#### Solving Inverse Problems Via Deep-Learning: Schroedinger Equations, Nuclear Matter EoS, and More

**Abstract:** There is a variety of challenging inverse problems in physics, in which the forward problem is mathematically straightforward whereas the backward one is not. As an example, let us consider a quantum system that can be described by solving the Schroedinger equation for a potential model. If the interaction potential is known, one can straightforwardly predict the microscopic properties, i.e., energy level and wave-functions. However, extracting the interaction potential from a given energy spectrum is a challenging but realistic problem.

In this seminar, I will present a new methodology to solve the inverse problem for the Schrödinger equation by exploiting Deep-Learning techniques. We represent the potential by a Deep Neural Network, which is essentially a piecewise interpolation and unbiasedly and universally describes any continuous functions. This technique is applied to extract the finite-temperature heavy-flavor potential in a QCD medium.

Another example is the extraction of the equation of state of nuclear matters with low temperature and high chemical potential. When the EoS is known, one can easily solve the Tolman–Oppenheimer–Volkoff equation and obtain the mass-radius relation. Nevertheless, extracting the EoS from a finite number of observations is challenging. We solve such a task by exploiting the aforementioned technique -- we express the EoS by a DNN and tune its parameters to reproduce the mass-radius relation. In this talk, I will first perform a closure test to validate the method. Then, preliminary results for EoS reconstructed from real observations will be presented.

Refs: PhysRevD.105.014017, 2201.01756