Chapter 7
Inverse Problems in Physics
Abstract First, we consider inverse problems in physics. In fact, inverse problems
are at the heart of revolutionary development in physics. What does it mean to
solve an inverse problem? What is the meaning of the phrase “machine learning
is good at solving inverse problems”? You will gain a comprehensive perspective
and significance in applying machine learning to theoretical physics.
There are many inverse problems in the world, and so there are in physics. In
this chapter, we will look at inverse problems 1 that can appear in physics from the
perspective of the machine learning, and learn the significance of them. 2 What is
an “inverse problem,” in the first place? In general, machine learning is often said
to be good at solving inverse problems. Why is that? By examining these, we can
concretely see how machine learning and deep learning can be applied to physics,
and at the same time we will learn how it makes sense to rely on machine learning
for individual physics problems that individual researchers have.
7.1 Inverse Problems and Learning
The inverse of the usual approach to solving a problem is commonly called the
inverse problem. The general nature of the inverse problem will be described later,
and let us start with a simple example.
Consider a classical mechanical problem that follows the evolution of time.
Given a differential equation that determines the time evolution, and then given the
initial state at a certain time t = 0, the state at any time t > 0 is determined. The
1 The inverse problem in mathematics is different from the generic inverse problem dealt with here.
In mathematics, when there is a theorem C that A is B, then the proposition that B is A is called the
inverse of theorem C. For example, one of the famous theorems proved by Kiyoshi Oka is Hartogs’
inverse problem, which is the inverse problem in that sense.
2 The contents of this chapter do not enumerate general solutions for inverse problems.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
A. Tanaka et al., Deep Learning and Physics, Mathematical Physics Studies,
https://doi.org/10.1007/978-981-33-6108-9_7
129
Inverse Problems in Physics
Abstract First, we consider inverse problems in physics. In fact, inverse problems
are at the heart of revolutionary development in physics. What does it mean to
solve an inverse problem? What is the meaning of the phrase “machine learning
is good at solving inverse problems”? You will gain a comprehensive perspective
and significance in applying machine learning to theoretical physics.
There are many inverse problems in the world, and so there are in physics. In
this chapter, we will look at inverse problems 1 that can appear in physics from the
perspective of the machine learning, and learn the significance of them. 2 What is
an “inverse problem,” in the first place? In general, machine learning is often said
to be good at solving inverse problems. Why is that? By examining these, we can
concretely see how machine learning and deep learning can be applied to physics,
and at the same time we will learn how it makes sense to rely on machine learning
for individual physics problems that individual researchers have.
7.1 Inverse Problems and Learning
The inverse of the usual approach to solving a problem is commonly called the
inverse problem. The general nature of the inverse problem will be described later,
and let us start with a simple example.
Consider a classical mechanical problem that follows the evolution of time.
Given a differential equation that determines the time evolution, and then given the
initial state at a certain time t = 0, the state at any time t > 0 is determined. The
1 The inverse problem in mathematics is different from the generic inverse problem dealt with here.
In mathematics, when there is a theorem C that A is B, then the proposition that B is A is called the
inverse of theorem C. For example, one of the famous theorems proved by Kiyoshi Oka is Hartogs’
inverse problem, which is the inverse problem in that sense.
2 The contents of this chapter do not enumerate general solutions for inverse problems.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
A. Tanaka et al., Deep Learning and Physics, Mathematical Physics Studies,
https://doi.org/10.1007/978-981-33-6108-9_7
129