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

Abstract

This work presents a recently developed approach based on physics-informed neural networks (PINNs) for the solution of initial value problems (IVPs), focusing on stiff chemical kinetic problems with governing equations of stiff ordinary differential equations (ODEs). The framework developed by the authors combines PINNs with the theory of functional connections and extreme learning machines in the so-called extreme theory of functional connections (X-TFC). While regular PINN methodologies appear to fail in solving stiff systems of ODEs easily, we show how our method, with a single-layer neural network (NN) is efficient and robust to solve such challenging problems without using artifacts to reduce the stiffness of problems. The accuracy of X-TFC is tested against several state-of-the-art methods, showing its performance both in terms of computational time and accuracy. A rigorous upper bound on the generalization error of X-TFC frameworks in learning the solutions of IVPs for ODEs is provided here for the first time. A significant advantage of this framework is its flexibility to adapt to various problems with minimal changes in coding. Also, once the NN is trained, it gives us an analytical representation of the solution at any desired instant in time outside the initial discretization. Learning stiff ODEs opens up possibilities of using X-TFC in applications with large time ranges, such as chemical dynamics in energy conversion, nuclear dynamics systems, life sciences, and environmental engineering.

Multiscale Physics-Informed Neural Networks for Stiff Chemical Kinetics

“…The equations and the values of the parameters can be found in ref . Based on the discussions in the previous section, we then fix the neural network architecture (i.e., 32 neurons × 3 layers), 10% GTD points to the data loss term, and use MPINNs to solve the stiff ODEs in this problem.…”

“…We then use MPINNs to solve the more complicated stiff problem�POLLU problem, which includes 20 chemical species and 25 reaction equations. The POLLU problem can be described by 20 different nonlinear ODEs, where details are provided by De Florio et al 18 In MPINNs, 20 chemical species are first divided into four categories according to their time scale orders of magnitude (see Table 3). The chemical species of each category are fitted with a separate PINN.…”

“…This method needs to use non-trivial artifacts (i.e., QSSA) to remove stiff chemical species in advance, limiting the application of this method to small stiff chemical kinetic problems. Very recently, an X-TFC framework 18 has been proposed to solve stiff chemical kinetics problems without artifacts. This framework expands the control functions and hardcodes the initial condition into the control function.…”

Stiff-PDEs and Physics-Informed Neural Networks

“…Recently, physics-informed neural networks (PINNs) have gained popularity due to the novel approach for solving forward [1][2][3][4] and inverse problems [5][6][7] involving PDEs using neural networks (NNs). Unlike conventional numerical techniques for solving PDEs, PINNs are non-data-driven meshless models that satisfy the prescribed initial (IC) and Llion Evans, Michelle Tindall, and Perumal Nithiarasu have contributed equally to this work.…”

Investigating molecular transport in the human brain from MRI with physics-informed neural networks

“…This challenge has been addressed by adaptive weighting strategies 8 – 11 , as well as theory of functional connections 12 , 13 . Despite these challenges, the effectiveness of the method has been demonstrated in a wide range of works, examples include turbulent flows 14 , heat transfer 15 , epidemiological compartmental models 16 or stiff chemical systems 17 .…”