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 to compute solutions. On the other hand, state-of-the-art machine learning (ML) based methods such as Neural ODE (NODE) 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 LiLaNs are the following key ideas: i) two encoder networks to encode 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 initial condition, parameters, and the physical time to latent times, one for each latent variable. Intuitively, this allows for the "stretching/squeezing" of time in the latent space, thereby allowing for varying levels of attention to different temporal scales in the solution. Finally, iii) a decoder network to decode the latent solutions into the physical solution at the corresponding physical time. LiLaN is thus a solution operator approach that instantly provides an approximate flow map solution of a system of nonlinear ODEs at any 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 LiLaNs is independent of \(\epsilon \) . Numerical results on "Robertson Stiff Chemical Kinetics Model" (Anantharaman et al. 2021) and "Plasma Collisional-Radiative Model" (Chung et al. High Energy Density Phys. 2005;1) suggest that LiLaNs outperformed state-of-the-art machine learning approaches for handling StODEs. Numerically, we also show that LiLaNs outperformed other machine learning methods on multiple partial differential equations (PDEs) with known stiff behaviors, such as the "Allen-Cahn" and "Cahn-Hilliard" PDEs (Montanelli and Bootland 2020). Furthermore, we show that LiLaNs is equally well-suited for non-stiff differential equations such as the 2D Navier-Stokes equation. Our numerical experiments on the 2D Navier-Stokes equation demonstrate that LiLaN achieves remarkable speedup and accuracy compared to state-of-the-art machine learning methods, further highlighting its versatility as a solution operator for complex fluid dynamics problems.
