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In deep learning, it is usually assumed that the shape of the loss surface is fixed. Differently, a novel concept of deformation operator is first proposed in this paper to deform the loss surface, thereby improving the optimization.…

Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. In this paper, we present a proof-of-principle sublinear-time algorithm for solving SDPs with low-rank…

Beam search is a desirable choice of test-time decoding algorithm for neural sequence models because it potentially avoids search errors made by simpler greedy methods. However, typical cross entropy training procedures for these models do…

Many applications in computational physics involve approximating problems with microstructure, characterized by multiple spatial scales in their data. However, these numerical solutions are often computationally expensive due to the need to…

Although projection-based reduced-order models (ROMs) for parameterized nonlinear dynamical systems have demonstrated exciting results across a range of applications, their broad adoption has been limited by their intrusivity: implementing…

A widely used strategy to reduce the computational cost in quantum-chemical calculations is to partition the system into an active subsystem, which is the focus of the computational efforts and an environment that is treated at a lower…

We address the problem of implementing bottleneck layers from classical pre-trained neural networks on a quantum computer, with the goal of exploring intrinsically quantum ansatz for representing large linear layers within hybrid…

A nonlinear-manifold reduced order model (NM-ROM) is a great way of incorporating underlying physics principles into a neural network-based data-driven approach. We combine NM-ROMs with domain decomposition (DD) for efficient computation.…

Koopman analysis of a general dynamics system provides a linear Koopman operator and an embedded eigenfunction space, enabling the application of standard techniques from linear analysis. However, in practice, deriving exact operators and…

This paper proposes a domain decomposition subspace neural network method for efficiently solving linear and nonlinear partial differential equations. By combining the principles of domain decomposition and subspace neural networks, the…

We consider the decomposition of bounded linear operators on Hilbert spaces in terms of functions forming frames. Similar to the singular-value decomposition, the resulting frame decompositions encode information on the structure and…

We propose a finite-element local basis-based operator learning framework for solving partial differential equations (PDEs). Operator learning aims to approximate mappings from input functions to output functions, where the latter are…

Optimizing machine learning algorithms that are used to solve the objective function has been of great interest. Several approaches to optimize common algorithms, such as gradient descent and stochastic gradient descent, were explored. One…

In this paper, we propose a refinement strategy to the well-known Physics-Informed Neural Networks (PINNs) for solving partial differential equations (PDEs) based on the concept of Optimal Transport (OT). Conventional black-box PINNs…

Convolution-type integral equations commonly occur in signal processing and image processing. Discretizing these equations yields large and ill-conditioned linear systems. While the classic multigrid method is effective for solving linear…

Restoring images degraded by spatially varying blur is a problem encountered in many disciplines such as astrophysics, computer vision or biomedical imaging. One of the main challenges to perform this task is to design efficient numerical…

Mathematical programming -- the task of expressing operations and decision-making problems in precise mathematical language -- is fundamental across domains, yet remains a skill-intensive process requiring operations research expertise.…

Optimal power flow (OPF) is an important problem in the operation of electric power systems. Due to the OPF problem's non-convexity, there may exist multiple local optima. Certifiably obtaining the global solution is important for certain…

Many engineering processes can be accurately modelled using partial differential equations (PDEs), but high dimensionality and non-convexity of the resulting systems pose limitations on their efficient optimisation. In this work, a model…

We propose a novel fine-tuning method to achieve multi-operator learning through training a distributed neural operator with diverse function data and then zero-shot fine-tuning the neural network using physics-informed losses for…