What question did this study set out to answer?

The study aims to enhance the accuracy of solving heat conduction problems using physics informed neural networks by integrating gradient augmentation and adaptive loss.

May 28, 2026Open Access

Gradient enhanced physics informed neural networks with adaptive loss for 2D heat equations with variable and non smooth conductivity

Key Points

The study aims to enhance the accuracy of solving heat conduction problems using physics informed neural networks by integrating gradient augmentation and adaptive loss.
Modified physics informed neural network framework utilizing gradient augmentation and adaptive weighted loss.
Employs automatic differentiation for computing solution gradients and incorporating them as auxiliary outputs.
Validation is performed on benchmark problems with statistical error analysis for robustness.
Proposed method achieves superior accuracy compared to standard PINN with fixed loss weights.
Improved training smoothness and convergence with fewer collocation points.
Reliability and scalability demonstrated for variable-coefficient heat conduction problems.

Abstract

This work presents a modified physics informed neural network (PINN) framework that integrates gradient augmentation and an adaptive weighted loss to solve heat conduction problems with time-dependent thermal conductivity and source term. The method uses automatic differentiation to compute solution gradients, which are incorporated as auxiliary outputs to improve training smoothness and accuracy. An adaptive weighted loss mechanism dynamically balances the contributions of the governing equation residual, initial conditions, and boundary conditions during training, thereby enhancing convergence, stability, and robustness. The proposed adaptive loss weighted gradient-enhanced PINN framework is validated on a benchmark problem, both with and without a known analytical solution and its performance is compared against a standard PINN using fixed loss weights. Results demonstrate that the proposed approach achieve superior accuracy, even when trained with fewer collocation points. To further assess robustness and stability, a statistical error analysis based on variance and standard deviation was conducted. The findings indicate that the proposed method provides a reliable and scalable computational framework for variable-coefficient heat conduction problems in nonlinear field theories.

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