Related papers: Mathematical artificial data for operator learning

The data-driven discovery of partial differential equations (PDEs) consistent with spatiotemporal data is experiencing a rebirth in machine learning research. Training deep neural networks to learn such data-driven partial differential…

Scientific machine learning is an emerging field that broadly describes the combination of scientific computing and machine learning to address challenges in science and engineering. Within the context of differential equations, this has…

Thanks to their universal approximation properties and new efficient training strategies, Deep Neural Networks are becoming a valuable tool for the approximation of mathematical operators. In the present work, we introduce Mesh-Informed…

Numerical algorithms and computational tools are instrumental in navigating and addressing complex simulation and data processing tasks. The exponential growth of metadata and parameter-driven simulations has led to an increasing demand for…

The quality, consistency, and information content of training data is often what determines the practical value of machine-learning models for atomistic simulations. Yet, many widely used electronic-structure databases are assembled having…

The successes of deep learning, variational inference, and many other fields have been aided by specialized implementations of reverse-mode automatic differentiation (AD) to compute gradients of mega-dimensional objectives. The AD…

PDE learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data. While deep learning models traditionally require copious amounts of training data, recent PDE learning…

Mathematical modeling is an essential step, for example, to analyze the transient behavior of a dynamical process and to perform engineering studies such as optimization and control. With the help of first-principles and expert knowledge, a…

The application of deep learning toward discovery of data-driven models requires careful application of inductive biases to obtain a description of physics which is both accurate and robust. We present here a framework for discovering…

This work develops an active learning framework to intelligently enrich data-driven reduced-order models (ROMs) of parametric dynamical systems, which can serve as the foundation of virtual assets in a digital twin. Data-driven ROMs are…

In the era of data-driven intelligence, the paradox of data abundance and annotation scarcity has emerged as a critical bottleneck in the advancement of machine learning. This paper gives a detailed overview of Active Learning (AL), which…

Metal additive manufacturing enables unprecedented design freedom and the production of customized, complex components. However, the rapid melting and solidification dynamics inherent to metal AM processes generate heterogeneous,…

Predictive student models are increasingly used in learning environments due to their ability to enhance educational outcomes and support stakeholders in making informed decisions. However, predictive models can be biased and produce unfair…

Modeling and simulation of electrical power systems are becoming increasingly important approaches for the development and operation of novel smart grid functionalities -- especially with regard to data-driven applications as data of…

Equations, particularly differential equations, are fundamental for understanding natural phenomena and predicting complex dynamics across various scientific and engineering disciplines. However, the governing equations for many complex…

Physics-informed machine learning (PIML) integrates mechanistic knowledge, typically in the form of partial differential equations (PDE), into data-driven models. Despite strong empirical performance, its statistical generalisation…

The use of derivative-based solvers to compute solutions to optimal control problems with non-differentiable cost or dynamics often requires reformulations or relaxations that complicate the implementation or increase computational…

Machines of all kinds from vehicles to industrial equipment are increasingly instrumented with hundreds of sensors. Using such data to detect anomalous behaviour is critical for safety and efficient maintenance. However, anomalies occur…

In situations where the solution of a high-fidelity dynamical system needs to be evaluated repeatedly, over a vast pool of parametric configurations and in absence of access to the underlying governing equations, data-driven model reduction…

Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…