid scale processes in coarse-resolution climate models if trained on high-resolution climate simulations. However, they may violate key physical…

Mechanical metamaterials are usually designed to show desired responses to prescribed forces. In some applications, the desired force-response relationship might be hard to specify exactly, although examples of forces and corresponding…