Inverse problems are a fundamental tool in many emerging technologies, enabling the recovery of important information from limited and noisy observations. In this survey, we aim to provide a comprehensive overview of the state of the art in inverse problems within emerging technologies. We focus on several important areas, including artificial intelligence, healthcare, machine learning, wireless communication, and energy modeling. We discuss the challenges and limitations of these inverse problems, as well as the recent advancements in their solution. Additionally, we highlight the significance of these inverse problems in each of these areas, and the potential impact they have on the development of new technologies. This survey provides a comprehensive overview of inverse problems in emerging technologies and serves as a valuable resource for researchers and practitioners in the field.
Mathematical modeling is a powerful tool to solve problems emerging from the development of
science and technology. The goal is frequently to gain quantities that can only be indirectly
detected through measurements. For instance, a direct scan of the human body’s interior is not
conceivable. However, there exist methods that can obtain a scan by measuring the remaining
intensity of X-rays that go through the body. Such indirect measurements are the result of
processes, which are usually mathematically modeled and depend on the quantities we wish to
ifnd. This type of problem is called an inverse problem, involve calculating causal factors from
observations [
Meanwhile, emerging technologies, such as Artificial Intelligence (AI), Internet of Things (IoT), medical imaging, augmented reality, and energy, are particularly well-suited to the use of inverse problems due to their need for fast and accurate solutions to complex problems. An example of an emerging technology problem that can be posed as an inverse problem is in the field of medical imaging. Medical imaging techniques, such as computed tomography (CT) and magnetic resonance imaging (MRI), allow doctors to visualize the internal structure of the human body. However, these images are often limited by factors such as noise and limited spatial resolution, which can reduce their accuracy and usefulness. One approach to improving the accuracy of these images is to treat the imaging problem as an inverse problem. In this approach, the goal is to estimate the internal structure of the body from the observed images, while taking into account the various limitations and uncertainties associated with the imaging process.
The purpose of this survey is to provide an overview of the current state of the field of inverse problems in emerging technologies. It covers the key concepts and theories in inverse problems, as well as the various applications and challenges associated with the use of inverse problems in emerging technologies. This survey aims to provide a comprehensive and accessible resource for researchers and practitioners interested in the use of inverse problems in emerging technologies. By highlighting the importance of inverse problems in this field and the potential for continued growth and innovation, this survey can help to encourage further research and development in this area.
The reminder of this paper is organized as follows. Section 2 covers the background behind the inverse problems. Section 3 discuss the employed research method. Section 4 provides the ifnding of our survey. Future directions are given in section 4. Finally, the conclusion to our work is given to section 5.
Inverse problems involve the recovery of information about an unknown system or process from observations or measurements of its outputs. A problem is called inverse if the solution of the first part is required to formulate the second part. In other words, it is the technique of discovering causes from the information of their efects. This process is illustrated in Figure 1.
Diferentiation of data is a common example of an inverse problem. Given a set of data points that represent a function, the goal of the diferentiation problem is to recover the derivative of the function from the data.
• Direct problem: let : [0.1] → R be a continuous function, compute () := ∫︁ 0
()d, ∈ [
• Inverse Problem: Given a diferentiable function : [0.1] → R, determine := ′ In practice, the diferentiation of data is an important problem that arises in many fields. Accurately diferentiating data is crucial for understanding the underlying dynamics and behavior of signals and systems, and for making informed decisions and predictions based on the data.
The three conditions for a problem to be well-posed in the sense of Hadamard [
Mathematically, we formulate the notion of well-posedness [
Let and be nomed spaces, : → a linear or nonlinear mapping. The equation = is called well-posed if the following holds: 1. Existence of solution: For every ∈ there is at least one ∈ such that = . 2. Uniqueness of a solution: For every ∈ there is at most one ∈ with = . 3. Continuous dependence of the solution on the data: The solution depends continuously on . That is, for every sequence () ⊂ with → ( → ∞), it follows that → ( → ∞).
These three conditions are important because they ensure that the problem is well-defined and that the solution is both predictable and stable. In many practical applications, it is essential to have a well-posed problem in order to make accurate predictions and ensure the stability of the solution. If any of the three condition are not satisfied, the problem is called improperlyposed or ill-posed.This ill-posed fundamental property makes inverse problems dificult and mathematically challenging. Ill-posed problems can pose significant dificulties when trying to solve them numerically, as small errors can result in arbitrarily large variations in the solution. To mitigate these dificulties, various regularization techniques can be used to stabilize the solution of an ill-posed problem. The idea behind regularization is to add additional information or constraints to the problem, in order to make it well-posed and to obtain a stable solution. This additional information is usually encoded as a penalty term in the objective function that is being minimized.
There are several types of regularization methods, including: 1) Tikhonov Regularization, 2) Morozov’s discrepancy principle and Bayesian Regularization.
Each of these regularization methods has its own strengths and weaknesses, and the choice of method depends on the specific problem and the available data. In some cases, it may be beneficial to use a combination of regularization methods, or to adapt the regularization parameters based on the data.
This study aims to investigate the inverse problems in emerging technologies to identify
innovative techniques that can help to address the challenges of existing methods. These
advancements can help to improve the accuracy and reliability of systems and applications.
