We introduce the mean inverse integrator (MII), a novel approach to increase the accuracy when training neural networks to approximate vector fields of dynamical systems from noisy data. This method can be used to average multiple trajectories obtained by numerical integrators such as Runge-Kutta methods. We show that the class of mono-implicit Runge-Kutta methods (MIRK) has particular advantages when used in connection with MII. When training vector field approximations, explicit expressions for the loss functions are obtained when inserting the training data in the MIRK formulae, unlocking symmetric and high-order integrators that would otherwise be implicit for initial value problems. The combined approach of applying MIRK within MII yields a significantly lower error compared to the plain use of the numerical integrator without averaging the trajectories. This is demonstrated with experiments using data from several (chaotic) Hamiltonian systems. Additionally, we perform a sensitivity analysis of the loss functions under normally distributed perturbations, supporting the favorable performance of MII.

翻译：我们提出均值逆积分器(MII)方法，这是一种在从含噪数据训练神经网络逼近动力系统向量场时提高精度的新方法。该方法可对通过龙格-库塔法等数值积分器获得的多条轨迹进行平均。研究表明，单隐式龙格-库塔法(MIRK)在与MII结合使用时具有特殊优势。在训练向量场逼近时，将训练数据代入MIRK公式可获得损失函数的显式表达式，从而解锁了本应对初值问题为隐式的对称和高阶积分器。将MIRK应用于MII的组合方法，相比直接使用数值积分器而不进行轨迹平均，可显著降低误差。通过多个(混沌)哈密顿系统的数据实验验证了该结论。此外，我们对正态分布扰动下的损失函数进行了灵敏度分析，进一步支持了MII的优异性能。