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As a method of universal approximation deep neural networks (DNNs) arecapable of finding approximate solutions to problems posed with little moreconstraints than a suitably-posed mathematical system and an objectivefunction. Consequently, DNNs have considerably more flexibility in applicationsthan classical numerical methods. On the other hand they offer an uncontrolledapproximation to the sought-after mathematical solution. This suggests thathybridization of classical numerical methods with DNN-based approximations maybe a desirable approach. In this work a DNN-based approximator inspired by thephysics-informed neural networks (PINNs) methodology is used to provide aninitial guess to a Newton's method iteration of a very-high order implicitRunge-Kutta (IRK) integration of a nonlinear system of ODEs, namely the Lorenzsystem. In the usual approach many explicit timesteps are needed to provide aguess to the implicit system's nonlinear solver, requiring enough work to makethe IRK method infeasible. The DNN-based approach described in this workenables large implicit time-steps to be taken to any desired degree of accuracyfor as much effort as it takes to converge the DNN solution to within a fewpercent accuracy. This work also develops a general formula for the matrixelements of the IRK method for an arbitrary quadrature order.

Arbitrarily high order implicit ODE integration by correcting a neural network approximation with Newton's method