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and more specifically in Morse theory, a gradient-like vector field is a generalization of gradient vector field. The primary motivation is as a technical
Gradient-like_vector_field
Multivariate derivative (mathematics)
In vector calculus, the gradient of a scalar-valued differentiable function f {\displaystyle f} of several variables is the vector field (or vector-valued
Gradient
Assignment of a vector to each point in a subset of Euclidean space
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space R n {\displaystyle
Vector_field
Evaluates a line integral through a gradient field using the original scalar field
\right)} where ∇φ denotes the gradient vector field of φ. The gradient theorem implies that line integrals through gradient fields are path-independent. In
Gradient_theorem
Vector differential operator
act on scalar fields by a formal scalar multiplication—to give a vector field called the gradient; second, it can act on vector fields by a formal dot
Del
Computer vision framework
Gradient vector flow (GVF), a computer vision framework introduced by Chenyang Xu and Jerry L. Prince, is the vector field that is produced by a process
Gradient_vector_flow
Broad concept generalizing scalars in mathematics and physics
codomain, Conservative vector field, a vector field that is the gradient of a scalar potential field Hamiltonian vector field, a vector field defined for any
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Four-vector analogue of the gradient operation
geometry, the four-gradient (or 4-gradient) ∂ {\displaystyle {\boldsymbol {\partial }}} is the four-vector analogue of the gradient ∇ → {\displaystyle
Four-gradient
Optimization algorithm
independent variable adjustments is proportional to the gradient vector of partial derivatives. The gradient descent can take many iterations to compute a local
Gradient_descent
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
and theoretical physics, a Killing vector field or Killing field (named after Wilhelm Killing) is a vector field on a Riemannian manifold or pseudo-Riemannian
Killing_vector_field
Machine learning technique
Gradient boosting is a machine learning technique based on boosting in a functional space, where the target is pseudo-residuals instead of residuals as
Gradient_boosting
Optimization algorithm
algorithm converges. In pseudocode, stochastic gradient descent can be presented as : Choose an initial vector of parameters w {\displaystyle w} and learning
Stochastic_gradient_descent
Set of methods for supervised statistical learning
In machine learning, a support vector machine (SVM) or support vector network is a supervised max-margin model with associated learning algorithms that
Support_vector_machine
Physical field surrounding an electric charge
electric field between atoms is the force responsible for chemical bonding that result in molecules. The electric field is defined as a vector field that
Electric_field
Local rate of change in potential with respect to displacement
curl, denoted ∇×, of the vector field vanishes. The force F {\displaystyle \mathbf {F} \,\!} is directly related to the gradient of the potential energy
Potential_gradient
Certain vector fields are the sum of an irrotational and a solenoidal vector field
theorem of vector calculus states that certain differentiable vector fields can be resolved into the sum of an irrotational (curl-free) vector field and a
Helmholtz_decomposition
Property of space that quantifies the magnetic influence at a given location
magnetic field may vary with location, it is described mathematically by assigning a vector to each point of space, making it a vector field. There are
Magnetic_field
Any Morse function f on a compact Riemannian manifold M defines a gradient vector field. If one imposes the condition that the unstable and stable manifolds
Morse–Smale_system
Quantity in electromagnetism
electromagnetism, magnetic vector potential (often denoted A) is the vector quantity defined so that its curl is equal to the magnetic field, B: ∇ × A = B {\textstyle
Magnetic_vector_potential
Field lines in a fluid flow
are field lines in a fluid flow. They differ only when the flow changes with time, that is, when the flow is not steady. Considering a velocity vector field
Streamlines, streaklines, and pathlines
Streamlines,_streaklines,_and_pathlines
Assignment of numbers to points in space
example Higgs-like fields. Vector fields, which associate a vector to every point in space. Some examples of vector fields include the air flow (wind)
Scalar_field
Vector operator in vector calculus
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters
Divergence
Use of coordinates for representing vectors
}{\partial z}}} With a scalar function f, the gradient is written as ∇ f , {\displaystyle \nabla f\,,} with a vector field F, the divergence is written as ∇ ⋅ F
Vector_notation
in solving non-linear equations like Euler's equations in computational fluid dynamics. Matrix-free conjugate gradient method has been applied in the non-linear
Matrix-free_methods
Magnetic resonance imaging sequence
bSSFP sequences demand a very high level of magnetic field homogeneity and control over gradient switching and shaping. The refocusing mechanism fails
Steady-state free precession imaging
Steady-state_free_precession_imaging
Machine learning model training problem
In machine learning, the vanishing gradient problem is the problem of greatly diverging gradient magnitudes between earlier and later layers encountered
