In
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
and
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural ansatzes
or, from German, ansätze ; ) is an
educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the solution by its results.
Use
An ansatz is the establishment of the starting equation(s), the theorem(s), or the value(s) describing a mathematical or physical problem or solution. It typically provides an initial estimate or framework to the solution of a mathematical problem,
and can also take into consideration the
boundary conditions (in fact, an ansatz is sometimes thought of as a "trial answer" and an important technique in solving differential equations
).
After an ansatz, which constitutes nothing more than an assumption, has been established, the equations are solved more precisely for the general function of interest, which then constitutes a confirmation of the assumption. In essence, an ansatz makes assumptions about the form of the solution to a problem so as to make the solution easier to find.
It has been demonstrated that machine learning techniques can be applied to provide initial estimates similar to those invented by humans and to discover new ones in case no ansatz is available.
Examples
Given a
set of experimental data that looks to be
clustered about a line, a linear ansatz could be made to find the
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s of the line by a
least squares curve fit.
Variational approximation methods use ansätze and then fit the parameters.
Another example could be the mass, energy, and entropy balance equations that, considered simultaneous for purposes of the elementary operations of
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mathemat ...
, are the ansatz to most basic problems of
thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
.
Another example of an ansatz is to suppose the solution of a homogeneous linear
differential equation to take an exponential form, or a power form in the case of a
difference equation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
. More generally, one can guess a particular solution of a system of equations, and test such an ansatz by directly substituting the solution into the system of equations. In many cases, the assumed form of the solution is general enough that it can represent arbitrary functions, in such a way that the set of solutions found this way is a full set of all the solutions.
See also
*
Method of undetermined coefficients
*
Bayesian inference
*
Bethe ansatz
*
Coupled cluster, a technique for solving the many-body problem that is based on an exponential Ansatz
*
Demarcation problem
In philosophy of science and epistemology, the demarcation problem is the question of how to distinguish between science and non-science. It also examines the boundaries between science, pseudoscience and other products of human activity, like ...
*
Guesstimate
*
Heuristic
A heuristic or heuristic technique (''problem solving'', '' mental shortcut'', ''rule of thumb'') is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless ...
*
Hypothesis
A hypothesis (: hypotheses) is a proposed explanation for a phenomenon. A scientific hypothesis must be based on observations and make a testable and reproducible prediction about reality, in a process beginning with an educated guess o ...
*
Trial and error
*
Train of thought
References
Bibliography
*
*
*
*{{Citation
, last1 = Karbach
, first1 = M.
, last2 = Hu
, first2 = K.
, last3 = Müller
, first3 = G.
, title = Introduction to the Bethe ansatz III.
, date = August 1, 2000
, url = http://www.phys.uri.edu/~gerhard/MSS/ms91.pdf
, access-date = 2008-10-25
, archive-url = https://web.archive.org/web/20060901073407/http://www.phys.uri.edu/~gerhard/MSS/ms91.pdf
, archive-date = September 1, 2006
, url-status = dead
Philosophy of physics
Concepts in physics
Mathematical terminology