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7 Inverse Problems in Physics
Schrödinger equation in quantum mechanics is considered the same way. On the
other hand, as its “inverse,” we can ask the following question: given the state at a
certain time t > 0, is it possible to determine the state at the time t = 0 to reproduce
it? This is what is called the initial value problem, and it is an inverse problem.
This example of a time evolution is easy to understand. More generally, the
inverse problem can be formulated as follows. Consider a polynomial function with
real number x as an argument:
f (x) =
n
k=0
J k x
k .
(7.1)
Given x, f (x) is determined, so the procedure to calculate f (x) from x can be
thought of as “time evolution” in which, given an initial value of x, after a while,
the system gives f (x). Thus, the inverse problem of “going back in time” ˙ I is that,
when a function f (x) is given and a value of y = f (a) is given, “solve a” is the
problem. Causally, it is a matter of estimating the cause from the result. 3
When considering the temporal order, there is a causal relationship between
the cause and the result. There are many inverse problems that are not about the
temporal order, but about spatial order. For example, nondestructive inspection
and X-ray tomography are often said to be typical examples of inverse problems.
This is a method of estimating whether there is a cavity inside an object by using
laws of conduction. Given the external shape of the object (outer surface boundary
condition, y = f (a) in the above equation) and the laws of conduction (e.g. Poisson
equation, wave equation, heat conduction equation, etc. which give the form of the
function f (x), that is, the coefficient J k ), the problem is to find the shape of the
internal cavity, that is, x. In this way, the problem of finding the original initial
conditions or boundary conditions from the values, for a given system of equations,
is an inverse problem.
Now, there is another kind of inverse problem, namely, when the function f (x)
itself is unknown, find the function f (x). In this case, the given condition is that
when the input data is x = x 1 , the output data is y 1 = f (x 1 ). Given such a condition,
the problem is to determine the unknown function f (x).
Of course, just a condition y 1 = f (x 1 ) cannot determine a general function.
This pair (x 1 , y 1 ) only has one constraint expression for the unknown coefficients
J 0 , · · · , J n of the polynomial function. Therefore, to determine the unknown
function f (x), generally the same number of input/output pairs (x i , y i ) (i =
1, 2, 3, · · · ) must be prepared.
Supervised machine learning is this latter problem. The pair of input data and
output data becomes the training data, and the unknown function becomes a neural
3 Of course, the Schrödinger equation is a linear equation, so positive time evolution and negative
time evolution are the same in terms of difficulty to solve, and should not be called inverse
problems.
7 Inverse Problems in Physics
Schrödinger equation in quantum mechanics is considered the same way. On the
other hand, as its “inverse,” we can ask the following question: given the state at a
certain time t > 0, is it possible to determine the state at the time t = 0 to reproduce
it? This is what is called the initial value problem, and it is an inverse problem.
This example of a time evolution is easy to understand. More generally, the
inverse problem can be formulated as follows. Consider a polynomial function with
real number x as an argument:
f (x) =
n
k=0
J k x
k .
(7.1)
Given x, f (x) is determined, so the procedure to calculate f (x) from x can be
thought of as “time evolution” in which, given an initial value of x, after a while,
the system gives f (x). Thus, the inverse problem of “going back in time” ˙ I is that,
when a function f (x) is given and a value of y = f (a) is given, “solve a” is the
problem. Causally, it is a matter of estimating the cause from the result. 3
When considering the temporal order, there is a causal relationship between
the cause and the result. There are many inverse problems that are not about the
temporal order, but about spatial order. For example, nondestructive inspection
and X-ray tomography are often said to be typical examples of inverse problems.
This is a method of estimating whether there is a cavity inside an object by using
laws of conduction. Given the external shape of the object (outer surface boundary
condition, y = f (a) in the above equation) and the laws of conduction (e.g. Poisson
equation, wave equation, heat conduction equation, etc. which give the form of the
function f (x), that is, the coefficient J k ), the problem is to find the shape of the
internal cavity, that is, x. In this way, the problem of finding the original initial
conditions or boundary conditions from the values, for a given system of equations,
is an inverse problem.
Now, there is another kind of inverse problem, namely, when the function f (x)
itself is unknown, find the function f (x). In this case, the given condition is that
when the input data is x = x 1 , the output data is y 1 = f (x 1 ). Given such a condition,
the problem is to determine the unknown function f (x).
Of course, just a condition y 1 = f (x 1 ) cannot determine a general function.
This pair (x 1 , y 1 ) only has one constraint expression for the unknown coefficients
J 0 , · · · , J n of the polynomial function. Therefore, to determine the unknown
function f (x), generally the same number of input/output pairs (x i , y i ) (i =
1, 2, 3, · · · ) must be prepared.
Supervised machine learning is this latter problem. The pair of input data and
output data becomes the training data, and the unknown function becomes a neural
3 Of course, the Schrödinger equation is a linear equation, so positive time evolution and negative
time evolution are the same in terms of difficulty to solve, and should not be called inverse
problems.