Vanishing_gradient_problem
Set of coordinates where the coordinate hypersurfaces all meet at right angles
coordinates. Quantities like the gradient and Laplacian follow through proper application of this operator. From dr and normalized basis vectors êi, the following
Orthogonal_coordinates
Device that measures magnetism
magnitude of the vector magnetic field. Magnetometers used to study the Earth's magnetic field may express the vector components of the field in terms of declination
Magnetometer
Relativistic vector field
relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential
Electromagnetic four-potential
Electromagnetic_four-potential
Integration over a non-flat region in 3D space
scalar field (that is, a function of position which returns a scalar as a value), or a vector field (that is, a function which returns a vector as value)
Surface_integral
Optimization and sampling technique
models. Like stochastic gradient descent, SGLD is an iterative optimization algorithm which uses minibatching to create a stochastic gradient estimator
Stochastic gradient Langevin dynamics
Stochastic_gradient_Langevin_dynamics
Field of engineering
normally solved using appropriate techniques from the field of optimization. These include gradient-based algorithms, population-based algorithms, or others
Multidisciplinary design optimization
Multidisciplinary_design_optimization
Line integral of the electric field
In classical electrostatics, the electrostatic field is a vector quantity expressed as the gradient of the electrostatic potential, which is a scalar
Electric_potential
Derivative of a function with multiple variables
function ∇f which takes the point a to the vector ∇f(a). Consequently, the gradient produces a vector field. A common abuse of notation is to define the
Partial_derivative
Time rate of change of some physical quantity of a material element in a velocity field
was considered for the scalar above. For a vector, the gradient becomes a tensor derivative; for tensor fields we may want to take into account not only
Material_derivative
Physical quantities taking values at each point in space and time
In science, a field or field quantity is a physical quantity – represented by a scalar, vector, spinor, or tensor – that has a value for each point in
Field_(physics)
dislocation, geometrically necessary dislocations are accumulated in strain gradient fields caused by geometrical constraints of the crystal lattice. In this case
Geometrically necessary dislocations
Geometrically_necessary_dislocations
Method of machine learning
Stochastic gradient descent Learning models Adaptive Resonance Theory Hierarchical temporal memory k-nearest neighbor algorithm Learning vector quantization
Online_machine_learning
Measurable property of a material or system
and F respectively. No symbol is necessarily required for the gradient of a scalar field, since only the nabla/del operator ∇ or grad needs to be written
Physical_quantity
numbers in a vector become larger: 1 m becomes 1000 mm. Covariant vectors, on the other hand, have units of one-over-distance (as in a gradient) and transform
Introduction to the mathematics of general relativity
Introduction_to_the_mathematics_of_general_relativity
Optimization algorithm for artificial neural networks
computes the gradient in weight space of a feedforward neural network, with respect to a loss function. Denote: x {\displaystyle x} : input (vector of features)
Backpropagation
Model of atmospheric motion
the effective pressure vector force is contrary to the pressure gradient, whence the minus sign before the gradient vector. Friction. This is a force
Balanced_flow
Energy related to Earth's gravity
of the potential energy per unit mass, so that the gravity vector is obtained as the gradient of the geopotential, without the negation. In addition to
Geopotential
Recurrent neural network architecture
type of recurrent neural network (RNN) aimed at mitigating the vanishing gradient problem commonly encountered by traditional RNNs. Its relative insensitivity
Long_short-term_memory
Analyzes the topology of a manifold by studying differentiable functions on that manifold
point on each critical level, which is usually proven by using gradient-like vector fields to rearrange the critical points. Morse theory can be used to
Morse_theory
Concept in physics
order terms}},} where ∇v the gradient of the velocity field, understood as a linear map that takes a displacement vector r to the corresponding change
Strain-rate_tensor
Change in speed due only to gravity
source. It is a vector oriented toward the field source, of magnitude measured in acceleration units. The gravitational acceleration vector depends only
Gravitational_acceleration
Theorem in calculus
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through
Divergence_theorem
Extension of the scalar spherical harmonics for use with vector fields
In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH
Vector_spherical_harmonics
Geometric object that has length and direction
physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude
Euclidean_vector
abstract field that, like known physical fields, decreases with increasing distance. Both scalar and vector fields are included in the models. The field(s)
Neighbour-sensing_model
Medical imaging technique
in k-space with the velocity vector of the trajectory proportional to the vector of the applied magnetic field gradient. By the term effective spin density
Physics of magnetic resonance imaging
