A neural network method for time-dependent inverse source problem with limited-aperture data

This paper is concerned with the mathematical design of a novel gesture-based input/instruction device with identification and prediction using a moving emitter. The emitter acts as a point source and there is a sensor monitoring the wave field emitted by the emitter away from it on a surface in real time. In practice, the emitter could be a ring worn on a finger of the human being who desires to interact/communicate with the computer, while the sensor could be mounted on a computer. The input/instruction can be recognized and predicted by identifying the moving trajectory of the emitter performed by the human being from the collected wave field data. The process can be modeled as an inverse moving source problem, that is, one identifies and predicts the trajectory of a moving point source by measuring the corresponding wave field. There are several salient features of our study. First, for the practical consideration, the dynamical wave field data are collected in a limited aperture and full aperture respectively. Second, we design a parameter inversion model by neural network (PIMNN) to reconstruct the trajectory of the moving point source. This model solves the problem of information loss caused by data acquisition in limited aperture and has certain robustness with respect to noise. The computing complexity of the PIMNN are calculated by the Multiply Accumulate. Third, we consider the trajectory prediction of the moving point source for the inverse source problem associated with the novel input/instruction approach, and construct a trajectory prediction model by neural network (TPMNN) to predict the trajectory of the moving point source. Numerical experiments show that the proposed device works effectively and efficiently in some practical scenarios.