An optimization-based method for identifying the speed profile of a moving surface from image data is studied. If the dynamic surface motion is modeled by a level set equation, the identification problem can be formulated as an optimization problem constrained with the level set equation whose (viscosity) solution, in general, has kinks. The non-differentiable solution prevents us from having a bounded gradient of the cost function of the optimization problem. To overcome this difficulty, we develop a novel identification approach using a regularized level set equation. The regularization guarantees the differentiability of the cost function and the boundedness of the gradient. Using numerical optimization techniques with the adjoint-based gradient, we solve the identification problem. We perform a numerical test to validate that the solution of an optimization problem with a regularized level set equation converges to the solution of the same optimization problem with an unregularized level set equation as the regularization factor tends to zero. The performance and usefulness of the method are demonstrated by a biological example in which we estimate the forces (per density) of actin and myosin in cell polarization.