Physics-informed neural networks and functional interpolation for stiff chemical kinetics

Area of Science:

• Computational Mathematics
• Chemical Engineering
• Applied Physics

Background:

• Stiff ordinary differential equations (ODEs) pose significant challenges for traditional numerical methods.
• Existing physics-informed neural network (PINN) approaches often struggle with the stiffness inherent in chemical kinetic problems.

Purpose of the Study:

• To develop and evaluate a novel physics-informed neural network framework for solving stiff ODE initial value problems (IVPs).
• To demonstrate the efficiency and robustness of the proposed method compared to state-of-the-art techniques.

Main Methods:

• The study introduces the extreme theory of functional connections (X-TFC), integrating PINNs with functional connections and extreme learning machines.
• A single-layer neural network (NN) is employed within the X-TFC framework.
• The method is tested on stiff ODE systems without employing stiffness reduction artifacts.

Main Results:

• X-TFC demonstrates efficiency and robustness in solving stiff ODEs, outperforming existing methods in computational time and accuracy.
• A rigorous upper bound on the generalization error for X-TFC in learning ODE solutions is established for the first time.
• The trained NN provides an analytical solution representation applicable beyond the initial discretization points.

Conclusions:

• X-TFC offers a flexible and effective approach for solving challenging stiff ODEs.
• This method has broad applicability in fields requiring large time-range simulations, including chemical dynamics, nuclear systems, life sciences, and environmental engineering.