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Using vanilla NeuralODEs to model large and/or complex systems often failsdue two reasons: Stability and convergence. NeuralODEs are capable ofdescribing stable as well as instable dynamic systems. Selecting an appropriatenumerical solver is not trivial, because NeuralODE properties change duringtraining. If the NeuralODE becomes more stiff, a suboptimal solver may need toperform very small solver steps, which significantly slows down the trainingprocess. If the NeuralODE becomes to instable, the numerical solver might notbe able to solve it at all, which causes the training process to terminate.Often, this is tackled by choosing a computational expensive solver that isrobust to instable and stiff ODEs, but at the cost of a significantly decreasedtraining performance. Our method on the other hand, allows to enforce ODEproperties that fit a specific solver or application-related boundaryconditions. Concerning the convergence behavior, NeuralODEs often tend to runinto local minima, especially if the system to be learned is highly dynamicand/or oscillating over multiple periods. Because of the vanishing gradient ata local minimum, the NeuralODE is often not capable of leaving it and convergeto the right solution. We present a technique to add knowledge of ODEproperties based on eigenvalues - like (partly) stability, oscillationcapability, frequency, damping and/or stiffness - to the training objective ofa NeuralODE. We exemplify our method at a linear as well as a nonlinear systemmodel and show, that the presented training process is far more robust againstlocal minima, instabilities and sparse data samples and improves trainingconvergence and performance.

Eigen-informed NeuralODEs: Dealing with stability and convergence issues of NeuralODEs