7.3 Inverse Problems and Physical Machine Learning
135
The boundary value problem does not work well when there is a stability
problem, among the ill-posedness according to Hadamard. As one of such examples,
we described the thermal diffusion equation. More generally, the inverse problem
of chaotic systems has stability problems. A chaotic system is a deterministic
dynamical system with bounded orbits and a small difference in initial conditions
shows an exponential increase. In other words, it is a system that “gets messed up
over time and does not remember the initial state.” Given the final state, one will not
be able to go back in time to build the initial state.
Nevertheless, the research of chaos is progressing in various ways. The reason
is that there are research perspectives regarding what kind of decision equation
of time evolution can produce chaos, and what is the source. For example, there
is a proof that chaos does not occur if the number of degrees of freedom is too
small: the Poincaré–Bendixson theorem. Therefore, even if the boundary value
inverse problem is ill-posed, the target of research could be fertile, putting the actual
problem of solvability aside.
For example, there are many studies that show that chaotic attractors are related
to the mechanism of memory in the brain, so research related to neural networks can
be expected in the future. In addition, the Lyapunov exponent, which characterizes
the strength of chaos (the Lyapunov exponent λ measures the extent to which
the difference exponentially amplifies as e λt ), gives a quantitative idea of how illposed the inverse problem is. The Lyapunov exponent of quantum systems is being
studied in relation to the AdS/CFT correspondence in string theory and quantum
information theory, and further development about the relationship with machine
learning is expected.
On the other hand, the inverse problem which is categorized as the system
determination problem is exactly where machine learning is applied. A system
determination problem that has been studied for a long time is a potential determination problem in quantum mechanics. Here we describe one example of the inverse
problem in quantum mechanics.
The quantum mechanics of a one-body system is defined by a Hamiltonian H .
Given the initial state wave function, the final state wave function can be obtained
by the time evolution operator e −iH t with the Hamiltonian H . This is a forward
problem. The inverse problem as a system determination problem is the case when
the Hamiltonian H is unknown. What we should regard as known is important to
solve this problem. As an example, let us describe one of the well-studied methods,
the inverse scattering method. In the inverse scattering method, when the unknown
part of the Hamiltonian is the potential V (x) and the potential is asymptotically flat,
the potential is determined from information such as scattering amplitudes. Suppose
a one-dimensional non-relativistic quantum mechanics is given by an unknown
potential V (x). Suppose also that we measured the S matrix that tells us how
much of a plane wave coming from the spatial infinity is transmitted and reflected.
In addition, suppose we know how the wave function bound to potential V (x)
approaches zero near infinity. A method for reconstructing the potential V (x) from
this information is the inverse scattering method. In the inverse scattering method,
135
The boundary value problem does not work well when there is a stability
problem, among the ill-posedness according to Hadamard. As one of such examples,
we described the thermal diffusion equation. More generally, the inverse problem
of chaotic systems has stability problems. A chaotic system is a deterministic
dynamical system with bounded orbits and a small difference in initial conditions
shows an exponential increase. In other words, it is a system that “gets messed up
over time and does not remember the initial state.” Given the final state, one will not
be able to go back in time to build the initial state.
Nevertheless, the research of chaos is progressing in various ways. The reason
is that there are research perspectives regarding what kind of decision equation
of time evolution can produce chaos, and what is the source. For example, there
is a proof that chaos does not occur if the number of degrees of freedom is too
small: the Poincaré–Bendixson theorem. Therefore, even if the boundary value
inverse problem is ill-posed, the target of research could be fertile, putting the actual
problem of solvability aside.
For example, there are many studies that show that chaotic attractors are related
to the mechanism of memory in the brain, so research related to neural networks can
be expected in the future. In addition, the Lyapunov exponent, which characterizes
the strength of chaos (the Lyapunov exponent λ measures the extent to which
the difference exponentially amplifies as e λt ), gives a quantitative idea of how illposed the inverse problem is. The Lyapunov exponent of quantum systems is being
studied in relation to the AdS/CFT correspondence in string theory and quantum
information theory, and further development about the relationship with machine
learning is expected.
On the other hand, the inverse problem which is categorized as the system
determination problem is exactly where machine learning is applied. A system
determination problem that has been studied for a long time is a potential determination problem in quantum mechanics. Here we describe one example of the inverse
problem in quantum mechanics.
The quantum mechanics of a one-body system is defined by a Hamiltonian H .
Given the initial state wave function, the final state wave function can be obtained
by the time evolution operator e −iH t with the Hamiltonian H . This is a forward
problem. The inverse problem as a system determination problem is the case when
the Hamiltonian H is unknown. What we should regard as known is important to
solve this problem. As an example, let us describe one of the well-studied methods,
the inverse scattering method. In the inverse scattering method, when the unknown
part of the Hamiltonian is the potential V (x) and the potential is asymptotically flat,
the potential is determined from information such as scattering amplitudes. Suppose
a one-dimensional non-relativistic quantum mechanics is given by an unknown
potential V (x). Suppose also that we measured the S matrix that tells us how
much of a plane wave coming from the spatial infinity is transmitted and reflected.
In addition, suppose we know how the wave function bound to potential V (x)
approaches zero near infinity. A method for reconstructing the potential V (x) from
this information is the inverse scattering method. In the inverse scattering method,