Scientific Machine Learning (SciML) is a merger of computational sciences and data-driven machine learning, implemented in software as a set of abstractions to leverage existing domain knowledge and physics models within learning schemes and accelerated computing platforms. Only now are the component technologies becoming mature and integrated, with the purpose of providing robust and reliable ML that encodes domain knowledge and yields interpretable solutions in nearly all sciences at any scale—from particle physics and protein folding, to epidemiology and economics, to climate and astrophysics.

While recent years have shown immense research progress in accelerating classical physics solvers and discovering new governing laws for complex physical systems, SciML methods and applications fall short of real-world utility. Across all physical and life sciences the technical maturity and validations necessary are lacking, failing, or unknown to SciML research. In the context of... (more)

The development and analysis of numerical methods for PDEs whose formulation or interpretation is derived from an underlying geometry is a persistent challenge in numerical analysis. Examples include PDEs posed on complicated manifolds or graphs, PDEs that describe interactions across complex interfaces, and equations derived from intrinsically geometric concepts such as curvature-driven flows or highly nonlinear Monge-Ampere equations arising in optimal transport. In recent years, these PDEs have gained significance in diverse areas such as machine learning, optical design problems, meteorology, medical imaging, and beyond. Hence, the development of numerical methods for this class of PDEs is poised to lead to breakthroughs for a wide range of timely problems. However, designing methods to accurately and efficiently solve these PDEs requires careful consideration of the interactions between discretization methods, the PDE operators, and the underlying geometric properties.

This... (more)

The purpose of the workshop is to advance toward the goal of making information relating to families of K3 surfaces and their elliptic fibrations into the LMFDB. We expect to begin with families of Picard number 19, building on the previous week's work on Shimura curves, and entering at least the following information: Picard lattice, frame lattices, transcendental lattice, 2-neighbour relations, elliptic fibrations, singular projective model. For some of these, code exists or is currently under construction; for others, it will need to be developed.

The workshop will consist of a small group of mostly experienced people. We expect that most of the time will be spent writing code in small groups, with at least one meeting a day to make sure that we are all going in the same direction while avoiding duplication of effort.

It is practically rare that a natural phenomenon or engineering problem can be accurately described by a single law of physics. The striking diversity of rules of life forces scientists to continuously increase the complexity of models to address the ever-growing requirements for their prediction capabilities. It remains a formidable challenge to derive and analyze numerical methods which are universal enough to handle complex multiphysics problems with the same ease and efficiency as traditional methods do for textbook PDEs.

The workshop will focus on recent trends in the field of numerical methods for multiphysics problems that include the development of monolithic approaches, structure preserving discretizations, geometrically unfitted methods, data-driven techniques, and modern algebraic methods for the resulting linear and nonlinear discrete systems. The topics of interest include models and discretizations for fluid - elastic structure interaction, non-Newtonian fluids, phase... (more)

This workshop focuses on connections between higher-order statistics and symmetric tensors, and their applications to machine learning, network science, and other domains. Higher-order statistics refers to the study of correlations between three or more covariates. This is in contrast to the usual mean and covariance, which are based on one and two covariates.

Higher-order statistics are needed to characterize complex data distributions, such as mixture models. Symmetric tensors, meanwhile, are multi-dimensional arrays. They generalize covariance matrices and affinity matrices and can be used to represent higher-order correlations. Tensor decompositions extend matrix factorizations from numerical linear algebra to multilinear algebra. Recently tensor-based approaches have become more practical, due to the availability of bigger datasets and new algorithms.

The workshop brings together applied mathematicians, statisticians, probabilists, machine learning experts, and computational... (more)

Single-cell assays provide a tool for investigating cellular heterogeneity and have led to new insights into a variety of biological processes that were not accessible with bulk sequencing technologies. Assays generate observations of many different molecular types and a grand mathematical challenge is to devise meaningful strategies to integrate data gathered across a variety of different sequencing modalities. The first-order approach to do this is to analyze the projected data by clustering. Keeping more refined shape information about the data enables more meaningful and accurate analysis. Geometric methods include (i) Manifold learning: Whereas classical approaches (PCA, metric MDS) assume projection to a low-dimensional Euclidean subspace, manifold learning finds coordinates that lie on a not necessarily flat or contractible manifold. (ii) Topological data analysis: Algebraic topology provides qualitative descriptors of global shape. Integrating these descriptors across feature... (more)

Estimators of various causal or statistical quantities are usually constructed with a particular target population in mind, that is, the population about which the investigators intend to draw inferences (e.g., decide on the implementation of a treatment strategy or use algorithm-derived predictions). Typically, however, the data used for estimation comes from a population that differs from the target population. How to ensure or evaluate whether the estimates generalize to the target population is a question that has received substantial attention in many scientific disciplines, but with the fields not always connecting with one another on overlapping challenges and solutions. This workshop will bring together experts from different disciplines to present state-of-the-science methods to address generalizability and discuss key challenges, and open problems.

The study of classical modular forms has emerged as a major topic in computational number theory, and the LMFDB is a convenient way to access data and knowledge on these objects.

This success calls for including higher-dimensional analogues, like Siegel modular forms, in the database as well. This workshop will gather experts on Siegel modular forms, algorithmic methods, and the LMFDB to take stock of where we are, discuss the next achievable goals and set further research directions.

Cracking the neural code is one of the longstanding questions in neuroscience. How does the activity of populations of neurons represent stimuli and perform neural computations? Decades of theoretical and experimental work have provided valuable clues about the principles of neural coding, as well as descriptive understandings of various neural codes. This raises a number of mathematical questions touching on algebra, combinatorics, probability, and geometry. This workshop will explore questions that arise from sensory perception and processing in olfactory, auditory, and visual coding, as well as properties of place field codes and grid cell codes, mechanisms for decoding population activity, and the role of noise and correlations. These questions may be tackled with techniques from information theory, mathematical coding theory, combinatorial commutative algebra, hyperplane arrangements, oriented matroids, convex geometry, statistical mechanics, and more.

In the last decade or so, applied topology and algebraic geometry have come into their own as vibrant areas of applied mathematics. At the same time, ideas and tools from topology and geometry have infiltrated theoretical and computational neuroscience. This kind of mathematics has shown itself to be a natural and useful language not only for analyzing neural data sets but also as a means of understanding principles of neural coding and computation. This workshop will bring together leading researchers at the interfaces of topology, geometry, and neuroscience to take stock of recent work and outline future directions. This includes a focus on topological data analysis (persistent homology and related methods), topological analysis of neural networks and their dynamics, topological decoding of neural activity, evolving topology of dynamic networks (e.g., networks that are changing as a result of learning), and analysis of connectome data. Related topics may include the geometry and... (more)

Limits of discrete random structures appear in different areas of probability, combinatorics, and machine learning. In statistical mechanics, probabilistic and combinatorial techniques are applied to rigorously describe the scaling limits of such random graphical models, which are closely related to phase transitions. In the vicinity of a phase transition, even a tiny change in some local parameter can result in dramatic changes in the macroscopic properties of the entire system. Random discrete structures are also useful mathematical models of large networks, which play a central role in our social and economic lives as the fabric over which we interact, form social connections, conduct economic transactions, transmit information, propagate disease, and much more.

The goal of this workshop is to integrate the algebraic combinatorics, probability, and machine learning paradigms of statistical mechanical models and to bring together researchers in related fields to discuss recent... (more)