Related papers: New directions in the applications of rough path t…

In this article, we present an efficient deep learning method called coupled deep neural networks (CDNNs) for coupled physical problems. Our method compiles the interface conditions of the coupled PDEs into the networks properly and can be…

Distribution regression refers to the supervised learning problem where labels are only available for groups of inputs instead of individual inputs. In this paper, we develop a rigorous mathematical framework for distribution regression…

The expected signature kernel arises in statistical learning tasks as a similarity measure of probability measures on path space. Computing this kernel for known classes of stochastic processes is an important problem that, in particular,…

Computational Fluid Dynamics (CFD) is central to science and engineering, but faces severe scalability challenges, especially in high-dimensional, multiscale, and turbulent regimes. Traditional numerical methods often become prohibitively…

Using a combination of recurrent neural networks and signature methods from the rough paths theory we design efficient algorithms for solving parametric families of path dependent partial differential equations (PPDEs) that arise in pricing…

Deep Neural Networks have shown tremendous success in the area of object recognition, image classification and natural language processing. However, designing optimal Neural Network architectures that can learn and output arbitrary graphs…

Development of algorithms and growth of computational resources in the past decades have enabled simulations of sediment transport processes with unprecedented fidelities. The Computational Fluid Dynamics--Discrete Element Method (CFD--DEM)…

Signature tensors of paths are a versatile tool for data analysis and machine learning. Recently, they have been applied to persistent homology, by embedding barcodes into spaces of paths. Among the different path embeddings, the…

In complex molecular systems, the reaction coordinate (RC) that characterizes transition pathways is essential to understand underlying molecular mechanisms. This review surveys a framework for identifying the RC by applying deep learning…

Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…

The current revolution in the field of machine learning (ML) is leading to many interesting developments in a wide range of areas, including fluid mechanics. Here we review recent and emerging possibilities in the context of predictions,…

Classic control techniques typically rely on a model of the system's response to external inputs, which is difficult to obtain from first principles especially if the unknown dynamics are nonlinear. In this paper, we address this issue by…

Solving large complex partial differential equations (PDEs), such as those that arise in computational fluid dynamics (CFD), is a computationally expensive process. This has motivated the use of deep learning approaches to approximate the…

In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…

The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is here examined through the lens of algebraic geometry.…

The rapidly advancing field of Fluid Mechanics has recently employed Deep Learning to solve various problems within that field. In that same spirit we try to perform Direct Numerical Simulation(DNS) which is one of the tasks in…

Deep learning, a branch of artificial intelligence, is a data-driven method that uses multiple layers of interconnected units or neurons to learn intricate patterns and representations directly from raw input data. Empowered by this…

Modeling the traffic dynamics is essential for understanding and predicting the traffic spatiotemporal evolution. However, deriving the partial differential equation (PDE) models that capture these dynamics is challenging due to their…

Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks in terms of both the input data and the (trained) network weights. As trained network weights are typically very rough when seen as…

Convolutional Neural Networks (CNNs) constitute a class of Deep Learning models which have been used in the recent past to resolve many problems in computer vision, in particular optical flow estimation. Measuring displacement and strain…