Summary In this study, we implement a physics-informed neural network (PINN) architecture to address the direct and inverse P-wave velocity problem in the context of a cross-well seismic scenario. The physics constraint to guide the learning of this algorithm is incorporated by including the first-order partial differential equations describing acoustic wave propagation as part of the loss function for training the network. We adopt an approach that exposes the network to only two snapshots of wavefield data during training, as well as a limited volume of velocity field measurements (seismograms) as observational data. To improve convergence and ensure efficient training, we employ a Neural Tangent Kernel (NTK) strategy to adaptively calibrate the weights of all components constituting the loss function throughout the training phase. The current numerical results demonstrate that this method is able to predict the seismograms and accurately estimate the three propagation wavefields, being effective in solving the breakthrough problem. The next step of this method will be the adjustment of the inverse problem to obtain the P-wave velocity.