Physics_of_magnetic_resonance_imaging
Representation of a tensor in Euclidean space
_{\text{z}}\end{aligned}}} be vector fields, in which all scalar and vector fields are functions of the position vector r and time t. The gradient operator in Cartesian
Cartesian_tensor
Geometric model of the physical space
integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary ∂Σ: ∬ Σ
Three-dimensional_space
Energy held by an object because of its position relative to other objects
defines a scalar potential field. In this case, the force can be defined as the negative of the vector gradient of the potential field. If the work for an applied
Potential_energy
Type of feedforward neural network
Amari reported the first multilayered neural network trained by stochastic gradient descent, was able to classify non-linearily separable pattern classes.
Multilayer_perceptron
Operator generalizing the Laplacian in differential geometry
its gradient vector field, which is the sum of the n pure second derivatives of f with respect to each vector of an orthonormal basis for Rn. Like the
Laplace–Beltrami_operator
Physical theory describing classical fields
constitutes a vector field. As the day progresses, the directions in which the vectors point change as the directions of the wind change. The first field theories
Classical_field_theory
Research field that lies at the intersection of machine learning and computer security
support vector machines and neural networks) might be robust to adversaries, until Battista Biggio and others demonstrated the first gradient-based attacks
Adversarial_machine_learning
the vector points parallel and perpendicular to the direction of the wavevector, respectively. The parallel component can be expressed as the gradient of
Scalar–vector–tensor decomposition
Scalar–vector–tensor_decomposition
Gradient of electrochemical potential, usually for an ion that can move across a membrane
electrochemical gradient is a gradient of electrochemical potential, usually for an ion that can move across a membrane. The gradient consists of two
Electrochemical_gradient
Type of machine learning model
the documents into vectors, then finding the documents with vectors (usually stored in a vector database) most similar to the vector of the query. The
Large_language_model
Class of artificial neural network
2001). "Gradient flow in recurrent nets: the difficulty of learning long-term dependencies". In Kolen, John F.; Kremer, Stefan C. (eds.). A Field Guide
Recurrent_neural_network
Action of a massive abelian gauge field
a generalized magnetic potential. The field B μ {\displaystyle B^{\mu }} transforms like a complex four-vector. The Lagrangian density is given by: L
Proca_action
Geometric model of the planar projection of the physical universe
} the formula for the Euclidean length of the vector. In a rectangular coordinate system, the gradient is given by ∇ f = ∂ f ∂ x i + ∂ f ∂ y j . {\displaystyle
Euclidean_plane
Vector field reconstruction is a method of creating a vector field from experimental or computer-generated data, usually with the goal of finding a differential
Vector_field_reconstruction
Vector in relativity
In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an element of a four-dimensional vector space object with four components
Four-vector
Fundamental construction of differential calculus
R (scalar fields), the Fréchet derivative corresponds to a vector field called the total derivative. This can be interpreted as the gradient but it is
Generalizations of the derivative
Generalizations_of_the_derivative
Mathematical model for describing material deformation under stress
body can be described by a displacement field. A displacement field is a vector field of all displacement vectors for all particles in the body, which relates
Finite_strain_theory
Algorithm for supervised learning of binary classifiers
is a function that can decide whether or not an input, represented by a vector of numbers, belongs to some specific class. It is a type of linear classifier
Perceptron
Equations of motion for viscous fluids
scalar field, ∇ ρ {\textstyle \nabla \rho \,} is the gradient of density ( ρ {\displaystyle \rho } ), which is the vector derivative of a scalar field, to
Navier–Stokes_equations
Feature descriptor used in computer vision
The histogram of oriented gradients (HOG) is a feature descriptor used in computer vision and image processing for the purpose of object detection. The
Histogram of oriented gradients
Histogram_of_oriented_gradients
Statement about integration on manifolds
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called
Generalized_Stokes_theorem
Satellite system
possible to freely pivot the craft around in a known local gradient of the magnetic field by only using electrical energy. The magnetic dipole generated
Magnetorquer
Technique for the generative modeling of a continuous probability distribution
(like a Brownian walker) and gradient descent down the potential well. The randomness is necessary: if the particles were to undergo only gradient descent
Diffusion_model
Machine learning technique
This is used to train the policy by gradient ascent on it, usually using a standard momentum-gradient optimizer, like the Adam optimizer. The original paper
Reinforcement learning from human feedback
Reinforcement_learning_from_human_feedback
Machine learning technique
{\displaystyle w} , which takes input x {\displaystyle x} and produces a vector of outputs ( w ( x ) 1 , . . . , w ( x ) n ) {\displaystyle (w(x)_{1},.
