Topic
Mathematical Operators
About: Mathematical Operators is a research topic. Over the lifetime, 943 publications have been published within this topic receiving 19003 citations.
Papers published on a yearly basis
1 / 2
Papers
TL;DR: A new deep neural network called DeepONet can lean various mathematical operators with small generalization error and can learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations.
Abstract: It is widely known that neural networks (NNs) are universal approximators of continuous functions. However, a less known but powerful result is that a NN with a single hidden layer can accurately approximate any nonlinear continuous operator. This universal approximation theorem of operators is suggestive of the structure and potential of deep neural networks (DNNs) in learning continuous operators or complex systems from streams of scattered data. Here, we thus extend this theorem to DNNs. We design a new network with small generalization error, the deep operator network (DeepONet), which consists of a DNN for encoding the discrete input function space (branch net) and another DNN for encoding the domain of the output functions (trunk net). We demonstrate that DeepONet can learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations. We study different formulations of the input function space and its effect on the generalization error for 16 different diverse applications. Neural networks are known as universal approximators of continuous functions, but they can also approximate any mathematical operator (mapping a function to another function), which is an important capability for complex systems such as robotics control. A new deep neural network called DeepONet can lean various mathematical operators with small generalization error.
1,667 citations
TL;DR: It is proved that every stringlike logical operator of this code can be deformed to a disjoint union of short segments, each of which is in the stabilizer group, and introduced a notion of "logical string segments" to avoid difficulties in defining one-dimensional objects in discrete lattices.
Abstract: We suggest concrete models for self-correcting quantum memory by reporting examples of local stabilizer codes in 3D that have no string logical operators. Previously known local stabilizer codes in 3D all have stringlike logical operators, which make the codes non-self-correcting. We introduce a notion of “logical string segments” to avoid difficulties in defining one-dimensional objects in discrete lattices. We prove that every stringlike logical operator of our code can be deformed to a disjoint union of short segments, each of which is in the stabilizer group. The code has surfacelike logical operators whose partial implementation has unsatisfied stabilizers along its boundary.
833 citations
TL;DR: This paper shows that connected operators work implicitly on a structured representation of the image made of flat zones, and proposes the max-tree as a suitable and efficient structure to deal with the processing steps involved in antiextensive connected operators.
Abstract: This paper deals with a class of morphological operators called connected operators. These operators filter the signal by merging its flat zones. As a result, they do not create any new contours and are very attractive for filtering tasks where the contour information has to be preserved. This paper shows that connected operators work implicitly on a structured representation of the image made of flat zones. The max-tree is proposed as a suitable and efficient structure to deal with the processing steps involved in antiextensive connected operators. A formal definition of the various processing steps involved in the operator is proposed and, as a result, several lines of generalization are developed. First, the notion of connectivity and its definition are analyzed. Several modifications of the traditional approach are presented. They lead to connected operators that are able to deal with texture. They also allow the definition of connected operators with less leakage than the classical ones. Second, a set of simplification criteria are proposed and discussed. They lead to simplicity-, entropy-, and motion-oriented operators. The problem of using a nonincreasing criterion is analyzed. Its solution is formulated as an optimization problem that can be very efficiently solved by a Viterbi (1979) algorithm. Finally, several implementation issues are discussed showing that these operators can be very efficiently implemented.
715 citations
TL;DR: In this article, a formalism is developed which allows overlap, kinetic energy, potential energy and electron repulsion integrals over cartesian Gaussian functions to be expressed in a very compact form involving easily computed auxiliary functions.
637 citations
TL;DR: In this paper, the authors derive upper bounds on the coefficients of the most general ΔF = 2 effective Hamiltonian, and these upper bounds can be translated into lower bounds on new physics that contributes to these low-energy effective interactions.
Abstract: We update the constraints on new-physics contributions to ΔF = 2 processes from the generalized unitarity triangle analysis, including the most recent experimental developments. Based on these constraints, we derive upper bounds on the coefficients of the most general ΔF = 2 effective Hamiltonian. These upper bounds can be translated into lower bounds on the scale of new physics that contributes to these low-energy effective interactions. We point out that, due to the enhancement in the renormalization group evolution and in the matrix elements, the coefficients of non-standard operators are much more constrained than the coefficient of the operator present in the Standard Model. Therefore, the scale of new physics in models that generate new ΔF = 2 operators, such as next-to-minimal flavour violation, has to be much higher than the scale of minimal flavour violation, and it most probably lies beyond the reach of direct searches at the LHC.
596 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 5 |
| 2020 | 4 |
| 2019 | 12 |
| 2018 | 13 |
| 2017 | 8 |
| 2016 | 11 |