Several forms for constructing novel physics-informed neural-networks (PINN) for the solution of partial-differential-algebraic equations based on derivative operator splitting are proposed, using the nonlinear Kirchhoff rod as a prototype for demonstration. The open-source DeepXDE is likely the most well documented framework with many examples. Yet, we encountered some pathological problems and proposed novel methods to resolve them. Among these novel methods are the PDE forms, which evolve from the lower-level form with fewer unknown dependent variables to higher-level form with more dependent variables, in addition to those from lower-level forms. Traditionally, the highest-level form, the balance-of-momenta form, is the starting point for (hand) deriving the lowest-level form through a tedious (and error prone) process of successive substitutions. The next step in a finite element method is to discretize the lowest-level form upon forming a weak form and linearization with appropriate interpolation functions, followed by their implementation in a code and testing. The time-consuming tedium in all of these steps could be bypassed by applying the proposed novel PINN directly to the highest-level form. We developed a script based on JAX. While our JAX script did not show the pathological problems of DDE-T (DDE with TensorFlow backend), it is slower than DDE-T. That DDE-T itself being more efficient in higher-level form than in lower-level form makes working directly with higher-level form even more attractive in addition to the advantages mentioned further above. Since coming up with an appropriate learning-rate schedule for a good solution is more art than science, we systematically codified in detail our experience running optimization through a normalization/standardization of the network-training process so readers can reproduce our results.

Recently, machine learning (ML) methods have been developed for increasing the accuracy of robot mechanisms. Complex mechanical issues such as non-linear friction, backlash, flexibility of structure transmission elements can cause these errors and they are hard to model. ML requires training data and the above mechanical phenomena are highly dependent on position of the robot in the workspace and also on its velocity, especially near zero velocity in both directions where non-linearities such as Streibek and Coulomb friction are most pronounced. It is well known that success of ML methods depends on amount of training data and it is expensive/time consuming to collect data from physical robot motion. We therefore address the problem of searching for trajectories in the 6D space of positions and velocities which collect the most information in the least amount of time. This reduces to a special case of the traveling-salesman problem in that the robot must be programmed to visit sampled points in the position-velocity phase space most efficiently. Two goals of this work are 1) Computationally study the difficulty of the TSP in this application by applying it to X, Y, Z motion in 3D space (6D phase space) and 2) assess the effectiveness of an extremely simple Nearest Neighbor search algorithm compared to random sampling of the search space. Results confirm that Nearest Neighbor heuristic searching produces significantly better trajectories than random sampling in this application.