Learning models of dynamical systems with external inputs, that may be, for example, nonsmooth or piecewise, is crucial for studying complex phenomena and predicting future state evolution, which is essential for applications such as safety guarantees and decision-making. In this work, we introduce \emph{Input Concomitant Neural ODEs (ICODEs)}, which incorporate precise real-time input information into the learning process of the models, rather than treating the inputs as hidden parameters to be learned. The sufficient conditions to ensure the model's contraction property are provided to guarantee that system trajectories of the trained model converge to a fixed point, regardless of initial conditions across different training processes. We validate our method through experiments on several representative real dynamics: Single-link robot, DC-to-DC converter, motion dynamics of a rigid body, Rabinovich-Fabrikant equation, Glycolytic-glycogenolytic pathway model, and heat conduction equation. The experimental results demonstrate that our proposed ICODEs efficiently learn the ground truth systems, achieving superior prediction performance under both typical and atypical inputs. This work offers a valuable class of neural ODE models for understanding physical systems with explicit external input information, with potential promising applications in fields such as physics and robotics.

Because of military drawdowns and the need for additional transportation lift requirements, the United States Marine Corps developed a concept that enabled it to modify a commercial container ship to support deployed aviation units. However, a problem soon emerged in that there were too few people who were expert enough to do the unique type of planning required for this ship. Additionally, once someone did develop some expertise, it was time for him/her to move on, retire, or leave active duty. There needed to be a way to capture this knowledge. Access modules are used to access secondand third-tier mobile facilities that are complexed below decks in support of IMA-level repair capability.

Because of military drawdowns and the need for additional transportation lift requirements, the United States Marine Corps developed a concept that enabled it to modify a commercial container ship to support deployed aviation units. However, a problem soon emerged in that there were too few people who were expert enough to do the unique type of planning required for this ship. Additionally, once someone did develop some expertise, it was time for him/her to move on, retire, or leave active duty. There needed to be a way to capture this knowledge. This condition was the impetus for the T-AVB AUTOMATED LOAD-PLANNING SYSTEM (TALPS) effort. TALPS is now a fielded, certified application for Marine Corps aviation.