LiLaN: A Linear Latent Network as the Solution Operator for Real-Time Solutions to Stiff Nonlinear Ordinary Differential Equations

TL;DR

LiLaN introduces a linear latent network that analytically solves stiff nonlinear ODEs and PDEs, significantly reducing computational costs and outperforming existing machine learning methods in handling timescale separation.

Contribution

The paper presents LiLaN, a novel neural network architecture that encodes and analytically solves stiff nonlinear differential equations in latent space, with a universal approximation guarantee.

Findings

LiLaN accurately approximates solutions to stiff nonlinear ODEs and PDEs.

LiLaN outperforms existing ML methods in efficiency and accuracy.

The latent dimension in LiLaN is independent of the approximation error epsilon.

Abstract

Solving stiff ordinary differential equations (StODEs) requires sophisticated numerical solvers, which are often computationally expensive. In general, traditional explicit time integration schemes with restricted time step sizes are not suitable for StODEs, and one must resort to costly implicit methods. On the other hand, state-of-the-art machine learning based methods, such as Neural ODE, poorly handle the timescale separation of various elements of the solutions to StODEs, while still requiring expensive implicit/explicit integration at inference time. In this work, we propose a linear latent network (LiLaN) approach in which the dynamics in the latent space can be integrated analytically, and thus numerical integration is completely avoided. At the heart of LiLaN are the following key ideas: i) two encoder networks to encode the initial condition together with parameters of the ODE to the slope and the initial condition for the latent dynamics, respectively. Since the latent dynamics, by design, are linear, the solution can be evaluated analytically; ii) a neural network to map the physical time to latent times, one for each latent variable. Finally, iii) a decoder network to decode the latent solution to the physical solution at the corresponding physical time. We provide a universal approximation theorem for the proposed LiLaN approach, showing that it can approximate the solution of any stiff nonlinear system on a compact set to any degree of accuracy epsilon. We also show an interesting fact that the dimension of the latent dynamical system in LiLaN is independent of epsilon. Numerical results on the "Robertson Stiff Chemical Kinetics Model," "Plasma Collisional-Radiative Model," and "Allen-Cahn" and "Cahn-Hilliard" PDEs suggest that LiLaN outperformed state-of-the-art machine learning approaches for handling stiff ordinary and partial differential equations.