Mixture_of_experts
Algorithm for modelling sequential data
gradient of F ( x ) {\displaystyle F(x)} is close to zero. Similarly to how the feedforward network modules are applied individually to each vector,
Transformer_(deep_learning)
Computer vision framework
with gradient angle θ = arctan ( C y C x ) , {\displaystyle \theta =\arctan \left({\frac {C_{y}}{C_{x}}}\right),} unit vectors along the gradient direction
Active_contour_model
Machine learning calibration technique
classes. The method was invented by John Platt in the context of support vector machines, replacing an earlier method by Vapnik, but can be applied to other
Platt_scaling
Interaction between electronic and nuclear vibrational motion in a molecule
the calculated coupling vector, although this error is usually tolerable. Evaluating derivative couplings with analytic gradient methods has the advantage
Vibronic_coupling
Transformation of a body from a reference configuration to a current configuration
differentiation of the displacement vector with respect to the material coordinates yields the material displacement gradient tensor ∇Xu. Thus we have: u (
Deformation_(physics)
Branch of theoretical physics
velocity and magnetic field vectors. The other vector is in the same direction as the electric field. The sum of these two vectors is the Lorentz force. Although
Classical_electromagnetism
Equations describing classical electromagnetism
magnetic field is a solenoidal vector field. The Maxwell–Faraday version of Faraday's law of induction describes how a time-varying magnetic field corresponds
Maxwell's_equations
Electromagnetic quantum-mechanical effect in regions of zero magnetic and electric field
was divergenceless, it was possible to express an electrostatic field as the gradient of a scalar potential (e.g. Coulomb's electrostatic potential, which
Aharonov–Bohm_effect
Machine learning model for vision processing
serializes each patch into a vector, and maps it to a smaller dimension with a single matrix multiplication. These vector embeddings are then processed
Vision_transformer
smallest. Therefore, growth in the error vector will cause a corresponding decrease in the tracer gradient and vice versa. This can be understood very
Chaotic_mixing
Model-free reinforcement learning algorithm
}\left(s_{t}\right)-{\hat {R}}_{t}\right)^{2}} typically via some gradient descent algorithm. Like all policy gradient methods, PPO is used for training an RL agent whose
Proximal_policy_optimization
Mathematical concept applicable to physics
property. In vector calculus, flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface
Flux
Canonical connection on a pseudo-Riemannian manifold
{\displaystyle \mathrm {grad} _{g}(\gamma )} is the gradient vector field of γ {\displaystyle \gamma } i.e. the vector field g {\displaystyle g} -dual to d γ {\displaystyle
Levi-Civita_connection
Smooth approximation of one-hot arg max
=(z_{1},\dotsc ,z_{K})\in \mathbb {R} ^{K}} and computes each component of vector σ ( z ) ∈ ( 0 , 1 ) K {\displaystyle \sigma (\mathbf {z} )\in (0,1)^{K}}
Softmax_function
Analysis of terrain
terrain surface is a vector ray that is perpendicular to the surface. The surface gradient ( ∇ f {\displaystyle \nabla f} ) is the vector ray that is tangent
Geomorphometry
Method in natural language processing
representation is a real-valued vector that encodes the meaning of the word in such a way that the words that are closer in the vector space are expected to be
Word_embedding
Memory unit used in neural networks
The GRU is like a long short-term memory (LSTM) with a gating mechanism to input or forget certain features, but lacks a context vector or output gate
