Stiff-PINN: Physics-Informed Neural Network for Stiff Chemical Kinetics (2011.04520v2)

Abstract: Recently developed physics-informed neural network (PINN) has achieved success in many science and engineering disciplines by encoding physics laws into the loss functions of the neural network, such that the network not only conforms to the measurements, initial and boundary conditions but also satisfies the governing equations. This work first investigates the performance of PINN in solving stiff chemical kinetic problems with governing equations of stiff ordinary differential equations (ODEs). The results elucidate the challenges of utilizing PINN in stiff ODE systems. Consequently, we employ Quasi-Steady-State-Assumptions (QSSA) to reduce the stiffness of the ODE systems, and the PINN then can be successfully applied to the converted non/mild-stiff systems. Therefore, the results suggest that stiffness could be the major reason for the failure of the regular PINN in the studied stiff chemical kinetic systems. The developed Stiff-PINN approach that utilizes QSSA to enable PINN to solve stiff chemical kinetics shall open the possibility of applying PINN to various reaction-diffusion systems involving stiff dynamics.

• The paper introduces Stiff-PINN, a novel Physics-Informed Neural Network approach that incorporates Quasi-Steady-State Assumptions to overcome the challenges of physical and numerical stiffness in solving stiff chemical kinetics.
• Results show Stiff-PINN significantly outperforms regular PINNs on benchmark problems like ROBER and POLLU, achieving loss reductions by several orders of magnitude where traditional methods fail.
• This framework opens avenues for solving stiff dynamics in various fields like environmental modeling and energy systems, suggesting further research into automated model reduction and optimization.

Stiff-PINN: Advancements in Physics-Informed Neural Networks for Stiff Chemical Kinetics

The paper "Stiff-PINN: Physics-Informed Neural Network for Stiff Chemical Kinetics" introduces an innovative approach to handling the challenges posed by numerical and physical stiffness in solving stiff chemical kinetic problems using Physics-Informed Neural Networks (PINNs). The paper conducts a thorough investigation into the limitations of traditional PINN methods when applied to systems characterized by stiff Ordinary Differential Equations (ODEs). Through these investigations, the authors identify stiffness as a primary factor contributing to the failure of conventional PINN methods, and propose the novel Stiff-PINN approach incorporating Quasi-Steady-State Assumptions (QSSA) to mitigate these challenges.

Key Concepts and Methodology

PINNs have gained considerable attention for their potential to leverage deep neural networks (DNNs) to solve differential equations that arise frequently in scientific and engineering domains. By embedding governing laws of physics directly into neural network architectures, PINNs can enforce these constraints as part of their training loss. This work explores the application of PINN methodologies to stiff chemical kinetics—characterized by an extreme disparity in timescales across different reacting species—posing significant challenges for traditional numerical solvers.

The Stiff-PINN framework innovatively modifies the typical PINN architecture by incorporating QSSA, which involves simplifying the kinetic model through the reduction of stiffness. Specifically by assuming species with fast timescales can be approximated using algebraic equations instead of ODEs, effective reduction of the stiffness in the kinetic systems can be achieved.

Results and Implications

The application of Stiff-PINN exhibits markedly improved performance over regular-PINN methods in two benchmark stiff chemical kinetic systems: the ROBER problem and the POLLU air pollution model. In these cases, Stiff-PINN successfully captures the dynamics of systems where regular-PINN fails due to physical stiffness. The numerical results demonstrate that the loss function value in Stiff-PINN is reduced by several orders of magnitude, highlighting its effectiveness in addressing multiscale challenges.

Implications for Future Research and Application

The development of Stiff-PINN paves the way for applications beyond chemical kinetics, potentially influencing areas such as environmental modeling, energy systems, and complex material engineering, where stiff dynamics are prevalent. Moreover, these findings suggest the value of further research into automated model reduction techniques and optimization strategies, which might enhance both PINN robustness and computational efficiency.

The proposed framework also sparks several open questions regarding handling stiffness in complex systems, deriving QSSA formulations for intricate models, and further optimizing neural network behavior in mildly stiff scenarios. Future advancements may integrate methodologies such as Computational Singular Perturbation (CSP) and Intrinsic Low-Dimensional Manifolds (ILDM) for an expanded approach to stiffness removal.

Overall, Stiff-PINN showcases significant promise for advancing the utility of neural networks in computational modeling of chemical systems, warranting exploration of similar hybrid methodologies across other domains characterized by multiscale challenges. As neural network optimization continues to evolve, the principles established in this paper could lead to broader incorporation of PINNs into the design and analysis of complex dynamic systems.