Simone Venturi

Ivan Zanardi, Simone Venturi, Marco Panesi

This work proposes a new machine learning (ML)-based paradigm aiming to enhance the computational efficiency of non-equilibrium reacting flow simulations while ensuring compliance with the underlying physics. The framework combines dimensionality reduction and neural operators through a hierarchical and adaptive deep learning strategy to learn the solution of multi-scale coarse-grained governing equations for chemical kinetics. The proposed surrogate's architecture is structured as a tree, with leaf nodes representing separate neural operator blocks where physics is embedded in the form of multiple soft and hard constraints. The hierarchical attribute has two advantages: i) It allows the simplification of the training phase via transfer learning, starting from the slowest temporal scales; ii) It accelerates the prediction step by enabling adaptivity as the surrogate's evaluation is limited to the necessary leaf nodes based on the local degree of non-equilibrium of the gas. The model is applied to the study of chemical kinetics relevant for application to hypersonic flight, and it is tested here on pure oxygen gas mixtures. In 0-D scenarios, the proposed ML framework can adaptively predict the dynamics of almost thirty species with a maximum relative error of 4.5% for a wide range of initial conditions. Furthermore, when employed in 1-D shock simulations, the approach shows accuracy ranging from 1% to 4.5% and a speedup of one order of magnitude compared to conventional implicit schemes employed in an operator-splitting integration framework. Given the results presented in the paper, this work lays the foundation for constructing an efficient ML-based surrogate coupled with reactive Navier-Stokes solvers for accurately characterizing non-equilibrium phenomena in multi-dimensional computational fluid dynamics simulations.

Simone Venturi, Tiernan Casey

Deep operator networks (DeepONets) are powerful architectures for fast and accurate emulation of complex dynamics. As their remarkable generalization capabilities are primarily enabled by their projection-based attribute, we investigate connections with low-rank techniques derived from the singular value decomposition (SVD). We demonstrate that some of the concepts behind proper orthogonal decomposition (POD)-neural networks can improve DeepONet's design and training phases. These ideas lead us to a methodology extension that we name SVD-DeepONet. Moreover, through multiple SVD analyses, we find that DeepONet inherits from its projection-based attribute strong inefficiencies in representing dynamics characterized by symmetries. Inspired by the work on shifted-POD, we develop flexDeepONet, an architecture enhancement that relies on a pre-transformation network for generating a moving reference frame and isolating the rigid components of the dynamics. In this way, the physics can be represented on a latent space free from rotations, translations, and stretches, and an accurate projection can be performed to a low-dimensional basis. In addition to flexibility and interpretability, the proposed perspectives increase DeepONet's generalization capabilities and computational efficiencies. For instance, we show flexDeepONet can accurately surrogate the dynamics of 19 variables in a combustion chemistry application by relying on 95% less trainable parameters than the ones of the vanilla architecture. We argue that DeepONet and SVD-based methods can reciprocally benefit from each other. In particular, the flexibility of the former in leveraging multiple data sources and multifidelity knowledge in the form of both unstructured data and physics-informed constraints has the potential to greatly extend the applicability of methodologies such as POD and PCA.

Ivan Zanardi, Simone Venturi, Marco Panesi

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.