Physics informed activation functions and loss functions for signal reconstruction and digital twinning

A coupled two-body mass Lagrangian is used to model a magnetic rail nScrypt printer extruder with a sensor attached to the side for signal reconstruction and digital twinning. The equations of motion (EQM) derived from the Lagrangian are then used to improve the accelerometer signal by incorporating them as activation functions in the layers of the neural network, then using the damping constant and frequency matrix as learnable parameters. This is compared to a modular neural network that uses the EQM as the loss residuals for backward propagation and the physical constants as neural nets, both methods were found to give a root-mean square error (RMSE) of ~ 14 mm to 0.75 mm from the original position signal respectively. The physics informed neural network (PINN) activation layer produced a better detailed fit, while the PINN loss function produced a better average line and standard deviation. The two methods are then combined to achieve a final result with an RMSE of 0.96 mm to 16.7 mm, and a standard deviation that drops significantly during training to ~ 1.6 mm, but comparable to the PINN activation of 44-827 mm for test samples.