This research employs a systematic review methodology known as mapping study or scoping
[
The prime question that leads the investigation of inverse problems in emerging technologies is "How can we find the unknown inputs or parameters that produce a given output or observation in modern emerging systems?" This question is at the heart of our work, and drives the search for new and improved solutions and techniques. To pipeline this systematic mapping review this key question was split into two research questions: 1. What are the diferent emerging technologies that uses inverse problem solutions? 2. How this technologies defined their problems as an inverse problem? The motivation behind understanding the diferent emerging technologies that use inverse problem solutions is to keep up-to-date with the latest developments and advancements in these ifelds. By having a clear understanding of the diferent technologies that use inverse problems, one can have a better appreciation of the diverse applications of this mathematical concept. This knowledge can be useful for researchers and engineers who are developing new algorithms and systems that use inverse problems. Overall, the motivation behind understanding the diferent emerging technologies that use inverse problem solutions is to stay informed and to have a deeper understanding of this important mathematical concept and its many applications.
The time frame covered by this study is from 2009 to 2022 inclusive. Additionally, the publications were re-searched using IEEEXplore, ACM Digital Library, Science Direct, and Google Scholar as the repositories.
In this section, we will try to answer the research questions posted in section 3. 1. What are the diferent emerging technologies that uses inverse problem solutions? There are several emerging technologies that use inverse problem solutions, including: Artificial Intelligence :
The importance of inverse problems in artificial intelligence is due to their ability to provide
valuable insights and solutions to complex problems. By combining mathematical models and
machine learning algorithms, AI systems can extract information from data, find hidden patterns
and relationships, and make predictions and decisions in real-time [
Health care:
In the field of healthcare, mathematical inverse problems have been used to address a variety of
problems, such as image analysis and interpretation, disease diagnosis, and treatment planning
[
One example of a research paper in this area is [
Overall, the use of mathematical inverse problems in healthcare has shown promise in the analysis and interpretation of medical data and has the potential to improve patient outcomes by providing more accurate and eficient diagnoses and treatments.
Cybersecurity:
The inverse problem in cybersecurity refers to the process of determining the cause of a
security breach or vulnerability by analyzing its efects and symptoms. Anomaly and misuse
detection in cybersecurity can be considered as an inverse problem, as it requires the identification
of unusual and malicious behavior in a complex network system. To tackle these challenges,
researchers are exploring a range of solutions, including machine learning algorithms, artificial
intelligence, and data analytics, to develop more accurate and efective methods for detecting
anomalies and misuses in cybersecurity. The inverse machine learning algorithm described in
[
Energy:
The use of inverse problems in energy systems is becoming increasingly important as we look
for more eficient and sustainable energy solutions. One example of the use of inverse problems
in energy systems is in the field of renewable energy sources such as wind and solar power
[
2. How this technologies defined their problems as an inverse problem?
There are several examples of emerging technologies that have defined their problems as inverse problems:
Intrusion detection systems:
One common approach to intrusion detection is to treat the problem as an inverse problem.
In this context, the inverse problem is to determine the underlying behavior patterns of the
network or system, and to identify deviations from these patterns that may indicate an intrusion
[
Internet of Things (IoT) systems: Internet of Things (IoT) systems can encounter various
problems that can be defined as inverse problems such as sensor calibration, localization [
Overall, the use of inverse problems in modern emerging technologies provides a powerful tool for solving a wide range of problems where limited information is available. By formulating the problem as an inverse problem, the unknowns can be found through mathematical modeling and numerical algorithms, leading to improved solutions and better outcomes.
Inverse problems have been a topic of research for many years and have been used in various emerging technologies. As these technologies continue to evolve, the future directions of inverse problems are expected to focus on several key areas. Some of these directions include:
Machine Learning and deep learning approaches: In machine and deep learning, the future direction is expected to focus on developing more eficient and scalable inverse problems solutions that can handle large datasets, as well as new machine learning models that can be integrated with existing inverse problem algorithms. This integration is likely to lead to the development of new hybrid approaches that can address even more challenging problems, such as non-linear inverse problems and multi-modal inverse problems. In addition, we can expect to see more applications of inverse problems in machine learning, particularly in natural language processing.
Real-time and big data applications: The increasing demand for real-time and big data applications in emerging technology is driving research eforts to develop methods that can eficiently solve inverse problems in these contexts. In real-time applications, inverse problems can be used to process large amounts of data in real-time and provide quick, accurate results. Big data applications, on the other hand, can benefit from the use of inverse problems by leveraging the scalability and eficiency of inverse problem algorithms. Inverse problems can be used to process large amounts of data and extract meaningful insights, even when the data is noisy or incomplete. In both real-time and big data applications, the use of inverse problems can help to improve the accuracy of results and reduce the amount of time and resources required to process large amounts of data.
Internet of Things (IoT): Inverse problems can be used to process and analyze the massive amounts of data generated by IoT devices, helping to uncover insights and make informed decisions. Energy: Inverse problems can be used to improve the eficiency and accuracy of energy systems, such as wind and solar power, by allowing engineers to better understand and model the underlying dynamics of these systems.
In conclusion, the use of inverse problems in emerging technologies is growing as more researchers and practitioners explore the potential applications of this field. Inverse problems are becoming increasingly relevant in areas such as artificial intelligence, internet of things, medical imaging, augmented reality, and energy, where they are being used to address problems such as model inversion, data analysis, and image and signal processing.
The future direction of the use of inverse problems in emerging technologies is likely to be driven by the need for fast and accurate solutions to complex problems in a wide range of applications. As new technologies emerge and existing ones continue to evolve, the use of inverse problems solutions is likely to play an increasingly important role in shaping the future of many industries and fields. Therefore, this survey highlights the importance of inverse problems in emerging technologies and the potential for continued growth and innovation in this field. By providing a comprehensive overview of the current state of the field, this survey can serve as a useful resource for researchers and practitioners interested in the use of inverse problems in emerging technologies.