Physics-Informed Neural Networks*

Methods of machine learning are starting to be widely used in

applications in hadronic physics. Neural networks provide a very

flexible parametrization of functions to be extracted from the

experimental data. They are used in hadronic physics to extract parton

distribution functions and other functions that describe the internal

structure of the nucleon. At the same time, the inverse problem

solutions must obey physical laws encoded in differential equations,

boundary conditions, etc. One of the goals of the inverse problem is

reliable estimation of the errors of the extractions and the posterior

probability density of the underlying parameters. The flexibility

provided by the neural networks is the origin of the pitfall of the

traditional neural networks, which is the extrapolation of the

results. The extrapolation needed for predictions for unmeasured

regions may become nonphysical if the appropriate constraints or

functional forms are not imposed. In our study, we will construct a

neural network that describes synthetic data and demonstrate potential

problems of its utilization. We will further study and apply methods

of the Physics-Informed Neural Networks (PINN) to tame the behavior of

the network and to guide the neural network to obey physical

laws. Using comprehensive examples, we will work towards the goal of

constructing a flexible PINN to handle the inverse problems that arise

in hadronic physics.