Gated_recurrent_unit
Exterior algebraic map taking tensors from p forms to n-p forms
Euclidean space, in which divergence of a vector field may be realized as the codifferential opposite to the gradient operator, and the Laplace operator on
Hodge_star_operator
Symbol used to indicate the del operator
electrodynamics, textbooks. The nabla is used in vector calculus as part of three distinct differential operators: the gradient (∇), the divergence (∇⋅), and the curl
Nabla_symbol
Equations of fluid dynamics
differentiation is left out, the result is a third order vector equation containing an unknown vector field (the gradient of pressure) that may be determined from the
Derivation of the Navier–Stokes equations
Derivation_of_the_Navier–Stokes_equations
Method of utilizing water in magnetic resonance imaging
of diffusion gradients (i.e. magnetic field variations in the MRI magnet) are applied that can determine at least 3 directional vectors (use of 6 different
Diffusion-weighted magnetic resonance imaging
Diffusion-weighted_magnetic_resonance_imaging
Method of improving artificial neural network
how quickly the network learns—without causing problems like vanishing or exploding gradients, where updates become too small or too large. It also appears
Batch_normalization
Millennium Prize Problem
velocity vector v and the gradient operator ∇. Because the gradient operator is a linear operator, the term (v · ∇)v is nonlinear in the velocity vector v.
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
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GRADIENT LIKE-VECTOR-FIELD
GRADIENT LIKE-VECTOR-FIELD
Male
English
English name derived from the vocabulary word, from Latin lacus, LAKE means "pond, lake."
Female
Hungarian
Hungarian pet form of Greek Eva, ÉVIKE means "life."
Male
English
Pet form of English Michael, MIKE means "who is like God?"
Male
English
Roman Latin name VICTOR means "conqueror."Â
Boy/Male
American, Australian, British, Chinese, Christian, Danish, Dutch, English, French, German, Hebrew, Irish, Italian, Jamaican, Swedish, Swiss
Who is Like God; Form of Michael
Male
Portuguese
Galician-Portuguese form of Roman Latin Victor, VITOR means "conqueror."
Male
Arthurian
, sir Hector de Maris; (defender).
Boy/Male
Spanish
Victor.
Boy/Male
Australian, British, Christian, English
Pond; Lake
Male
French
French form of Roman Latin Gratian, GRATIEN means "pleasing, agreeable."
Boy/Male
Greek, Indian, Swedish
Victory; Useful; Bringer of Victory
Male
English
 Anglicized form of Scottish Gaelic Eachann, HECTOR means "brown horse." Compare with another form of Hector.
Boy/Male
American, British, Christian, Danish, Dutch, English, Finnish, French, German, Greek, Hindu, Indian, Irish, Jamaican, Latin, Romanian, Slovenia, Spanish, Swedish, Swiss, Tamil, Ukrainian
Victorious; Conqueror; Winner; Champion; One who Conquers; Victory
Male
Hawaiian
Hawaiian name LIKO means "bud."
Male
Portuguese
Portuguese form of Latin Hector, HEITOR means "defend; hold fast."
Male
Scandinavian
 Scandinavian form of Roman Latin Victor, VIKTOR means "conqueror." Compare with another form of Viktor.
Boy/Male
English American
Doctor; teacher.
Girl/Female
Greek
In Greek mythology Nike was the goddess of victory.
Male
Native American
Native American Miwok name LISE means "salmon head rising above water." Compare with feminine Lise.
Girl/Female
Dutch, French, German, Greek
Victory
GRADIENT LIKE-VECTOR-FIELD
GRADIENT LIKE-VECTOR-FIELD
GRADIENT LIKE-VECTOR-FIELD
GRADIENT LIKE-VECTOR-FIELD
GRADIENT LIKE-VECTOR-FIELD
GRADIENT LIKE-VECTOR-FIELD
GRADIENT LIKE-VECTOR-FIELD
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