Related papers: Neural Operators for Bypassing Gain and Control Co…

The operator learning has received significant attention in recent years, with the aim of learning a mapping between function spaces. Prior works have proposed deep neural networks (DNNs) for learning such a mapping, enabling the learning…

We study the derivative-informed learning of nonlinear operators between infinite-dimensional separable Hilbert spaces by neural networks. Such operators can arise from the solution of partial differential equations (PDEs), and are used in…

We present a unified theoretical framework for analyzing the stability and consistency of Physics-Informed Neural Networks (PINNs), grounded in operator coercivity, variational formulations, and non-asymptotic perturbation theory. PINNs…

The numerical reconstruction of controls for nonlinear partial differential equations (PDEs) remains a challenging and relatively underdeveloped problem, despite the extensive literature on controllability theory. In this work, we introduce…

The growing demand for accurate, efficient, and scalable solutions in computational mechanics highlights the need for advanced operator learning algorithms that can efficiently handle large datasets while providing reliable uncertainty…

Scientific machine learning has enabled the extraction of physical insights and data-driven modeling of high-dimensional spatiotemporal data, yet achieving physically interpretable latent representations and computationally efficient…

Learning underlying dynamics from data is important and challenging in many real-world scenarios. Incorporating differential equations (DEs) to design continuous networks has drawn much attention recently, however, most prior works make…

Operator learning has emerged as a powerful tool in scientific computing for approximating mappings between infinite-dimensional function spaces. A primary application of operator learning is the development of surrogate models for the…

Solving high-dimensional partial differential equations (PDEs) is a major challenge in scientific computing. We develop a new numerical method for solving elliptic-type PDEs by adapting the Q-learning algorithm in reinforcement learning.…

Deep neural networks, despite their success in numerous applications, often function without established theoretical foundations. In this paper, we bridge this gap by drawing parallels between deep learning and classical numerical analysis.…

Although neural operators are widely used in data-driven physical simulations, their training remains computationally expensive. Recent advances address this issue via downstream learning, where a model pretrained on simpler problems is…

Deep operator networks (DeepONets) are receiving increased attention thanks to their demonstrated capability to approximate nonlinear operators between infinite-dimensional Banach spaces. However, despite their remarkable early promise,…

Optimal control schemes have achieved remarkable performance in numerous engineering applications. However, they typically require high computational cost, which has limited their use in real-world engineering systems with fast dynamics…

When neural networks (NNs) are used as a type of nonlinear parametric representation to solve partial differential equations (PDEs), they often display frequency-dependent learning dynamics that can differ from those seen in direct function…

Deep neural networks (DNNs) recently emerged as a promising tool for analyzing and solving complex differential equations arising in science and engineering applications. Alternative to traditional numerical schemes, learning-based solvers…

Autonomous driving technologies have received notable attention in the past decades. In autonomous driving systems, identifying a precise dynamical model for motion control is nontrivial due to the strong nonlinearity and uncertainty in…

Koopman operator theory is a kind of data-driven modelling approach that accurately captures the nonlinearities of mechatronic systems such as vehicles against physics-based methods. However, the infinite-dimensional Koopman operator is…

We analyze Neural Ordinary Differential Equations (NODEs) from a control theoretical perspective to address some of the main properties and paradigms of Deep Learning (DL), in particular, data classification and universal approximation.…

The neural operator (NO) framework has emerged as a powerful tool for solving partial differential equations (PDEs). Recent NOs are dominated by the Transformer architecture, which offers NOs the capability to capture long-range…

Operator regression provides a powerful means of constructing discretization-invariant emulators for partial-differential equations (PDEs) describing physical systems. Neural operators specifically employ deep neural networks to approximate…