The Performance of Approximating Ordinary Differential Equations by Neural Nets

The dynamics of many systems are described by ordinary differential equations (ODE). Solving ODE´s with standard methods (i.e. numerical integration) needs a high amount of computing time but only a small amount of storage memory. For some applications, e.g. short time weather forecast or real time robot control, long computation times are prohibitive. Is there a method which uses less computing time (but has drawbacks in other aspects, e.g. memory), so that the computation of ODE´s gets faster? We will try to discuss this question for the method of a neural network which was trained on ODE dynamics and compare both methods using the same approximation error. In many cases, as for physics engines used in computer games, the shape of the approximation curve is important and not the exact values of the approximation. Therefore, we introduce as error measure the subjective error based on the Total Least Square Error (TLSE) which gives more consistent results than the standard error. Finally, we derive a method to evaluate where neural nets are advantageous over numerical ODE integration and where this is not the case.