Physics-Informed Neural Operators (PINOs)
Physics-informed neural operators (PINOs) are a class of machine learning models designed to learn mappings between function spaces, unlike conventional neural networks which learn mappings between finite-dimensional vector spaces. While classical NNs approximate input-output relationships at discrete points, neural operators learn how entire input functions map to output functions, making them well-suited to modeling physical systems described by differential equations. PINOs go further by embedding governing physical laws directly into the architecture or loss function, enforcing residuals from PDEs or ODEs during training. This physics-constrained approach improves generalization, reduces data requirements, and ensures physically consistent predictions.
My research applies PINOs to the surrogate modeling of thermochemical reactive operators in nonequilibrium, multicomponent plasma flows. These systems are governed by stiff, coupled ODEs describing species population dynamics and energy exchanges due to collisional-radiative kinetics and thermochemical nonequilibrium. Conventionally, such source terms are integrated using implicit ODE solvers at every grid point—a method that is accurate but computationally intensive and limits scalability in high-fidelity 2D and 3D simulations. PINOs address this by learning fast surrogates that approximate these operators with high accuracy while preserving key physical structure. They offer orders-of-magnitude speedups and enable tight coupling with unsteady CFD solvers, unlocking new capabilities for real-time simulations.
🔬 Physics-Embedded Learning
In this context, PINOs are not simply data-driven regressors—they function as physics-preserving surrogates. The models are explicitly designed to:
- Enforce mass and energy conservation laws
- Maintain positivity of species concentrations and temperatures
- Preserve the equilibrium distribution function of species’ quantum energy levels
This is not achieved by merely augmenting the loss function with physics-based residuals (e.g., from ODE constraints). Instead, the proposed PINO architectures embed the governing equations directly into the network design itself, yielding substantial improvements in accuracy while ensuring physically consistent behavior—even under strong extrapolation regimes.
đź“š Selected Publications
I. Zanardi, S. Venturi, and M. Panesi.
“Adaptive physics‑informed neural operator for coarse‑grained non‑equilibrium flows”.
In: Scientific Reports 13 (Sept. 2023).
DOI
I. Zanardi, S. Venturi, and M. Panesi.
“MENO: Hybrid Matrix Exponential-based Neural Operator for Stiff ODEs. Application to Thermochemical Kinetics”.
arXiv preprint (July 2025).
DOI
I. Zanardi, S. Venturi, and M. Panesi.
“Towards Efficient Simulations of Non‑Equilibrium Chemistry in Hypersonic Flows: Application of Neural Operators in Multidimensional CFD Simulations”.
In: AIAA SCITECH 2024 Forum, American Institute of Aeronautics and Astronautics (Jan. 2024).
DOI
đź’» Scientific Softwares
PyCOMET
A modular, TensorFlow-based platform for physics-informed scientific machine learning. It enables rapid prototyping of PINO models, with:
- Operator learning modules targeting ODE systems
- Constraints enforcing physics-based invariants
- GPU acceleration for large-scale training and inference
TF2
TF2 enables direct deployment of TensorFlow models in C++/Fortran codes without TensorFlow installation. It allows PINOs to be embedded into legacy CFD solvers for fast, physics-informed surrogate inference during reactive flow simulations.