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

2025-07-18

AI Summary

Stiff ordinary differential equation (ODE) systems arising in thermochemical kinetics exhibit sparse nonlinear structure—most variables evolve linearly, with nonlinearity confined to only a few components. Method: We propose a physics-informed hybrid neural operator framework that decomposes the system into a time-varying linear subsystem (exactly modeled via matrix exponentials) and a sparse nonlinear subsystem (parameterized by a graph neural network as a learnable correction term), explicitly embedding governing equation structure. Contribution/Results: This design ensures physical consistency, markedly improving extrapolation capability and training efficiency. Evaluated on multiple real-world thermochemical systems, the model achieves <2% relative error, attains up to 4800× GPU speedup and 185× CPU speedup over conventional solvers, and demonstrates high accuracy, strong generalization, and potential for real-time simulation.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Mixture of Experts (MoE)
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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$ imes$ on GPU and 185$ imes$ 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.

Problem

• Efficiently solves stiff ODEs with sparse nonlinear structure
• Decomposes system into nonlinear and linear time-varying components
• Ensures physical consistency and computational efficiency in simulations

Innovation

• Hybrid neural operator for stiff ODEs
• Matrix exponential for linear subsystems
• Physics-embedded architecture ensures consistency