What is an Inverse Problem?
Definition
When there is a physical phenomenon expressed by a formula, the process of solving the formula based on a cause to find information about the result is called a direct problem or forward problem.
Conversely, the process of finding information about the cause from information about the result is called an inverse problem.
Explanation
Many problems are direct problems, for example, when an object with an initial velocity of $v_{0}$ moves in a parabolic motion at a launch angle of $\theta$, finding the horizontal range is one such problem. Here, the inverse problem would be finding the initial velocity and launch angle when the horizontal range is known.
Differential Equations
Consider the initial value problem of the following wave equation.
$$ \begin{cases} \Delta u - u_{tt} = 0 &\text{in } \mathbb{R}^{n} \times (0, \infty) \\ u = f, u_{t} = 0 &\text{on } \mathbb{R}^{n} \times \left\{ t=0 \right\} \end{cases} $$
Here, solving the initial value problem, that is, finding the solution $y$ when the initial value $f$ is given, is a direct problem. Conversely, finding the initial value $f$ when some solution $y$ is given is an inverse problem.
If the solution $y$ is uniquely determined by the initial value $f$, then the initial value problem itself can be considered as an operator and expressed as $W f = y$. Solving the inverse problem of the given wave equation means finding the inverse operator $W^{-1}$ that satisfies $f = W^{-1}y$. Then, whenever the solution $y$ is known, the initial value $f$ can be found.
Integral Equations
$$ a = \int_{x_{0}}^{x_{1}} f(x)dx $$
In the integral equation given above, computing the definite integral $a$ whenever the function $f$ is given is a direct problem. Conversely, finding the function $f$ that satisfies the equation when the integral value $a$ is given is an inverse problem. As can be seen, solving integral equations is quite difficult.
Can One Hear the Shape of a Drum?
A famous example is the problem addressed in the paper Can One Hear the Shape of a Drum? published by mathematician Mark Kac in 1966, which is about guessing the shape of a drum from its sound. If one considers the shape of the drum as the cause (initial value) and the sound as the result (solution), then figuring out the shape of the drum by its sound is an inverse problem.