MENO: Hybrid Matrix Exponential-based Neural Operator for Stiff ODEs. Application to Thermochemical Kinetics

TL;DR

MENO is a hybrid neural operator framework that efficiently solves stiff ODE systems with linear and nonlinear components, ensuring physical consistency and achieving significant computational speedups in thermochemical applications.

Contribution

MENO introduces a novel neural matrix exponential formulation integrated with neural operators for modeling linear subsystems within stiff ODEs, enhancing accuracy and efficiency.

Findings

Achieves below 2% relative error in zero-dimensional models

Provides up to 4,800× speedup on GPU

Maintains accuracy in multidimensional extrapolations

Abstract

We introduce MENO (''Matrix Exponential-based Neural Operator''), a hybrid surrogate modeling framework for efficiently solving stiff systems of ordinary differential equations (ODEs) that exhibit a sparse nonlinear structure. In such systems, only a few variables contribute nonlinearly to the dynamics, while the majority influence the equations linearly. MENO exploits this property by decomposing the system into two components: the low-dimensional nonlinear part is modeled using conventional neural operators, while the linear time-varying subsystem is integrated using a novel neural matrix exponential formulation. This approach combines the exact solution of linear time-invariant systems with learnable, time-dependent graph-based corrections applied to the linear operators. Unlike black-box or soft-constrained physics-informed (PI) models, MENO embeds the governing equations directly into its architecture, ensuring physical consistency (e.g., steady states), improved robustness, and more efficient training. We validate MENO on three complex thermochemical systems: the POLLU atmospheric chemistry model, an oxygen mixture in thermochemical nonequilibrium, and a collisional-radiative argon plasma in one- and two-dimensional shock-tube simulations. MENO achieves relative errors below 2% in trained zero-dimensional settings and maintains good accuracy in extrapolatory multidimensional regimes. It also delivers substantial computational speedups, achieving up to 4 800$\times$ on GPU and 185$\times$ on CPU compared to standard implicit ODE solvers. Although intrusive by design, MENO's physics-based architecture enables superior generalization and reliability, offering a scalable path for real-time simulation of stiff reactive systems.