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

The concept of the path-dependent partial differential equation (PPDE) was first introduced in the context of path-dependent derivatives in financial markets. Its semilinear form was later identified as a non-Markovian backward stochastic…

(Partial) differential equations (PDEs) are fundamental tools for describing natural phenomena, making their solution crucial in science and engineering. While traditional methods, such as the finite element method, provide reliable…

Deep neural networks have demonstrated superior performance in artificial intelligence applications, but the opaqueness of their inner working mechanism is one major drawback in their application. The prevailing unit-based interpretation is…

Signed distance map (SDM) is a common representation of surfaces in medical image analysis and machine learning. The computational complexity of SDM for 3D parametric shapes is often a bottleneck in many applications, thus limiting their…

The optimal design of compounds through manipulating properties at the molecular level is often the key to considerable scientific advances and improved process systems performance. This paper highlights key trends, challenges, and…

Recent techniques have been successful in reconstructing surfaces as level sets of learned functions (such as signed distance fields) parameterized by deep neural networks. Many of these methods, however, learn only closed surfaces and are…

Attempts from different disciplines to provide a fundamental understanding of deep learning have advanced rapidly in recent years, yet a unified framework remains relatively limited. In this article, we provide one possible way to align…

Providing fast and accurate solutions to partial differential equations is a problem of continuous interest to the fields of applied mathematics and physics. With the recent advances in machine learning, the adoption learning techniques in…

Deep neural networks (DNN) have been used to model nonlinear relations between physical quantities. Those DNNs are embedded in physical systems described by partial differential equations (PDE) and trained by minimizing a loss function that…

We consider rough differential equations whose coefficients contain path-dependent bounded variation terms and prove the existence and a priori estimate of solutions. These equations include classical path-dependent SDEs containing running…

We present a novel approach (DyNODE) that captures the underlying dynamics of a system by incorporating control in a neural ordinary differential equation framework. We conduct a systematic evaluation and comparison of our method and…

Neural controlled differential equations (NCDEs), which are continuous analogues to recurrent neural networks (RNNs), are a specialized model in (irregular) time-series processing. In comparison with similar models, e.g., neural ordinary…

Networks are one of the most powerful structures for modeling problems in the real world. Downstream machine learning tasks defined on networks have the potential to solve a variety of problems. With link prediction, for instance, one can…

Designing a controller for autonomous vehicles capable of providing adequate performance in all driving scenarios is challenging due to the highly complex environment and inability to test the system in the wide variety of scenarios which…

The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively…

We propose a kinematic wave-based Deep Convolutional Neural Network (Deep CNN) to estimate high-resolution traffic speed fields from sparse probe vehicle trajectories. We introduce two key approaches that allow us to incorporate kinematic…

The dynamics of physical systems that require high-dimensional representation can often be captured in a few meaningful degrees of freedom called collective variables (CVs). However, identifying CVs is challenging and constitutes a…

On the forefront of scientific computing, Deep Learning (DL), i.e., machine learning with Deep Neural Networks (DNNs), has emerged a powerful new tool for solving Partial Differential Equations (PDEs). It has been observed that DNNs are…

Chemically reacting flows are common in engineering, such as hypersonic flow, combustion, explosions, manufacturing processes and environmental assessments. For combustion, the number of reactions can be significant (over 100) and due to…

Signature kernels have emerged as a powerful tool within kernel methods for sequential data. In the paper "The Signature Kernel is the solution of a Goursat PDE", the authors identify a kernel trick that demonstrates that, for continuously…