Physics-informed neural networks framework for solving the highly nonlinear Bratu equation arising in combustion theory

Saurabh Tomar; Higinio Ramos; Saurabh Tomar; Higinio Ramos

doi:10.3934/math.2025972

AIMS Mathematics

2025, Volume 10, Issue 9: 21853-21872. doi: 10.3934/math.2025972

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Physics-informed neural networks framework for solving the highly nonlinear Bratu equation arising in combustion theory

Saurabh Tomar ^1,2,
Higinio Ramos ^{3
,
,}

1.
C3I Center, Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, Kanpur, UP 208016, India
2.
Department of Applied Sciences, Rajiv Gandhi National Aviation University, Amethi, UP 229302, India
3.
Scientific Computing Group, Universidad de Salamanca, Plaza de la Merced, Salamanca 37008, Spain

Received: 20 July 2025 Revised: 09 September 2025 Accepted: 12 September 2025 Published: 19 September 2025
MSC : 34B16, 65L10, 65L60

This research introduced a physics informed neural network (PINN) framework designed to effectively solve the highly nonlinear Bratu equation, which arises in various physical contexts such as chemical reaction theory and combustion processes. PINNs provide a mesh-free solution by embedding physical laws directly into the loss function of the neural network and utilizing automatic differentiation for accurate derivative calculations. However, standard PINNs often face challenges in strictly enforcing boundary conditions (BCs), resulting in numerical inaccuracies and slow convergence. To overcome this, we proposed an innovative method that precisely enforces BCs through a transformation, thereby eliminating residual errors and significantly improving the reliability and performance of the PINN framework. Numerical experiments validated the effectiveness of the proposed approach, showing improved accuracy, faster convergence, and more stable training dynamics. Detailed analyses were conducted to investigate the influence of key hyperparameters, such as activation functions, network architecture, and learning rates, on the model's performance.
- Bratu problem,
- physics-informed neural networks,
- machine learning,
- neural network
Citation: Saurabh Tomar, Higinio Ramos. Physics-informed neural networks framework for solving the highly nonlinear Bratu equation arising in combustion theory[J]. AIMS Mathematics, 2025, 10(9): 21853-21872. doi: 10.3934/math.2025972

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Abstract

This research introduced a physics informed neural network (PINN) framework designed to effectively solve the highly nonlinear Bratu equation, which arises in various physical contexts such as chemical reaction theory and combustion processes. PINNs provide a mesh-free solution by embedding physical laws directly into the loss function of the neural network and utilizing automatic differentiation for accurate derivative calculations. However, standard PINNs often face challenges in strictly enforcing boundary conditions (BCs), resulting in numerical inaccuracies and slow convergence. To overcome this, we proposed an innovative method that precisely enforces BCs through a transformation, thereby eliminating residual errors and significantly improving the reliability and performance of the PINN framework. Numerical experiments validated the effectiveness of the proposed approach, showing improved accuracy, faster convergence, and more stable training dynamics. Detailed analyses were conducted to investigate the influence of key hyperparameters, such as activation functions, network architecture, and learning rates, on the model's performance.

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