Accelerating the Simulation of Ordinary Differential Equations Through Physics-Preserving Neural Networks

Abstract: Numerical simulation of ordinary differential equations (ODEs) can be challenging when the system exhibits high accelerations and rapidly changing dynamics. Under these conditions the ODE solver often needs to take very small time steps in order to resolve the solution accurately, resulting in increased computational cost. In order to accelerate the simulation of these ODEs we present a novel methodology that uses a pseudo-invertible neural network to map system states into a high-dimensional latent-space. The network is then trained so that the dynamics in this learned latent space are slow, and can be simulated with relatively few function calls. Unlike existing neural methods, the latent dynamic equations are not learned from trajectory data, but derived from the original system equations and the chain rule. This allows the method to generalize better than existing approaches because the derived equations are correct by construction. In this work, we derive latent state equations of motion for any general ODE, and describe the loss function used to enforce slow time evolution of the latent states. We then apply this technique to multiple example ODEs and show that these problems can be solved with $3$x to $20$x fewer function calls for the same accuracy when simulating in the learned latent space. This reduction in cost could decrease computational demands for scientific simulations across engineering and physics applications.