Mean-field game (MFG) systems offer a framework for modeling multi-agent dynamics, but unknown parameters pose challenges. In this work, we tackle an inverse problem, recovering MFG parameters from limited, noisy boundary observations. Despite the problem's ill-posed nature, we aim to efficiently retrieve these parameters to understand population dynamics. Our focus is on recovering running cost and interaction energy in MFG equations from boundary measurements. We formalize the problem as an constrained optimization problem with L1 regularization. We then develop a fast and robust operator splitting algorithm to solve the optimization using techniques including harmonic extensions, three-operator splitting scheme, and primal-dual hybrid gradient method.  Numerical experiments illustrate the effectiveness and robustness of the algorithm. This is a joint work with Yat Tin Chow (UCR), Samy Wu Fung (Colorado School of Mines), Levon Nurbekyan (Emory), and Stanley J.
  Osher (
 UCLA).